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geomstats.test.conf module
-N_SHAPE_POINTS = [1, 2]#
+N_SHAPE_POINTS = [1, 3]#
-N_TIME_POINTS = [1, 2]#
+N_TIME_POINTS = [1, 4]#
-N_VEC_REPS = [3]#
+N_VEC_REPS = [4]#
diff --git a/api/geomstats.test_cases.backend.html b/api/geomstats.test_cases.backend.html index f3e38fd0d..004e7077f 100644 --- a/api/geomstats.test_cases.backend.html +++ b/api/geomstats.test_cases.backend.html @@ -52,7 +52,7 @@ - + diff --git a/api/geomstats.test_cases.datasets.html b/api/geomstats.test_cases.datasets.html index 104f13a63..28a640d8c 100644 --- a/api/geomstats.test_cases.datasets.html +++ b/api/geomstats.test_cases.datasets.html @@ -52,7 +52,7 @@ - + diff --git a/api/geomstats.test_cases.distributions.html b/api/geomstats.test_cases.distributions.html index ff7b32f13..ca6f92d9a 100644 --- a/api/geomstats.test_cases.distributions.html +++ b/api/geomstats.test_cases.distributions.html @@ -52,7 +52,7 @@ - + diff --git a/api/geomstats.test_cases.geometry.html b/api/geomstats.test_cases.geometry.html index fadbe8a56..e5a07150e 100644 --- a/api/geomstats.test_cases.geometry.html +++ b/api/geomstats.test_cases.geometry.html @@ -52,7 +52,7 @@ - + diff --git a/api/geomstats.test_cases.geometry.stratified.html b/api/geomstats.test_cases.geometry.stratified.html index 0a2ff8410..b4ceb9bb3 100644 --- a/api/geomstats.test_cases.geometry.stratified.html +++ b/api/geomstats.test_cases.geometry.stratified.html @@ -52,7 +52,7 @@ - + diff --git a/api/geomstats.test_cases.html b/api/geomstats.test_cases.html index c75231e5a..5e569f5b5 100644 --- a/api/geomstats.test_cases.html +++ b/api/geomstats.test_cases.html @@ -52,7 +52,7 @@ - + diff --git a/api/geomstats.test_cases.information_geometry.html b/api/geomstats.test_cases.information_geometry.html index 7766df54b..9b3fc85d9 100644 --- a/api/geomstats.test_cases.information_geometry.html +++ b/api/geomstats.test_cases.information_geometry.html @@ -52,7 +52,7 @@ - + diff --git a/api/geomstats.test_cases.learning.html b/api/geomstats.test_cases.learning.html index 7e9d84225..a59074cad 100644 --- a/api/geomstats.test_cases.learning.html +++ b/api/geomstats.test_cases.learning.html @@ -52,7 +52,7 @@ - + diff --git a/api/geomstats.test_cases.numerics.html b/api/geomstats.test_cases.numerics.html index 054add590..048d14a70 100644 --- a/api/geomstats.test_cases.numerics.html +++ b/api/geomstats.test_cases.numerics.html @@ -52,7 +52,7 @@ - + diff --git a/api/geomstats.visualization.html b/api/geomstats.visualization.html index 845f71c22..2f066eff5 100644 --- a/api/geomstats.visualization.html +++ b/api/geomstats.visualization.html @@ -54,7 +54,7 @@ - + diff --git a/api/index.html b/api/index.html index 96a07eae7..a589408c3 100644 --- a/api/index.html +++ b/api/index.html @@ -54,7 +54,7 @@ - + diff --git a/api/modules.html b/api/modules.html index e0d857186..17659cfa2 100644 --- a/api/modules.html +++ b/api/modules.html @@ -54,7 +54,7 @@ - + diff --git a/contributing/index.html b/contributing/index.html index 1ac3c96c3..9dcbff56b 100644 --- a/contributing/index.html +++ b/contributing/index.html @@ -52,7 +52,7 @@ - + diff --git a/explanation/index.html b/explanation/index.html index 7d8f8b48e..5a042b07a 100644 --- a/explanation/index.html +++ b/explanation/index.html @@ -55,7 +55,7 @@ - + diff --git a/genindex.html b/genindex.html index 218f5bb0b..e63d0190a 100644 --- a/genindex.html +++ b/genindex.html @@ -49,7 +49,7 @@ - + diff --git a/getting_started/examples.html b/getting_started/examples.html index f6f7941fe..985d9c72a 100644 --- a/getting_started/examples.html +++ b/getting_started/examples.html @@ -52,7 +52,7 @@ - + diff --git a/getting_started/first-steps.html b/getting_started/first-steps.html index 18b0bde9a..449c14525 100644 --- a/getting_started/first-steps.html +++ b/getting_started/first-steps.html @@ -52,7 +52,7 @@ - + diff --git a/getting_started/index.html b/getting_started/index.html index ede1658bb..d8c366468 100644 --- a/getting_started/index.html +++ b/getting_started/index.html @@ -52,7 +52,7 @@ - + diff --git a/governance.html b/governance.html index 98989e24b..3225cc8d1 100644 --- a/governance.html +++ b/governance.html @@ -52,7 +52,7 @@ - + diff --git a/gsod.html b/gsod.html index e9ff3d2da..852ef1037 100644 --- a/gsod.html +++ b/gsod.html @@ -52,7 +52,7 @@ - + diff --git a/hackathons.html b/hackathons.html index 7c76dbbe6..531781002 100644 --- a/hackathons.html +++ b/hackathons.html @@ -54,7 +54,7 @@ - + diff --git a/hackathons/ihp_10_2022.html b/hackathons/ihp_10_2022.html index 3f733930d..ba972edd6 100644 --- a/hackathons/ihp_10_2022.html +++ b/hackathons/ihp_10_2022.html @@ -53,7 +53,7 @@ - + diff --git a/index.html b/index.html index bd4ceabf3..97e494815 100644 --- a/index.html +++ b/index.html @@ -53,7 +53,7 @@ - + diff --git a/notebooks/00_foundations__introduction_to_geomstats.html b/notebooks/00_foundations__introduction_to_geomstats.html index 02de9534c..c385a8cff 100644 --- a/notebooks/00_foundations__introduction_to_geomstats.html +++ b/notebooks/00_foundations__introduction_to_geomstats.html @@ -54,7 +54,7 @@ - + @@ -460,7 +460,7 @@

What is the motivation for analyzing data on manifolds?#

Many data sets lie on a manifold. The definition of manifold will be introduced later, but for now we can think of a manifold as being something like a surface, such as the surface (2-dimensional sphere) in the figure below.

-

67ae256f59dc46c9b46e2536b9d639b2

+

5a7b7bc420114027a6d021d6eafb4cce

Analyzing data that lie on manifolds is often possible without Riemannian Geometry, but choosing to analyze data on manifolds is advantageous for three reasons:

  1. Analyzing data on the manifold it lies on allows you to reduce the degrees of freedom of the system, which makes computations less complicated and more intuitive and interpretable.

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    1) Analyzing data on a manifold reduces the degrees of freedom of the system

    The number of \(\textbf{Degrees of Freedom}\) a system has is equivalent to the number of variables needed to describe the system completely. For example, an object moving freely in three dimensions will require three variables to describe it completely \((x,y,z)\); or \((\phi,\theta, r)\) in spherical coordinates- see Figure (a), below. However, if you know that the point lies on the surface of a sphere of known radius, then you would only need to use two variables, \((\theta,\phi)\), to specify its position- see Figure (b), below.

    Knowing that the point lies on the surface of a manifold (a sphere) allows us to use fewer variables to record its location, which is computationally more efficient in terms of memory requirements, and less mentally taxing (if you are solving a problem on paper).

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    Reducing the number of variables is one of the major motivations behind using manifolds to analyze data. Of course, it is sometimes possible to analyze data without manifolds and Riemannian Geometry, but this will be more complicated and computationally intensive.

2) Knowing the manifold that a data set belongs to may give you a deeper understanding of your data’s evolution or “why your data looks/evolves the way it does”#

Objects travelling along a manifold often follow geodesics on that manifold. When the manifold is equipped with a so-called “Riemannian metric” (which will be introduced in a later notebook), we can define geodesics on the manifold. The geodesic is the path of shortest distance that a particle can travel in the space that it is in. For example, geodesics in 2D and 3D space are straight lines are the shortest way to get from one point to another. The figure below shows paths between two points in cartesian space. One path (\(\gamma\)) is a geodesic in cartesian coordinates, and the other path (\(\gamma^{'}\)) is not a geodesic. Figure (a) shows this in 2D, and Figure (b) shows this in 3D.

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However, when an object lies in a higher dimensional curved space, its geodesic will not follow a straight line. For example, if an object is constrained to move along the surface of a sphere, the shortest path between points is not a straight line, but rather, a curve. A straight line in 3D space would not lie on the surface of a sphere, so it cannot be the shortest path along the sphere. The figure below shows \(\gamma\), which is a geodesic on the sphere, along with \(\gamma^{'}\), which is not a geodesic on the sphere.

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If you did not know that the object was moving along the surface of the sphere, you would wonder why it is taking such an “irratic” path instead of just going straight. The motion of the particles in your system might seem random because you do not understand the space they are moving in. However, if you learn more about the space they are moving in (the surface of a sphere), you would realize that the particles are following very reasonable and predictable paths along geodesics, and this would give you \(\textit{not only}\) a better understanding of how particles have moved in the past but \(\textit{also}\) predictive power to determine how particles will move in the future.

3) Knowing the manifold a data set lives in will help you give you better predictive power and extract the “signal” from a noisy data set or a data set with very few datapoints.#

Let’s begin with a simpler example in cartesian coordinates. Let’s say that you are measuring the position of a car moving at constant velocity, but you dropped your position-measuring-device after only one measurement, and your data looks like this:

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How can you get any information from this? It would be very difficult to get information from this if you don’t have an exact model for what the data \(\textit{should}\) look like. Luckily, it is well known that objects moving with constant velocity \(v\) can be modelled by the equation \(x_f = x_i + v\Delta t\), so even though you do not have many data points, you will still be able to accurately predict your car’s future position. However, if you did not know that your data should follow a straight line such as in Figure (c) or Figure (d), or you did not know the car’s velocity v such as in Figure (b), then you might try to fit the data using incorrect models, and your (incorrect) model would not be able to accurately predict the car’s future position.

In the following figure, the green point in each graph shows the single measured point (the car’s only measured position), and the purple points show future points (the car’s future positions). Note that we are only able to accurately predict future points with a correct model.

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Similarly, knowing the exact manifold your data lies on can help you analyze your current data points and predict future data. For example, let’s say that your data lives on a 2-dimensional sphere, as shown below.

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If you did not know that your data live on the surface of a sphere or you did not, then you might try to fit your data with a line (such as in Figure (b) below) instead of using Geomstats to analyze your data, which appears to follow a geodesic curve on a sphere (shown in Figure (a) below).

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\(\textbf{A note about Geomstat's current capabilities:}\)

Our current version of Geomstats cannot tell you \(\textit{which}\) manifold your data lives on, and it also cannot perform “manifold fitting” (adjusting the general parameters of a manifold to fit your data set). For example, Geomstats cannot tell you if your data lives on the surface of a sphere, and if you \(\textit{know}\) that your data falls on the surface of a sphere, Geomstats cannot tell you what the radius of that sphere is.

However, if you \(\textit{know}\) the exact manifold that your data lives on (e.g. you know that your data lives on a sphere of radius R=3), then you can \(\textit{use}\) Geomstats to unlock the predictive power that differential geometry can provide you.

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What will you learn in these tutorials?

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In the next notebook, we will discuss the Manifold class.

diff --git a/notebooks/01_foundations__manifolds.html b/notebooks/01_foundations__manifolds.html index 676bf54cf..7d2b9ebeb 100644 --- a/notebooks/01_foundations__manifolds.html +++ b/notebooks/01_foundations__manifolds.html @@ -55,7 +55,7 @@ - + @@ -481,7 +481,7 @@

1. Introduction

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2. What is a Manifold?#

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2.2 Imprecise “Layman’s Terms” Definition:\(f: (x_1,...,x_d) \to f(x_1,...,x_d)\). This local graph cannot be applied globally (to the whole manifold) because a function \(f\) must have only one output per set of inputs, and if we were to try to describe the whole manifold with such a graph, then many manifolds would have more than one output for one set of inputs.

The figure below provides a visualization of each of these definitions. The first figure is a visualization of “local parametrization”, the second is a visualization of “local implicit function” and the third is a visualization of “local graph”:

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Each one of these definitions of a manifold is important because each one gives us an additional way to describe a manifold. This is useful because some manifolds are much easier described (or computationally implemented) by one of these definitions than the others. For example, any three of these definitions can be used to describe a hypersphere, but a hypersphere is most easily implemented using definition (2).

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2.3 Hypersphere example:#

A hypersphere is one type of manifold, but there are many other types of manifolds that are commonly used and seen in nature. We will show a few examples here to help you build intuition about what a manifold is and what a manifold can look like.

For example, a torus (the surface of a donut) is a manifold.

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A klein bottle (the surface of the shape shown below) is also a manifold.

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Perhaps a more familiar example for those in physics: Minkowski space (or Minkowski spacetime) is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold (where x, y, z, and time are the dimensions of the space). This manifold is theorized to take one of three forms. A 4-dimensional manifold of: (a) flat curvature, (b) positive curvature, or (c) negative curvature.

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3. Why do we care about manifolds?#

\(\textbf{Manifolds are important because data in nature "naturally falls on manifolds"}\), and as we discussed in the introduction section, knowing the manifold that a data set belongs to may give you more predictive power and a better understanding of the data’s evolution.

\(\textbf{What does it mean for data to "naturally fall on a manifold"?}\) In nature, data are often subject to constraints, and these constraints force the data to lie on manifolds. For example, consider the position of cities on the earth.

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The cities are subject to the following constraints: 1) they cannot fly above the surface of the earth because gravity holds them down and 2) they cannot sink down into the earth because the surface of the earth holds them up. Therefore, they are constrained to move (or not move) on the surface of a 3-dimensional sphere. This space that they are confined to exist in should sound familiar– it is the space of a 2-dimensional hypersphere!

Other data falls on manifolds in similar ways: the data is subject to certain constraints, which forces it to fall on a manifold.

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4. What does the #

Here we will provide a brief reminder of tangent vectors as a way to lay the groundwork for discussing tangent spaces.

A \(\textbf{tangent vector}\) is a vector that is “tangent” to a curve or surface at a given point. When a vector is “tangent” to a curve, this means that the vector has the same slope as the curve does at that point.

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4.2 Tangent Spaces#

Now we will introduce the concept of a “tangent space”. The tangent space at a certain point on a manifold is comprised of all of the possible tangent vectors that exist at that point. For example, if you are considering the tangent space on a curve, then the only possible tangent vectors are tangent vectors that point forward and backward along the line (a). However, if you are considering the tangent space at a point on a surface, then the tangent vectors can point forward, backward, left, right, and everywhere in between, and the set of all tangent vectors forms a plane (b).

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Thus, the tangent space of a 1-dimensional manifold (curve) is also one dimensional, and the tangent space of a 2-dimensional manifold (a 2-dimensional surface) is also 2-dimensional.

Similarly, for every n-dimensional manifold, there exists an n-dimensional tangent space at each point on the manifold, and the tangent space is comprised of all possible tangent vectors on that manifold.

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4.2 Tangent Spaces

5. How is the Manifold class structured?#

The hierarchical structure of the classes inheriting from the Manifold parent class is as follows (this Figure is a courtesy of Nicolas Guigui):

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As discussed in the previous section, one of the primary purposes of the Manifold class is to hold information about various types of manifolds. Rules that are universally true for all manifolds are implemented in methods in the parent class Manifold. Rules that are true for some types of manifolds are implemented in the subclasses of Manifold: LevelSet, VectorSpaceOpenSet, FiberBundle, ProductManifold, VectorSpace, MatrixLieAlgebra, and MatrixLieGroup. Specific types of manifolds are described in methods within these subclasses.

In this notebook, we will focus on describing the subclasses pertinent to the geometry module of geomstats: LevelSet, VectorSpaceOpenSet, ProductManifold and VectorSpace.

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5.2 VectorSpace

5.2.1 What is an Open Set?#

Intuitively, an open set is a group of numbers that does not include points on the boundary of whatever they are describing. For example, if you were to take the set of all points between a and b but \(\textbf{not}\) include the values a and b, this would be an open set (shown in figure a). If you were to take the set of all points between a and b and include the values a and b, this would be a closed set (shown in figure b)

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The above examples showed open and closed sets in one dimension. Similarly, in two dimensions, open sets can be defined as sets which do not contain their boundaries. For example, the inside of the sphere, i.e. the ball without its boundary, is a manifold that is an open set. The image below shows an example of an open set (a) and a closed set (b) in two dimensions.

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5.2.2 What Methods are Implemented in VectorSpaceOpenSet?#

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5.3.1 What is a Level Set?\(\textbf{Level Sets}\). A level set is the set of values \(x\) for which a function f(x) is equal to a given constant. In other words, a level set is a set of curves for which the function describing a manifold is constant along that curve.

In the same way that VectorSpaceOpenSet is an implementation of the first definition of a manifold (Local Parametrization), LevelSet is an implementation of the second definition of a manifold (Local Implicit Function). a level set is a set of points for which the function \(f\) takes the exact same value. This value is called the “level”, and does not need to be a scalar, it could also be a vector.

For example, consider a hypersphere in three dimensional space. Each of the concentric spheres is a 2-dimensional manifold, each corresponding to a different level \((r1, r2,r3)\).

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You can see here that the subclass Hypersphere is indeed implemented as a LevelSet.

diff --git a/notebooks/02_foundations__connection_riemannian_metric.html b/notebooks/02_foundations__connection_riemannian_metric.html index 284213d71..c4e5f350c 100644 --- a/notebooks/02_foundations__connection_riemannian_metric.html +++ b/notebooks/02_foundations__connection_riemannian_metric.html @@ -55,7 +55,7 @@ - + @@ -465,7 +465,7 @@

  • Discuss the RiemannianMetric subclass of the Connection Class

    • This notebook provides an overview of the \(\textbf{Connection}\) class in Geomstats. Intuitively: a connection is a mathematical operator that tells you how much a vector will change when you move it along a manifold in the direction of another vector. The symbol for connection is \(\nabla\), and \(\nabla_{\vec{u}} \vec{v}\) denotes how much vector \(\vec{v}\) would change if you moved it an infinitesimal distance in the direction of vector \(\vec{u}\) (see figure below).

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    1. What is a Connection?#

    We will start by defining the connection \(\textit{mathematically}\), and then we will define the connection again in a more \(\textit{conceptual and intuitive}\) way with examples.

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    1.1 General Mathematical Definition of the Connection“\(\nabla:\)” - this part is saying that it is the connection (denoted \(\nabla\)) that is performing the transformation.

    “\(T M\)” - this symbolizes the \(\textbf{tangent bundle}\) of the manifold \(M\). A tangent bundle is the collection of all tangent spaces on the manifold \(M\). A tangent bundle is itself also a manifold.

    “\(\Gamma(T M)\)” - \(\Gamma(T M)\) denotes the space of vector fields. A single element of \(\Gamma(T M)\) is a single vector field on the manifold \(M\). A vector field is comprised of a set of vectors on the manifold, where there is a vector at each point, and that vector is an element of the tangent space at that point (shown below).

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    Putting all of this together, \(\nabla: \Gamma (T M) \times \Gamma (T M) \to \Gamma (T M)\) is simply saying that the connection is a transformation which takes two arguments, both of which are vector fields (comprised of tangent vectors), and the result of the transformation is another vector field (comprised of tangent vectors).

    \(\textbf{2. Now, let's consider the conditions that the connection must satisfy.}\)

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      1.1 General Mathematical Definition of the Connection

      1.2 What does the Connection actually do? – a more friendly description of the connection, with an example#

      Consider a single vector \(\vec{a_p}\), tangent to the manifold at point \(p\) (shown in the figure below).

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      Now, suppose you want to see what vector \(\vec{a_p}\) looks like when it is translated to a different point on the manifold. You may be asking why we are placing importance on this question– because for a vector in cartesian coordinates, this question would be trivial. If you translated vector \(\vec{a_p}\) to a different point in cartesian coordinates, the vector itself would not change (as shown in the figure below).

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      A spatially translated vector in cartesian coordinates does not change because the basis vectors in cartesian coordinates are translationally invariant, meaning all points in cartesian coordinates have the same set of basis vectors.

      This is not, however, necessarily true for points on a manifold– if you move a tangent vector from one point to another (i.e. from one tangent space to another), the vector will not necessarily look the same after this transformation. (The figure below shows that when \(\vec{a_p}\) is translated to a different point on the manifold, it could be pointing in a direction different from its original direction.)

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      This is why the connection is such an important concept as we analyze data on manifolds. The connection helps us quantify how much a vector will change when we move it from one point to another on a manifold.

      More specifically, the connection is able to tell us how one vector will change if it is moved in the direction of another vector. For example, if vectors \(\vec{a}\) and \(\vec{b}\) exist in the same tangent plane, then the connection \(\nabla_{\vec{b}} \vec{a}\) tells us how much the vector \(\vec{a}\) will change if it is moved an infinitesimally small distance in the direction of \(\vec{b}\) (shown in the figure below).

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    2.1 Parallel Vector Fields\(\textbf{X}\) infinitesimally along the curve \(\gamma\) in the direction of \(\gamma\)’s tangent vector \(\dot{\gamma}\), the vector field \(\textbf{X}\) would not change. Note that this means that in differential geometry, we can only define the term “parallel” when we define it \(\textit{along a curve}\). The figure below shows that at each point \(t\) along the curve \(\gamma\), the vector field satisfies \(\nabla_{\dot{\gamma}(t)} X(t)=0\), which means that the vector field shown in pink is a parallel vector field.

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    2.2 Parallel Transport#

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    2.2 Parallel Transport\(T_{\gamma(a)}M\)) to another tangent space (\(T_{\gamma(s)}M\)).

    Conceptually, this is exactly like solving a differential equation: we can find a “solution” (in this case: the transported vector) to the equation \(\nabla_{\dot \gamma(t)} X(\gamma(t)) = 0\) if we are given initial conditions (in this case: an initial point and tangent vector). Note that \(X(\gamma(t))\) in the differential equation simply indicates that the vector field \(X\) is parallel along the curve \(\gamma(t)\). In other words, the parallel transport calculates the parallel vector field along the curve \(\gamma(t)\) and then selects the element of the vector field that lies at the final point \(\gamma(s)\). The figure below shows this process.

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    Note that the connection was able to tell us how a vector would change infinitesimally if we move a vector in the direction of another vector, but here, we are able to use parallel transport to calculate how a vector will change if it is translated a \(\textbf{non-infinitesimal}\) distance.

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    3.3 Building Intuition with Physics\(\textit{four-dimensional}\) “fabric” that makes up the universe, and it is a manifold. Spacetime curves around masses (like planets, stars, black holes), which means that in those areas, we cannot approximate the space as being Euclidean. It is in these spaces that we must use geodesics to predict the paths that objects will follow– because free particles (and light) follow geodesics in space.

    In fact, telescopes have confirmed that light bends around very massive objects like black holes (shown below).

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    This is just one example of why geodesics are crucial to understanding real-world data. Other examples exist in areas like biophysics where the trajectory of cells along a manifold may follow geodesics.

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    4. Description of Connection in Local Coordinates (using Christoffel Symbols

    4.1 Math Notation: \(\mathbf{ \partial_i}\) Basis Vector#

    \({\partial_i}\) is equivalent to \(\vec{e}_i\). For example, in three dimensions, one could express the basis vectors \(\hat{x}, \hat{y}, \hat{z}\) as either \(\vec{e}_x, \vec{e}_y, \vec{e}_z\) or as \(\partial_x, \partial_y, \partial_z\).

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    Note/practicality: basis vectors on a manifold may differ at each point, which means that basis vectors are actually dependent on the tangent space \(T_pM\) to which they belong. Thus, we “should” be writing \(\vec{e}_i(p)\), but in practice, the point \(p\) is omitted.

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    5. Additional Important Mathematical concepts utilized by

    Exponentials on Manifolds#

    In Euclidean space, \(\textit{addition}\) is a tool which takes two points \(p_1, p_2\), “adds” them, and generates a third, larger point \(p_3\) (as shown in the figure below). Addition gives us a way to “move forward” in Euclidean space.

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    On manifolds, the \(\textit{exponential}\) provides a tool which “takes the exponential of the tangent vector at point \(p\)” to generate a third point on the manifold. The exponential does this by 1) identifying the unique geodesic \(\gamma\) that goes through \(p\) and \(v_p\), 2) identifying the “length” \(l\) of the tangent vector \(v_p\), and 3) calculating another point \(p'\) along \(\gamma(t)\) that is a “distance” \(l\) from the initial point \(p\) (see figure below). Note again that the notion of “length” and “distance” is different on a manifold than it is in Euclidean space and that quantifying length is not something that we will be able to do without specifying a metric (discussed in RiemannianMetric section).

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    The exponential is often described as being the “differential geometry version” of addition.

    Logarithms on Manifolds#

    In Euclidean space, \(\textit{subtraction}\) is an operation which allows us to take the third point \(p_3\) and one of the initial points \(p_1\) and extract the other initial point \(p_2\) (as shown in the figure below).

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    Similarly, the \(\textit{logarithm}\) allows us to take the final point \(p'\) and the initial point \(p\) to extract the tangent vector \(v_p\) at the initial point. The logarithm is able to do this by 1) identifying the unique geodesic \(\gamma\) that connects the two points 2) calculating the “length” of that geodesic 3) generating the unique tangent vector at \(p\), with a “length” equal to that of the geodesic (shown in the figure below). Again, remember that “length” is not something that we can quantify without specifying a metric, which we will not be able to do until the RiemannianMetric section.

    A key point here is that if you know a point and a tangent vector at that point, you can calculate a unique geodesic that goes through that point, and similarly, if you know the point and geodesic, you should be able to extract the unique tangent vector that produced that geodesic.

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    The logarithm is the inverse map of the exponential and is often described as being the “differential geometry version” of subtraction.

    6. Geomstats Documentation: The Connection Class#

    The Connection class in geomstats has four methods and one subclass, whose structure is shown below:

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    The methods of the Connection class also equip the RiemannianMetric subclass with the tools it needs to perform calculations on manifolds (we will discuss the RiemannianMetric class later in this notebook).

    We will first print the key methods of the Connection class, and then at the end we will print the entire connection class for anyone who is interested.

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    7.2 How is the Riemannian Metric Connected with the Connection?

    7.3 RiemannianMetric in Geomstats#

    The RiemannianMetric subclass is the class that actually performs calculations on manifolds. This is because we cannot perform “general” calculations like logarithm, exponential, parallel transport, and calculating geodesics without first being equip with a metric. The RiemannianMetric subclass contains a collection of metrics that Geomstats users may use to perform calculations on manifolds. The structure of the RiemannianMetric class is shown below.

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     In [7]:
    diff --git a/notebooks/03_practical_methods__data_on_manifolds.html b/notebooks/03_practical_methods__data_on_manifolds.html index d127a95cc..3a07d4739 100644 --- a/notebooks/03_practical_methods__data_on_manifolds.html +++ b/notebooks/03_practical_methods__data_on_manifolds.html @@ -55,7 +55,7 @@ - + diff --git a/notebooks/04_practical_methods__from_vector_spaces_to_manifolds.html b/notebooks/04_practical_methods__from_vector_spaces_to_manifolds.html index 098a422ee..d15f4e422 100644 --- a/notebooks/04_practical_methods__from_vector_spaces_to_manifolds.html +++ b/notebooks/04_practical_methods__from_vector_spaces_to_manifolds.html @@ -55,7 +55,7 @@ - + diff --git a/notebooks/05_practical_methods__simple_machine_learning_on_tangent_spaces.html b/notebooks/05_practical_methods__simple_machine_learning_on_tangent_spaces.html index 26ba3a710..dfd3f2754 100644 --- a/notebooks/05_practical_methods__simple_machine_learning_on_tangent_spaces.html +++ b/notebooks/05_practical_methods__simple_machine_learning_on_tangent_spaces.html @@ -55,7 +55,7 @@ - + diff --git a/notebooks/06_practical_methods__riemannian_frechet_mean_and_tangent_pca.html b/notebooks/06_practical_methods__riemannian_frechet_mean_and_tangent_pca.html index c0ee87444..91adaeec0 100644 --- a/notebooks/06_practical_methods__riemannian_frechet_mean_and_tangent_pca.html +++ b/notebooks/06_practical_methods__riemannian_frechet_mean_and_tangent_pca.html @@ -55,7 +55,7 @@ - + diff --git a/notebooks/07_practical_methods__riemannian_kmeans.html b/notebooks/07_practical_methods__riemannian_kmeans.html index 97af01663..b61884555 100644 --- a/notebooks/07_practical_methods__riemannian_kmeans.html +++ b/notebooks/07_practical_methods__riemannian_kmeans.html @@ -55,7 +55,7 @@ - + diff --git a/notebooks/08_practical_methods__information_geometry.html b/notebooks/08_practical_methods__information_geometry.html index d2f75fc15..4e1ef173a 100644 --- a/notebooks/08_practical_methods__information_geometry.html +++ b/notebooks/08_practical_methods__information_geometry.html @@ -55,7 +55,7 @@ - + diff --git a/notebooks/09_practical_methods__implement_your_own_riemannian_geometry.html b/notebooks/09_practical_methods__implement_your_own_riemannian_geometry.html index 5470fc32b..5263ff566 100644 --- a/notebooks/09_practical_methods__implement_your_own_riemannian_geometry.html +++ b/notebooks/09_practical_methods__implement_your_own_riemannian_geometry.html @@ -55,7 +55,7 @@ - + diff --git a/notebooks/10_practical_methods__shape_analysis.html b/notebooks/10_practical_methods__shape_analysis.html index b898e8ee1..30b0c3818 100644 --- a/notebooks/10_practical_methods__shape_analysis.html +++ b/notebooks/10_practical_methods__shape_analysis.html @@ -55,7 +55,7 @@ - + diff --git a/notebooks/11_real_world_applications__cell_shapes_analysis.html b/notebooks/11_real_world_applications__cell_shapes_analysis.html index d08ee150a..21d837865 100644 --- a/notebooks/11_real_world_applications__cell_shapes_analysis.html +++ b/notebooks/11_real_world_applications__cell_shapes_analysis.html @@ -55,7 +55,7 @@ - + @@ -476,7 +476,7 @@

    Shape Analysis of Cancer CellsICLR 2021 Computational Geometry and Topology challenge.

    -

    793a874d256f45ac87ccadbfd0410f84

    +

    fb7f806036104cffa55f477004d53fa0

    Figure 1: Representative images of the cell lines using fluorescence microscopy, studied in this notebook (Image credit : Ashok Prasad). The cells nuclei (blue), the actin cytoskeleton (green) and the lipid membrane (red) of each cell are stained and colored. We only focus on the cell shape in our analysis.

    @@ -510,7 +510,7 @@

    2. Dataset DescriptionWe study a dataset of mouse Osteosarcoma imaged cells (AXCFP2019). The dataset contains two different cancer cell lines : DLM8 and DUNN, respectively representing a more agressive and a less agressive cancer. Among these cells, some have also been treated with different single drugs that perturb the cellular cytoskeleton. Overall, we can label each cell according to their cell line (DLM8 and DUNN), and also if it is a control cell (no treatment), or has been treated with one of the following drugs : Jasp (jasplakinolide) and Cytd (cytochalasin D).

    Each cell comes from a raw image containing a set of cells, which was thresholded to generate binarized images.

    -

    64ea55a60ea049a48d412c8313e1619e

    +

    d33fbfa6ed65421fa4a6f264707690d7

    After binarizing the images, contouring was used to isolate each cell, and to extract their boundaries as a counter-clockwise ordered list of 2D coordinates, which corresponds to the representation of discrete curve in Geomstats. We load these discrete curves into the notebook.

     In [2]:
diff --git a/notebooks/12_real_world_applications__emg_sign_classification_in_spd_manifold.html b/notebooks/12_real_world_applications__emg_sign_classification_in_spd_manifold.html
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@@ -55,7 +55,7 @@
   
   
   
-    
+    
   
   
   
@@ -479,7 +479,7 @@ 
    Hand gesture classification with EMG data using Riemannian metrics
    
+
    003678d872f54008bedab028f4bfcc8c
    
 
    
 
    Context#
    
 
    The data are acquired from somOS-interface: an sEMG armband that allows you to interact via bluetooth with an Android smartphone (you can contact Marius Guerard (marius.guerard@gmail.com) or Renaud Renault (renaud.armand.renault@gmail.com) for more info on how to make this kind of armband yourself).
    
diff --git a/notebooks/13_real_world_applications__graph_embedding_and_clustering_in_hyperbolic_space.html b/notebooks/13_real_world_applications__graph_embedding_and_clustering_in_hyperbolic_space.html
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@@ -55,7 +55,7 @@
   
   
   
-    
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@@ -553,7 +553,7 @@ 
    Parameters and Initialization Some information about the dataset is displayed to provide insight into its
+
    The Zachary karate club network was collected from the members of a university karate club by Wayne Zachary in 1977. Each node represents a member of the club, and each edge represents an undirected relation between two members. An often discussed problem using this dataset is to find the two groups of people into which the karate club split after an argument between two teachers. ab5f01c14acf491b8b45da76cf20e541 Some information about the dataset is displayed to provide insight into its
 complexity.
    
 
    
 
     In [3]:
@@ -579,7 +579,7 @@ 
    Parameters and Initialization\(V\) as the set of nodes and \(E \subset V\times V\) the set of edges. The goal of embedding GSD is to provide a faithful and exploitable representation of the graph structure. It is mainly achieved by preserving first-order proximity that enforces nodes sharing edges to be close to each other. It can additionally preserve second-order proximity that enforces two nodes sharing the same context (i.e., nodes that share neighbors but are not necessarily directly connected) to be
 close. Let \(\mathbb{B}^m\) be the Poincaré Ball of dimension \(m\) equipped with the distance function \(d\). The below figure shows geodesics between pairs of points on \(\mathbb{B}^2\). Geodesics are the shortest path between two points. The distance function \(d\) of two points is the length of the geodesic that links them.
    
-
    66460f12e6f841dd8c80cc464e00f4b3
    
+
    af5bc2aa2058440a9d07e5cd6d1f2f9d
    
 
    Declaring an instance of the PoincareBall manifold of two dimensions in geomstats is straightforward:
    
 
    
 
     In [4]:
@@ -597,7 +597,7 @@ 
    Loss function.
 \[\mathcal{L} = - \sum_{v_i\in V} \sum_{v_j \in C_i} \bigg[ log(\sigma(-d^2(\phi_i, \phi_j'))) + \sum_{v_k\sim \mathcal{P}_n} log(\sigma(d^2(\phi_i, \phi_k')))  \bigg]\]
    
 
    where \(\sigma(x)=\frac{1}{1+e^{-x}}\) is the sigmoid function and \(\phi_i \in \mathbb{B}^m\) is the embedding of the \(i\)-th node of \(V\), \(C_i\) the nodes in the context of the \(i\)-th node, \(\phi_j'\in \mathbb{B}^m\) the embedding of \(v_j\in C_i\) and \(\mathcal{P}_n\) the negative sampling distribution over \(V\): \(\mathcal{P}_n(v)=\frac{deg(v)^{3/4}}{\sum_{v_i\in V}deg(v_i)^{3/4}}\). Intuitively one can see that to minimizing \(L\),
-the distance between \(v_i\) and \(v_j\) should get smaller, while the one between \(v_i\) and \(v_k\) would get larger. ba63c137ffcd43d09c8a52d5626d2530
    
+the distance between \(v_i\) and \(v_j\) should get smaller, while the one between \(v_i\) and \(v_k\) would get larger. b8d28814764549f98f39ed32c291937d
    
 
    
 
    
 
    Riemannian optimization.#
    
diff --git a/notebooks/14_real_world_applications__hand_poses_analysis_in_kendall_shape_space.html b/notebooks/14_real_world_applications__hand_poses_analysis_in_kendall_shape_space.html
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    Classifying hands poses with Kendall shape spaces
 
    We wish to investigate if there is a difference in these two groups.
    
 
    The hand poses are represented as the coordinates of 22 joints in 3D:
    
-
    c69c0fff853848678e4d61cfe2a1fd61
    
+
    f6eb3cebc1f149bdbe24aee0b2ed9a70
    
 
    
 
     In [1]:
 
    
diff --git a/notebooks/15_real_world_applications__optic_nerve_heads_analysis_in_kendall_shape_space.html b/notebooks/15_real_world_applications__optic_nerve_heads_analysis_in_kendall_shape_space.html
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    Computing with shapes of landmarks in Kendall shape spaces
    
+
    cfb9e523d41d4245bd20fed889dcebf8
    
 
    
 
     In [1]:
 
    
diff --git a/notebooks/16_real_world_applications__visualizations_in_kendall_shape_spaces.html b/notebooks/16_real_world_applications__visualizations_in_kendall_shape_spaces.html
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diff --git a/notebooks/17_foundations__stratified_spaces.html b/notebooks/17_foundations__stratified_spaces.html
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    Spider#
    Spider: the space defined by the number of rays
    
 
  • Spider Geometry: by chosing a metric on the rays, we can define a metric on the whole space
    • 
 
-
    5e2d0697164f4f01ac967435ba145738
    
+
    07dbdc44ebae4c77878b2994597b88bd
    
 
    
 
     In [1]:
 
    
diff --git a/notebooks/18_real_world_applications__sao_paulo_traffic_optimization.html b/notebooks/18_real_world_applications__sao_paulo_traffic_optimization.html
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    Optimization of Sao Paulo traffic
    
+
    52dfaa610fd4415689c5cfa000dd0021
    
 
    Figure 1: Sao Paulo: A city with 180km traffic jams – BBC News
    
 
    
 
     In [1]:
@@ -1250,7 +1250,7 @@ 
    3.1.2. Model\(r\), we note \(T_r\) the congestion time that cars will have to wait in the case the traffic is jammed: \(T_r \rightsquigarrow G(k_r, \nu_r)\), with distribution:
    
 
    
 \[\forall t>0, \, \mathbb{f}(t) = \frac{\nu_r^{k_r}}{\Gamma(k_r)} t^{k_r-1} e^{-\nu_r t}.\]
    
-
    ec1746efbaaf4a8bbac04a9609a36c5c
    
+
    d2af5e309b124cbb8fbd6d4e379995dc
    
 
    As a road \(x_r\) can be represented by two parameters \(k_r\) and \(\nu_r\), we can consider our space of study to be the space of such parameters (i.e. \(\mathbb{(R_+^*)^2}\)).
    
 
    For the following, we denote Gamma distributions’ parameters by \((\kappa_r, \gamma_r)\), where \(\kappa_r\)=\(k_r\) (expected jam size) and \(\gamma_r\)=\(\frac{k_r}{\nu_r}\) is the expected congestion time (mean of the Gamma distribution). The space of study is still \(\mathbb{(R_+^*)^2}\), and we are instantiating it in the next cell.
    
 
    
diff --git a/notebooks/19_practical_methods__aac.html b/notebooks/19_practical_methods__aac.html
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diff --git a/roadmap.html b/roadmap.html
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-Search.setIndex({"alltitles": {"1) Analyzing data on a manifold reduces the degrees of freedom of the system, making computations less complicated.": [[32, "1)-Analyzing-data-on-a-manifold-reduces-the-degrees-of-freedom-of-the-system,-making-computations-less-complicated."]], "1. A first example in 2D": [[42, "1.-A-first-example-in-2D"]], "1. Introduction": [[33, null]], "1. Introduction and Motivation": [[43, "1.-Introduction-and-Motivation"], [50, "1.-Introduction-and-Motivation"]], "1. What is a Connection?": [[34, null]], "1.1 General Mathematical Definition of the Connection": [[34, "1.1-General-Mathematical-Definition-of-the-Connection"]], "1.2 What does the Connection actually do? \u2013 a more friendly description of the connection, with an example": [[34, "1.2-What-does-the-Connection-actually-do?----a-more-friendly-description-of-the-connection,-with-an-example"]], "2) Knowing the manifold that a data set belongs to may give you a deeper understanding of your data\u2019s evolution or \u201cwhy your data looks/evolves the way it does\u201d": [[32, "2)-Knowing-the-manifold-that-a-data-set-belongs-to-may-give-you-a-deeper-understanding-of-your-data's-evolution-or-%22why-your-data-looks/evolves-the-way-it-does%22"]], "2. Dataset Description": [[43, "2.-Dataset-Description"]], "2. Dataset description": [[50, "2.-Dataset-description"]], "2. Focus on quotienting out reparametrizations": [[42, "2.-Focus-on-quotienting-out-reparametrizations"]], "2. What is a Manifold?": [[33, "2.-What-is-a-Manifold?"]], "2. Why is the Connection important (big picture)? \u2013 parallel transport, parallel vector fields": [[34, "2.-Why-is-the-Connection-important-(big-picture)?----parallel-transport,-parallel-vector-fields"]], "2.1 Parallel Vector Fields": [[34, "2.1-Parallel-Vector-Fields"]], "2.1 Precise Mathematical Definition": [[33, "2.1-Precise-Mathematical-Definition"]], "2.1. Comparing parametrized curves": [[42, "2.1.-Comparing-parametrized-curves"]], "2.2 Imprecise \u201cLayman\u2019s Terms\u201d Definition:": [[33, "2.2-Imprecise-%22Layman's-Terms%22-Definition:"]], "2.2 Parallel Transport": [[34, "2.2-Parallel-Transport"]], "2.2. Comparing unparametrized curves": [[42, "2.2.-Comparing-unparametrized-curves"]], "2.3 Hypersphere example:": [[33, "2.3-Hypersphere-example:"]], "2.4 More examples of manifolds:": [[33, "2.4-More-examples-of-manifolds:"]], "3) Knowing the manifold a data set lives in will help you give you better predictive power and extract the \u201csignal\u201d from a noisy data set or a data set with very few datapoints.": [[32, "3)-Knowing-the-manifold-a-data-set-lives-in-will-help-you-give-you-better-predictive-power-and-extract-the-%22signal%22-from-a-noisy-data-set-or-a-data-set-with-very-few-datapoints."]], "3. Defining Geodesics with the Connection": [[34, "3.-Defining-Geodesics-with-the-Connection"]], "3. Example in 3D": [[42, "3.-Example-in-3D"]], "3. Mathematical modeling": [[50, "3.-Mathematical-modeling"]], "3. Preprocessing": [[43, "3.-Preprocessing"]], "3. 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Description of Connection in Local Coordinates (using Christoffel Symbols)": [[34, "4.-Description-of-Connection-in-Local-Coordinates-(using-Christoffel-Symbols)"]], "4. What does the Manifold class do?": [[33, "4.-What-does-the-Manifold-class-do?"]], "4.1 Math Notation: \\mathbf{ \\partial_i} Basis Vector": [[34, "4.1-Math-Notation:-\\mathbf{-\\partial_i}-Basis-Vector"]], "4.1 Tangent Vectors": [[33, "4.1-Tangent-Vectors"]], "4.2 Einstein Notation": [[34, "4.2-Einstein-Notation"]], "4.2 Tangent Spaces": [[33, "4.2-Tangent-Spaces"]], "4.3 What does a Christoffel Symbol represent?": [[34, "4.3-What-does-a-Christoffel-Symbol-represent?"]], "4.4 The Geodesic Equation in local coordinates": [[34, "4.4-The-Geodesic-Equation-in-local-coordinates"]], "4.5 Why should we care about Christoffel Symbols and this representation of the connection?": [[34, "4.5-Why-should-we-care-about-Christoffel-Symbols-and-this-representation-of-the-connection?"]], "5. Additional Important Mathematical concepts utilized by Connection Class: Logarithm, Exponential": [[34, "5.-Additional-Important-Mathematical-concepts-utilized-by-Connection-Class:-Logarithm,-Exponential"]], "5. How is the Manifold class structured?": [[33, "5.-How-is-the-Manifold-class-structured?"]], "5. Solving the problem at hand": [[50, "5.-Solving-the-problem-at-hand"]], "5.1 The Parent Class: Manifold": [[33, "5.1-The-Parent-Class:-Manifold"]], "5.1.1 Examples of Using Manifold\u2019s Attributes and Methods in the Subclass: Hypersphere": [[33, "5.1.1-Examples-of-Using-Manifold's-Attributes-and-Methods-in-the-Subclass:-Hypersphere"]], "5.1.2 The Full Manifold Class Code": [[33, "5.1.2-The-Full-Manifold-Class-Code"]], "5.2 VectorSpaceOpenSet": [[33, "5.2-VectorSpaceOpenSet"]], "5.2.1 What is an Open Set?": [[33, "5.2.1-What-is-an-Open-Set?"]], "5.2.2 What Methods are Implemented in VectorSpaceOpenSet?": [[33, "5.2.2-What-Methods-are-Implemented-in-VectorSpaceOpenSet?"]], "5.3 LevelSet": [[33, "5.3-LevelSet"]], "5.3.1 What is a Level Set?": [[33, "5.3.1-What-is-a-Level-Set?"]], "5.3.2 LevelSet in Geomstats": [[33, "5.3.2-LevelSet-in-Geomstats"]], "5.4 VectorSpace": [[33, "5.4-VectorSpace"]], "5.5 ProductManifold": [[33, "5.5-ProductManifold"]], "5.5.1 Example": [[33, "5.5.1-Example"]], "6. Geomstats Documentation: The Connection Class": [[34, "6.-Geomstats-Documentation:-The-Connection-Class"]], "6.1 method: parallel_transport()": [[34, "6.1-method:-parallel_transport()"]], "6.2 method: geodesic_equation()": [[34, "6.2-method:-geodesic_equation()"]], "6.3 method: exp() (exponential)": [[34, "6.3-method:-exp()-(exponential)"]], "6.4 method: log() (logarithm)": [[34, "6.4-method:-log()-(logarithm)"]], "6.5 The full Connection class": [[34, "6.5-The-full-Connection-class"]], "7. RiemannianMetric Sub-Class": [[34, "7.-RiemannianMetric-Sub-Class"]], "7.1 Riemannian Metrics": [[34, "7.1-Riemannian-Metrics"]], "7.2 How is the Riemannian Metric Connected with the Connection?": [[34, "7.2-How-is-the-Riemannian-Metric-Connected-with-the-Connection?"]], "7.3 RiemannianMetric in Geomstats": [[34, "7.3-RiemannianMetric-in-Geomstats"]], "8. Conclusion": [[34, "8.-Conclusion"]], "A primer in space, metric and aligners": [[51, "A-primer-in-space,-metric-and-aligners"]], "A. Test on the same session and user as Training/Calibration": [[44, "A.-Test-on-the-same-session-and-user-as-Training/Calibration"]], "A.0. Using Logistic Regression on the vectorized Matrix (Euclidean Method)": [[44, "A.0.-Using-Logistic-Regression-on-the-vectorized-Matrix-(Euclidean-Method)"]], "A.1. Using MLP on the vectorized Matrix (Euclidean Method)": [[44, "A.1.-Using-MLP-on-the-vectorized-Matrix-(Euclidean-Method)"]], "A.2. Using Tangent space projection + Logistic Regression": [[44, "A.2.-Using-Tangent-space-projection-+-Logistic-Regression"]], "A.3. Using Tangent space projection + MLP": [[44, "A.3.-Using-Tangent-space-projection-+-MLP"]], "A.4. Using Euclidean MDM": [[44, "A.4.-Using-Euclidean-MDM"]], "A.5. Using Riemannian MDM": [[44, "A.5.-Using-Riemannian-MDM"]], "API Reference": [[20, null]], "About the organization": [[28, "about-the-organization"]], "About the project": [[28, "about-the-project"]], "Additional information": [[28, "additional-information"]], "Align all and Compute for Graphs": [[51, null]], "Alignment": [[43, "Alignment"]], "Analysis": [[28, "analysis"]], "Analyze Distances to the \u201cGlobal\u201d Mean Shape": [[43, "Analyze-Distances-to-the-%22Global%22-Mean-Shape"]], "Answering questions": [[22, "answering-questions"]], "Backends": [[22, "backends"]], "Brain connectomes: data on the manifold of Symmetric Positive Definite (SPD) matrices": [[35, "Brain-connectomes:-data-on-the-manifold-of-Symmetric-Positive-Definite-(SPD)-matrices"]], "Breach": [[27, "breach"]], "Budget": [[28, "budget"]], "Build the docs": [[22, "build-the-docs"]], "Building the Documentation": [[22, "building-the-documentation"]], "Capturing the graph structure": [[45, "Capturing-the-graph-structure"]], "Changing the Governance Documents": [[27, "changing-the-governance-documents"]], "Changing the Metric on the Space of Cell Shapes": [[43, "Changing-the-Metric-on-the-Space-of-Cell-Shapes"]], "Classifying hands poses with Kendall shape spaces": [[46, null]], "Code Examples": [[24, null]], "Code Review Guidelines": [[22, "code-review-guidelines"]], "Coding Style Guidelines": [[22, "coding-style-guidelines"]], "Commenting on alternative solutions": [[22, "commenting-on-alternative-solutions"]], "Communication team": [[27, "communication-team"]], "Computing with shapes of landmarks in Kendall shape spaces": [[47, null]], "Computing with triangular shapes in Kendall framework": [[48, null]], "Conclusion": [[33, "Conclusion"], [37, "Conclusion"], [50, "Conclusion"]], "Conflict of Interest": [[27, "conflict-of-interest"]], "Context": [[44, "Context"]], "Contributing Guide": [[22, null]], "Contributor Experience Team": [[27, "contributor-experience-team"]], "Contributors": [[27, "contributors"]], "Core developers": [[27, "core-developers"]], "Covariances plot of the Frechet Mean of the affine invariant metric": [[44, "Covariances-plot-of-the-Frechet-Mean-of-the-affine-invariant-metric"]], "Covariances plot of the euclidean average": [[44, "Covariances-plot-of-the-euclidean-average"]], "Create or choose an issue for new contributors": [[22, "create-or-choose-an-issue-for-new-contributors"]], "Data init: Generate clusters randomly on the sphere": [[39, "Data-init:-Generate-clusters-randomly-on-the-sphere"]], "Data on Manifolds": [[35, null]], "Decision Making Process": [[27, "decision-making-process"]], "Defining the Heisenberg sub-Riemannian structure via a frame-field": [[53, "Defining-the-Heisenberg-sub-Riemannian-structure-via-a-frame-field"]], "Dependencies and a virtual environment": [[22, "dependencies-and-a-virtual-environment"]], "Design": [[31, "design"]], "Design and Organize Scattered Documentation in Geomstats": [[28, "design-and-organize-scattered-documentation-in-geomstats"]], "Detailed Goals": [[54, "detailed-goals"]], "Development Setup": [[22, "development-setup"]], "Distance of the Cell Shapes to their Own Mean Shape": [[43, "Distance-of-the-Cell-Shapes-to-their-Own-Mean-Shape"]], "Docstring Examples": [[22, "docstring-examples"]], "Documentation": [[22, "documentation"]], "Equip with metric": [[30, "equip-with-metric"]], "Examples of data on manifolds": [[35, "Examples-of-data-on-manifolds"]], "Explanation": [[23, null]], "Exponentials on Manifolds": [[34, "Exponentials-on-Manifolds"]], "First steps": [[25, null]], "Folder Structure": [[22, "folder-structure"]], "Frechet Mean": [[51, "Frechet-Mean"]], "From addition to exponential map": [[36, "From-addition-to-exponential-map"]], "From data on linear spaces to data on manifolds": [[35, "From-data-on-linear-spaces-to-data-on-manifolds"]], "From substraction to logarithm map": [[36, "From-substraction-to-logarithm-map"]], "From vector spaces to manifolds": [[36, null], [36, "id1"]], "Fr\u00e9chet Mean and Tangent PCA": [[38, null]], "Fr\u00e9chet mean": [[38, "Fr\u00e9chet-mean"], [38, "id1"]], "Fr\u00e9chet mean of beta distributions": [[40, "Fr\u00e9chet-mean-of-beta-distributions"]], "Future changes": [[30, "future-changes"]], "GGRegression Fifa 2018 Player Passing Networks": [[52, "GGRegression-Fifa-2018-Player-Passing-Networks"]], "Generate data on the hyperbolic plane": [[38, "Generate-data-on-the-hyperbolic-plane"]], "Generate data on the sphere": [[38, "Generate-data-on-the-sphere"]], "Geodesic": [[48, "Geodesic"]], "Geodesics": [[36, "Geodesics"], [48, "Geodesics"]], "Geodesics on the hyperbolic plane": [[36, "Geodesics-on-the-hyperbolic-plane"]], "Geodesics on the special euclidean group SE(3)": [[36, "Geodesics-on-the-special-euclidean-group-SE(3)"]], "Geometry of beta distributions": [[40, "Geometry-of-beta-distributions"]], "Geometry of normal distributions": [[40, "Geometry-of-normal-distributions"]], "Geomstats": [[31, null]], "Geomstats governance and decision-making": [[27, null]], "Getting Started": [[26, null]], "Getting the code": [[22, "getting-the-code"]], "Google Season of Docs: Case Study": [[28, null]], "Google Season of Docs: Project Page": [[28, "google-season-of-docs-project-page"]], "Graph": [[49, "Graph"]], "Graph Space": [[49, "Graph-Space"], [52, null]], "Graph Space Metric": [[49, "Graph-Space-Metric"]], "Graph to Geodesic Alignment": [[49, "Graph-to-Geodesic-Alignment"]], "Graph to Graph Alignment": [[49, "Graph-to-Graph-Alignment"]], "Hackathon at Institut Henri Poincar\u00e9, 17th-21st Oct 2022": [[30, null]], "Hackathons": [[29, null]], "Hand gesture classification with EMG data using Riemannian metrics": [[44, null]], "Hands shapes": [[46, "Hands-shapes"]], "Hyperbolic Embedding of Graphs and Clustering": [[45, null]], "Hypothesis Testing on Equality of Means": [[43, "Hypothesis-Testing-on-Equality-of-Means"]], "Immersed manifolds: the example of a surface defined by its graph": [[41, "Immersed-manifolds:-the-example-of-a-surface-defined-by-its-graph"]], "Immersed manifolds: the example of the 2-sphere": [[41, "Immersed-manifolds:-the-example-of-the-2-sphere"]], "Implement your own Riemannian Geometry": [[41, null]], "Important links for technical writers": [[28, "important-links-for-technical-writers"]], "In the Hyperbolic plane": [[38, "In-the-Hyperbolic-plane"]], "Information geometry": [[40, null]], "Interpolation: Encoding Discrete Curves With Same Number of Points": [[43, "Interpolation:-Encoding-Discrete-Curves-With-Same-Number-of-Points"]], "Intrinsic vs extrinsic coordinates (connection)": [[30, "intrinsic-vs-extrinsic-coordinates-connection"]], "Intrinsic vs extrinsic coordinates (manifold)": [[30, "intrinsic-vs-extrinsic-coordinates-manifold"]], "Intro to Docstrings": [[22, "intro-to-docstrings"]], "Introduction": [[40, "Introduction"], [45, "Introduction"]], "Issue Triaging": [[22, "issue-triaging"]], "Issue tracker": [[22, "issue-tracker"]], "K-Means clustering of normal distributions": [[40, "K-Means-clustering-of-normal-distributions"]], "K-Means clustering on a Riemannian Manifold": [[39, null]], "Labelled or Unlabelled?": [[49, "Labelled-or-Unlabelled?"]], "Labelling and assigning the issue": [[22, "labelling-and-assigning-the-issue"]], "Learn More": [[31, "learn-more"]], "Learn differential geometry": [[24, "learn-differential-geometry"]], "Learning on Tangent Data": [[37, null]], "Learning on spaces of distributions": [[40, "Learning-on-spaces-of-distributions"]], "Logarithms on Manifolds": [[34, "Logarithms-on-Manifolds"]], "Loss function.": [[45, "Loss-function."]], "Making changes": [[22, "making-changes"]], "Mammals Grooming": [[52, "Mammals-Grooming"]], "Measuring project\u2019s success": [[28, "measuring-projects-success"]], "Metrics": [[28, "metrics"]], "Module contents": [[0, "module-geomstats"], [1, "module-geomstats.datasets"], [2, "module-geomstats.distributions"], [3, "module-geomstats.geometry"], [4, "module-geomstats.geometry.stratified"], [5, "module-geomstats.information_geometry"], [6, "module-geomstats.learning"], [7, "module-geomstats.numerics"], [8, "module-geomstats.numerics.optimization"], [9, "module-geomstats.test"], [10, "module-geomstats.test_cases"], [11, "module-contents"], [12, "module-geomstats.test_cases.datasets"], [13, "module-geomstats.test_cases.distributions"], [14, "module-geomstats.test_cases.geometry"], [15, "module-geomstats.test_cases.geometry.stratified"], [16, "module-geomstats.test_cases.information_geometry"], [17, "module-geomstats.test_cases.learning"], [18, "module-geomstats.test_cases.numerics"], [19, "module-geomstats.visualization"]], "Monkey\u2019s optical nerve heads: Data as landmarks in 3D": [[35, "Monkey's-optical-nerve-heads:-Data-as-landmarks-in-3D"]], "Motivation and Dataset": [[52, "Motivation-and-Dataset"], [52, "id1"]], "New structure": [[30, "new-structure"]], "No dynamic behavior": [[30, "no-dynamic-behavior"]], "Numerically optimizing the loss function": [[45, "Numerically-optimizing-the-loss-function"]], "Objectives": [[31, "objectives"]], "On the sphere": [[38, "On-the-sphere"]], "Operate Riemannian K-Means clustering": [[39, "Operate-Riemannian-K-Means-clustering"]], "Optimization of Sao Paulo traffic": [[50, null]], "Packages & Modules": [[20, null]], "Parallel transport": [[48, "Parallel-transport"], [48, "id2"]], "Parameter": [[6, "parameter"]], "Parameters": [[44, "Parameters"]], "Parameters and Initialization": [[45, "Parameters-and-Initialization"]], "Participants": [[28, "participants"]], "Past and future hackathons": [[29, "past-and-future-hackathons"]], "Plotting results": [[45, "Plotting-results"]], "Points shape": [[30, "points-shape"]], "Pose of objects in pictures: data on the Lie group of 3D rotations": [[35, "Pose-of-objects-in-pictures:-data-on-the-Lie-group-of-3D-rotations"]], "Practical methods": [[55, "practical-methods"]], "Pre-processing the dataset": [[50, "Pre-processing-the-dataset"]], "Preprocessing into covariance matrices": [[44, "Preprocessing-into-covariance-matrices"]], "Principal Components": [[51, "Principal-Components"]], "Principal geodesic analysis of normal distributions": [[40, "Principal-geodesic-analysis-of-normal-distributions"]], "Problem Statement": [[28, "problem-statement"]], "Project Description": [[28, "project-description"]], "Proposal Abstract": [[28, "proposal-abstract"]], "Proposed Budget": [[28, "proposed-budget"]], "Pull Request Checklist": [[22, "pull-request-checklist"]], "Pullback via diffeomorphism: Warping of the plane \\mathbb{R}^2": [[41, "Pullback-via-diffeomorphism:-Warping-of-the-plane-\\mathbb{R}^2"]], "Purpose of this document": [[54, "purpose-of-this-document"]], "Real world applications": [[55, "real-world-applications"]], "Reference:": [[35, "Reference:"]], "References": [[37, "References"], [40, "References"], [42, "References"], [45, "References"], [46, "References"], [47, "References"], [48, "References"]], "Regression": [[51, "Regression"]], "Related events": [[29, "related-events"]], "Remarks:": [[42, "Remarks:"]], "Remove duplicate samples in curves": [[43, "Remove-duplicate-samples-in-curves"]], "Reporting bugs and features": [[22, "reporting-bugs-and-features"]], "Reproducing issues": [[22, "reproducing-issues"]], "Riemannian optimization.": [[45, "Riemannian-optimization."]], "Roadmap": [[54, null]], "Roles And Responsibilities": [[27, "roles-and-responsibilities"]], "Run learning algorithms": [[24, "run-learning-algorithms"]], "Run the tests": [[22, "run-the-tests"]], "Running tests": [[22, "running-tests"]], "Set of Graphs: GraphSpace": [[49, "Set-of-Graphs:-GraphSpace"]], "Shape Analysis of Cancer Cells": [[43, null]], "Shape analysis of curves with the Square Root Velocity metric": [[42, null]], "Sign Classification": [[44, "Sign-Classification"]], "Social networks: data on the hyperbolic space": [[35, "Social-networks:-data-on-the-hyperbolic-space"]], "Source control with Git": [[22, "source-control-with-git"]], "Spider": [[49, "Spider"]], "Stalled Pull Requests": [[22, "stalled-pull-requests"]], "Statement of purpose: Geomstats in 2022": [[54, "statement-of-purpose-geomstats-in-2022"]], "Stratified spaces": [[49, null]], "Sub-Riemannian structures as limits of Riemannian structures": [[53, "Sub-Riemannian-structures-as-limits-of-Riemannian-structures"]], "Submodules": [[0, "submodules"], [1, "submodules"], [2, "submodules"], [3, "submodules"], [4, "submodules"], [5, "submodules"], [6, "submodules"], [7, "submodules"], [9, "submodules"], [10, "submodules"], [11, "submodules"], [12, "submodules"], [13, "submodules"], [14, "submodules"], [15, "submodules"], [16, "submodules"], [17, "submodules"], [18, "submodules"], [19, "submodules"]], "Subpackages": [[0, "subpackages"], [3, "subpackages"], [7, "subpackages"], [10, "subpackages"], [14, "subpackages"]], "Summary": [[28, "summary"]], "Summary plots": [[44, "Summary-plots"]], "TDD with pytest": [[22, "tdd-with-pytest"]], "Table of contents:": [[40, "Table-of-contents:"]], "Tangent PCA (at the Fr\u00e9chet mean)": [[38, "Tangent-PCA-(at-the-Fr\u00e9chet-mean)"], [38, "id2"]], "Technical Committee": [[27, "technical-committee"]], "Template of a bug/issue report": [[22, "template-of-a-bug-issue-report"]], "Test Driven Development": [[22, "test-driven-development"]], "Testing": [[22, "testing"]], "The Anatomy of a Docstring": [[22, "the-anatomy-of-a-docstring"]], "The Data": [[37, "The-Data"], [44, "The-Data"]], "The Heisenberg group": [[53, "The-Heisenberg-group"]], "The Manifold": [[37, "The-Manifold"]], "The Transformer": [[37, "The-Transformer"]], "The problem": [[28, "the-problem"]], "The project\u2019s scope": [[28, "the-projects-scope"]], "The space \\Sigma_2^3 of 2D triangles": [[48, "The-space-\\Sigma_2^3-of-2D-triangles"]], "The space \\Sigma_3^3 of 3D triangles": [[48, "The-space-\\Sigma_3^3-of-3D-triangles"]], "Timeline and Results": [[28, "timeline-and-results"]], "Title": [[28, "id4"]], "Tutorial: Sub-Riemannian geometry on the Heisenberg group": [[53, null]], "Tutorials": [[55, null]], "Uniform distribution": [[48, "Uniform-distribution"], [48, "id1"]], "Usecase : optical nerve heads": [[48, "Usecase-:-optical-nerve-heads"]], "Vectorization": [[3, "vectorization"]], "Visualization of Interpolated Dataset of Curves": [[43, "Visualization-of-Interpolated-Dataset-of-Curves"]], "Visualization of the Mean of each Treatment": [[43, "Visualization-of-the-Mean-of-each-Treatment"]], "What is the motivation for analyzing data on manifolds?": [[32, null]], "What will you learn in these tutorials?": [[32, "What-will-you-learn-in-these-tutorials?"]], "Workflow": [[29, "workflow"]], "Workflow of a contribution": [[22, "workflow-of-a-contribution"]], "World cities: data on the sphere": [[35, "World-cities:-data-on-the-sphere"]], "Writing Docstrings": [[22, "writing-docstrings"]], "Writing tests for geomstats": [[22, "writing-tests-for-geomstats"]], "geomstats": [[21, null]], "geomstats package": [[0, null]], "geomstats.algebra_utils module": [[0, "module-geomstats.algebra_utils"]], "geomstats.datasets package": [[1, null]], "geomstats.datasets.prepare_emg_data module": [[1, "module-geomstats.datasets.prepare_emg_data"]], "geomstats.datasets.prepare_graph_data module": [[1, "module-geomstats.datasets.prepare_graph_data"]], "geomstats.datasets.utils module": [[1, "module-geomstats.datasets.utils"]], "geomstats.distributions package": [[2, null]], "geomstats.distributions.brownian_motion module": [[2, "module-geomstats.distributions.brownian_motion"]], "geomstats.distributions.lognormal module": [[2, "module-geomstats.distributions.lognormal"]], "geomstats.errors module": [[0, "module-geomstats.errors"]], "geomstats.exceptions module": [[0, "module-geomstats.exceptions"]], "geomstats.geometry package": [[3, null]], "geomstats.geometry.base 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"geomstats.geometry.poincare_half_space": [[3, "module-geomstats.geometry.poincare_half_space", false]], "geomstats.geometry.poincare_polydisk": [[3, "module-geomstats.geometry.poincare_polydisk", false]], "geomstats.geometry.positive_lower_triangular_matrices": [[3, "module-geomstats.geometry.positive_lower_triangular_matrices", false]], "geomstats.geometry.positive_reals": [[3, "module-geomstats.geometry.positive_reals", false]], "geomstats.geometry.pre_shape": [[3, "module-geomstats.geometry.pre_shape", false]], "geomstats.geometry.product_hpd_and_siegel_disks": [[3, "module-geomstats.geometry.product_hpd_and_siegel_disks", false]], "geomstats.geometry.product_manifold": [[3, "module-geomstats.geometry.product_manifold", false]], "geomstats.geometry.product_positive_reals_and_poincare_disks": [[3, "module-geomstats.geometry.product_positive_reals_and_poincare_disks", false]], "geomstats.geometry.pullback_metric": [[3, "module-geomstats.geometry.pullback_metric", false]], "geomstats.geometry.quotient_metric": [[3, "module-geomstats.geometry.quotient_metric", false]], "geomstats.geometry.rank_k_psd_matrices": [[3, "module-geomstats.geometry.rank_k_psd_matrices", false]], "geomstats.geometry.riemannian_metric": [[3, "module-geomstats.geometry.riemannian_metric", false]], "geomstats.geometry.sasaki_metric": [[3, "module-geomstats.geometry.sasaki_metric", false]], "geomstats.geometry.scalar_product_metric": [[3, "module-geomstats.geometry.scalar_product_metric", false]], "geomstats.geometry.siegel": [[3, "module-geomstats.geometry.siegel", false]], "geomstats.geometry.skew_symmetric_matrices": [[3, "module-geomstats.geometry.skew_symmetric_matrices", false]], "geomstats.geometry.spd_matrices": [[3, "module-geomstats.geometry.spd_matrices", false]], "geomstats.geometry.special_euclidean": [[3, "module-geomstats.geometry.special_euclidean", false]], "geomstats.geometry.special_orthogonal": [[3, "module-geomstats.geometry.special_orthogonal", false]], "geomstats.geometry.stiefel": [[3, "module-geomstats.geometry.stiefel", false]], "geomstats.geometry.stratified": [[4, "module-geomstats.geometry.stratified", false]], "geomstats.geometry.stratified.bhv_space": [[4, "module-geomstats.geometry.stratified.bhv_space", false]], "geomstats.geometry.stratified.graph_space": [[4, "module-geomstats.geometry.stratified.graph_space", false]], "geomstats.geometry.stratified.point_set": [[4, "module-geomstats.geometry.stratified.point_set", false]], "geomstats.geometry.stratified.quotient": [[4, "module-geomstats.geometry.stratified.quotient", false]], "geomstats.geometry.stratified.spider": [[4, "module-geomstats.geometry.stratified.spider", false]], "geomstats.geometry.stratified.trees": [[4, "module-geomstats.geometry.stratified.trees", false]], "geomstats.geometry.stratified.vectorization": [[4, "module-geomstats.geometry.stratified.vectorization", false]], "geomstats.geometry.stratified.wald_space": [[4, 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"tangent_submersion() (geomstats.geometry.full_rank_correlation_matrices.fullrankcorrelationmatrices method)": [[3, "geomstats.geometry.full_rank_correlation_matrices.FullRankCorrelationMatrices.tangent_submersion", false]], "tangent_submersion() (geomstats.geometry.grassmannian.grassmannian method)": [[3, "geomstats.geometry.grassmannian.Grassmannian.tangent_submersion", false]], "tangent_submersion() (geomstats.geometry.hyperboloid.hyperboloid method)": [[3, "geomstats.geometry.hyperboloid.Hyperboloid.tangent_submersion", false]], "tangent_submersion() (geomstats.geometry.lower_triangular_matrices.strictlylowertriangularmatrices method)": [[3, "geomstats.geometry.lower_triangular_matrices.StrictlyLowerTriangularMatrices.tangent_submersion", false]], "tangent_submersion() (geomstats.geometry.positive_lower_triangular_matrices.pltunitdiagmatrices method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.PLTUnitDiagMatrices.tangent_submersion", false]], "tangent_submersion() (geomstats.geometry.pre_shape.preshapespace method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpace.tangent_submersion", false]], "tangent_submersion() (geomstats.geometry.stiefel.stiefel method)": [[3, "geomstats.geometry.stiefel.Stiefel.tangent_submersion", false]], "tangent_submersion() (geomstats.geometry.symmetric_matrices.nullrowsumssymmetricmatrices method)": [[3, "geomstats.geometry.symmetric_matrices.NullRowSumsSymmetricMatrices.tangent_submersion", false]], "tangent_submersion() (geomstats.geometry.symmetric_matrices.symmetrichollowmatrices method)": [[3, "geomstats.geometry.symmetric_matrices.SymmetricHollowMatrices.tangent_submersion", false]], "tangent_submersion() (geomstats.information_geometry.multinomial.multinomialdistributions method)": [[5, "geomstats.information_geometry.multinomial.MultinomialDistributions.tangent_submersion", false]], "tangent_translation_map() (geomstats.geometry.lie_group.liegroup method)": [[3, 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false]], "to_tangent() (geomstats.geometry.lie_group.liegroup method)": [[3, "geomstats.geometry.lie_group.LieGroup.to_tangent", false]], "to_tangent() (geomstats.geometry.lie_group.matrixliegroup method)": [[3, "geomstats.geometry.lie_group.MatrixLieGroup.to_tangent", false]], "to_tangent() (geomstats.geometry.manifold.manifold method)": [[3, "geomstats.geometry.manifold.Manifold.to_tangent", false]], "to_tangent() (geomstats.geometry.nfold_manifold.nfoldmanifold method)": [[3, "geomstats.geometry.nfold_manifold.NFoldManifold.to_tangent", false]], "to_tangent() (geomstats.geometry.positive_lower_triangular_matrices.pltunitdiagmatrices method)": [[3, "geomstats.geometry.positive_lower_triangular_matrices.PLTUnitDiagMatrices.to_tangent", false]], "to_tangent() (geomstats.geometry.pre_shape.preshapespace method)": [[3, "geomstats.geometry.pre_shape.PreShapeSpace.to_tangent", false]], "to_tangent() (geomstats.geometry.product_manifold.productmanifold method)": [[3, 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+Search.setIndex({"alltitles": {"1) Analyzing data on a manifold reduces the degrees of freedom of the system, making computations less complicated.": [[32, "1)-Analyzing-data-on-a-manifold-reduces-the-degrees-of-freedom-of-the-system,-making-computations-less-complicated."]], "1. A first example in 2D": [[42, "1.-A-first-example-in-2D"]], "1. Introduction": [[33, null]], "1. Introduction and Motivation": [[43, "1.-Introduction-and-Motivation"], [50, "1.-Introduction-and-Motivation"]], "1. What is a Connection?": [[34, null]], "1.1 General Mathematical Definition of the Connection": [[34, "1.1-General-Mathematical-Definition-of-the-Connection"]], "1.2 What does the Connection actually do? \u2013 a more friendly description of the connection, with an example": [[34, "1.2-What-does-the-Connection-actually-do?----a-more-friendly-description-of-the-connection,-with-an-example"]], "2) Knowing the manifold that a data set belongs to may give you a deeper understanding of your data\u2019s evolution or \u201cwhy your data looks/evolves the way it does\u201d": [[32, "2)-Knowing-the-manifold-that-a-data-set-belongs-to-may-give-you-a-deeper-understanding-of-your-data's-evolution-or-%22why-your-data-looks/evolves-the-way-it-does%22"]], "2. Dataset Description": [[43, "2.-Dataset-Description"]], "2. Dataset description": [[50, "2.-Dataset-description"]], "2. Focus on quotienting out reparametrizations": [[42, "2.-Focus-on-quotienting-out-reparametrizations"]], "2. What is a Manifold?": [[33, "2.-What-is-a-Manifold?"]], "2. Why is the Connection important (big picture)? \u2013 parallel transport, parallel vector fields": [[34, "2.-Why-is-the-Connection-important-(big-picture)?----parallel-transport,-parallel-vector-fields"]], "2.1 Parallel Vector Fields": [[34, "2.1-Parallel-Vector-Fields"]], "2.1 Precise Mathematical Definition": [[33, "2.1-Precise-Mathematical-Definition"]], "2.1. Comparing parametrized curves": [[42, "2.1.-Comparing-parametrized-curves"]], "2.2 Imprecise \u201cLayman\u2019s Terms\u201d Definition:": [[33, "2.2-Imprecise-%22Layman's-Terms%22-Definition:"]], "2.2 Parallel Transport": [[34, "2.2-Parallel-Transport"]], "2.2. Comparing unparametrized curves": [[42, "2.2.-Comparing-unparametrized-curves"]], "2.3 Hypersphere example:": [[33, "2.3-Hypersphere-example:"]], "2.4 More examples of manifolds:": [[33, "2.4-More-examples-of-manifolds:"]], "3) Knowing the manifold a data set lives in will help you give you better predictive power and extract the \u201csignal\u201d from a noisy data set or a data set with very few datapoints.": [[32, "3)-Knowing-the-manifold-a-data-set-lives-in-will-help-you-give-you-better-predictive-power-and-extract-the-%22signal%22-from-a-noisy-data-set-or-a-data-set-with-very-few-datapoints."]], "3. Defining Geodesics with the Connection": [[34, "3.-Defining-Geodesics-with-the-Connection"]], "3. Example in 3D": [[42, "3.-Example-in-3D"]], "3. Mathematical modeling": [[50, "3.-Mathematical-modeling"]], "3. Preprocessing": [[43, "3.-Preprocessing"]], "3. Why do we care about manifolds?": [[33, "3.-Why-do-we-care-about-manifolds?"]], "3.1 Intuitive Definition of a Geodesic": [[34, "3.1-Intuitive-Definition-of-a-Geodesic"]], "3.1. Road representation: introduction to the Gamma manifold.": [[50, "3.1.-Road-representation:-introduction-to-the-Gamma-manifold."]], "3.1.1. Hypotheses": [[50, "3.1.1.-Hypotheses"]], "3.1.2. Model": [[50, "3.1.2.-Model"]], "3.2 Defining a geodesic with the connection": [[34, "3.2-Defining-a-geodesic-with-the-connection"]], "3.2. Mathematical representation of renovation efforts": [[50, "3.2.-Mathematical-representation-of-renovation-efforts"]], "3.2.1. Hypotheses": [[50, "3.2.1.-Hypotheses"]], "3.2.2. Model": [[50, "3.2.2.-Model"]], "3.3 Building Intuition with Physics": [[34, "3.3-Building-Intuition-with-Physics"]], "3.3. Optimization problem": [[50, "3.3.-Optimization-problem"]], "4 Data Analysis": [[43, "4-Data-Analysis"]], "4. Dataset processing": [[50, "4.-Dataset-processing"]], "4. Description of Connection in Local Coordinates (using Christoffel Symbols)": [[34, "4.-Description-of-Connection-in-Local-Coordinates-(using-Christoffel-Symbols)"]], "4. What does the Manifold class do?": [[33, "4.-What-does-the-Manifold-class-do?"]], "4.1 Math Notation: \\mathbf{ \\partial_i} Basis Vector": [[34, "4.1-Math-Notation:-\\mathbf{-\\partial_i}-Basis-Vector"]], "4.1 Tangent Vectors": [[33, "4.1-Tangent-Vectors"]], "4.2 Einstein Notation": [[34, "4.2-Einstein-Notation"]], "4.2 Tangent Spaces": [[33, "4.2-Tangent-Spaces"]], "4.3 What does a Christoffel Symbol represent?": [[34, "4.3-What-does-a-Christoffel-Symbol-represent?"]], "4.4 The Geodesic Equation in local coordinates": [[34, "4.4-The-Geodesic-Equation-in-local-coordinates"]], "4.5 Why should we care about Christoffel Symbols and this representation of the connection?": [[34, "4.5-Why-should-we-care-about-Christoffel-Symbols-and-this-representation-of-the-connection?"]], "5. Additional Important Mathematical concepts utilized by Connection Class: Logarithm, Exponential": [[34, "5.-Additional-Important-Mathematical-concepts-utilized-by-Connection-Class:-Logarithm,-Exponential"]], "5. How is the Manifold class structured?": [[33, "5.-How-is-the-Manifold-class-structured?"]], "5. Solving the problem at hand": [[50, "5.-Solving-the-problem-at-hand"]], "5.1 The Parent Class: Manifold": [[33, "5.1-The-Parent-Class:-Manifold"]], "5.1.1 Examples of Using Manifold\u2019s Attributes and Methods in the Subclass: Hypersphere": [[33, "5.1.1-Examples-of-Using-Manifold's-Attributes-and-Methods-in-the-Subclass:-Hypersphere"]], "5.1.2 The Full Manifold Class Code": [[33, "5.1.2-The-Full-Manifold-Class-Code"]], "5.2 VectorSpaceOpenSet": [[33, "5.2-VectorSpaceOpenSet"]], "5.2.1 What is an Open Set?": [[33, "5.2.1-What-is-an-Open-Set?"]], "5.2.2 What Methods are Implemented in VectorSpaceOpenSet?": [[33, "5.2.2-What-Methods-are-Implemented-in-VectorSpaceOpenSet?"]], "5.3 LevelSet": [[33, "5.3-LevelSet"]], "5.3.1 What is a Level Set?": [[33, "5.3.1-What-is-a-Level-Set?"]], "5.3.2 LevelSet in Geomstats": [[33, "5.3.2-LevelSet-in-Geomstats"]], "5.4 VectorSpace": [[33, "5.4-VectorSpace"]], "5.5 ProductManifold": [[33, "5.5-ProductManifold"]], "5.5.1 Example": [[33, "5.5.1-Example"]], "6. Geomstats Documentation: The Connection Class": [[34, "6.-Geomstats-Documentation:-The-Connection-Class"]], "6.1 method: parallel_transport()": [[34, "6.1-method:-parallel_transport()"]], "6.2 method: geodesic_equation()": [[34, "6.2-method:-geodesic_equation()"]], "6.3 method: exp() (exponential)": [[34, "6.3-method:-exp()-(exponential)"]], "6.4 method: log() (logarithm)": [[34, "6.4-method:-log()-(logarithm)"]], "6.5 The full Connection class": [[34, "6.5-The-full-Connection-class"]], "7. RiemannianMetric Sub-Class": [[34, "7.-RiemannianMetric-Sub-Class"]], "7.1 Riemannian Metrics": [[34, "7.1-Riemannian-Metrics"]], "7.2 How is the Riemannian Metric Connected with the Connection?": [[34, "7.2-How-is-the-Riemannian-Metric-Connected-with-the-Connection?"]], "7.3 RiemannianMetric in Geomstats": [[34, "7.3-RiemannianMetric-in-Geomstats"]], "8. Conclusion": [[34, "8.-Conclusion"]], "A primer in space, metric and aligners": [[51, "A-primer-in-space,-metric-and-aligners"]], "A. Test on the same session and user as Training/Calibration": [[44, "A.-Test-on-the-same-session-and-user-as-Training/Calibration"]], "A.0. Using Logistic Regression on the vectorized Matrix (Euclidean Method)": [[44, "A.0.-Using-Logistic-Regression-on-the-vectorized-Matrix-(Euclidean-Method)"]], "A.1. Using MLP on the vectorized Matrix (Euclidean Method)": [[44, "A.1.-Using-MLP-on-the-vectorized-Matrix-(Euclidean-Method)"]], "A.2. Using Tangent space projection + Logistic Regression": [[44, "A.2.-Using-Tangent-space-projection-+-Logistic-Regression"]], "A.3. Using Tangent space projection + MLP": [[44, "A.3.-Using-Tangent-space-projection-+-MLP"]], "A.4. Using Euclidean MDM": [[44, "A.4.-Using-Euclidean-MDM"]], "A.5. Using Riemannian MDM": [[44, "A.5.-Using-Riemannian-MDM"]], "API Reference": [[20, null]], "About the organization": [[28, "about-the-organization"]], "About the project": [[28, "about-the-project"]], "Additional information": [[28, "additional-information"]], "Align all and Compute for Graphs": [[51, null]], "Alignment": [[43, "Alignment"]], "Analysis": [[28, "analysis"]], "Analyze Distances to the \u201cGlobal\u201d Mean Shape": [[43, "Analyze-Distances-to-the-%22Global%22-Mean-Shape"]], "Answering questions": [[22, "answering-questions"]], "Backends": [[22, "backends"]], "Brain connectomes: data on the manifold of Symmetric Positive Definite (SPD) matrices": [[35, "Brain-connectomes:-data-on-the-manifold-of-Symmetric-Positive-Definite-(SPD)-matrices"]], "Breach": [[27, "breach"]], "Budget": [[28, "budget"]], "Build the docs": [[22, "build-the-docs"]], "Building the Documentation": [[22, "building-the-documentation"]], "Capturing the graph structure": [[45, "Capturing-the-graph-structure"]], "Changing the Governance Documents": [[27, "changing-the-governance-documents"]], "Changing the Metric on the Space of Cell Shapes": [[43, "Changing-the-Metric-on-the-Space-of-Cell-Shapes"]], "Classifying hands poses with Kendall shape spaces": [[46, null]], "Code Examples": [[24, null]], "Code Review Guidelines": [[22, "code-review-guidelines"]], "Coding Style Guidelines": [[22, "coding-style-guidelines"]], "Commenting on alternative solutions": [[22, "commenting-on-alternative-solutions"]], "Communication team": [[27, "communication-team"]], "Computing with shapes of landmarks in Kendall shape spaces": [[47, null]], "Computing with triangular shapes in Kendall framework": [[48, null]], "Conclusion": [[33, "Conclusion"], [37, "Conclusion"], [50, "Conclusion"]], "Conflict of Interest": [[27, "conflict-of-interest"]], "Context": [[44, "Context"]], "Contributing Guide": [[22, null]], "Contributor Experience Team": [[27, "contributor-experience-team"]], "Contributors": [[27, "contributors"]], "Core developers": [[27, "core-developers"]], "Covariances plot of the Frechet Mean of the affine invariant metric": [[44, "Covariances-plot-of-the-Frechet-Mean-of-the-affine-invariant-metric"]], "Covariances plot of the euclidean average": [[44, "Covariances-plot-of-the-euclidean-average"]], "Create or choose an issue for new contributors": [[22, "create-or-choose-an-issue-for-new-contributors"]], "Data init: Generate clusters randomly on the sphere": [[39, "Data-init:-Generate-clusters-randomly-on-the-sphere"]], "Data on Manifolds": [[35, null]], "Decision Making Process": [[27, "decision-making-process"]], "Defining the Heisenberg sub-Riemannian structure via a frame-field": [[53, "Defining-the-Heisenberg-sub-Riemannian-structure-via-a-frame-field"]], "Dependencies and a virtual environment": [[22, "dependencies-and-a-virtual-environment"]], "Design": [[31, "design"]], "Design and Organize Scattered Documentation in Geomstats": [[28, 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manifolds": [[36, null], [36, "id1"]], "Fr\u00e9chet Mean and Tangent PCA": [[38, null]], "Fr\u00e9chet mean": [[38, "Fr\u00e9chet-mean"], [38, "id1"]], "Fr\u00e9chet mean of beta distributions": [[40, "Fr\u00e9chet-mean-of-beta-distributions"]], "Future changes": [[30, "future-changes"]], "GGRegression Fifa 2018 Player Passing Networks": [[52, "GGRegression-Fifa-2018-Player-Passing-Networks"]], "Generate data on the hyperbolic plane": [[38, "Generate-data-on-the-hyperbolic-plane"]], "Generate data on the sphere": [[38, "Generate-data-on-the-sphere"]], "Geodesic": [[48, "Geodesic"]], "Geodesics": [[36, "Geodesics"], [48, "Geodesics"]], "Geodesics on the hyperbolic plane": [[36, "Geodesics-on-the-hyperbolic-plane"]], "Geodesics on the special euclidean group SE(3)": [[36, "Geodesics-on-the-special-euclidean-group-SE(3)"]], "Geometry of beta distributions": [[40, "Geometry-of-beta-distributions"]], "Geometry of normal distributions": [[40, "Geometry-of-normal-distributions"]], "Geomstats": [[31, null]], "Geomstats governance and decision-making": [[27, null]], "Getting Started": [[26, null]], "Getting the code": [[22, "getting-the-code"]], "Google Season of Docs: Case Study": [[28, null]], "Google Season of Docs: Project Page": [[28, "google-season-of-docs-project-page"]], "Graph": [[49, "Graph"]], "Graph Space": [[49, "Graph-Space"], [52, null]], "Graph Space Metric": [[49, "Graph-Space-Metric"]], "Graph to Geodesic Alignment": [[49, "Graph-to-Geodesic-Alignment"]], "Graph to Graph Alignment": [[49, "Graph-to-Graph-Alignment"]], "Hackathon at Institut Henri Poincar\u00e9, 17th-21st Oct 2022": [[30, null]], "Hackathons": [[29, null]], "Hand gesture classification with EMG data using Riemannian metrics": [[44, null]], "Hands shapes": [[46, "Hands-shapes"]], "Hyperbolic Embedding of Graphs and Clustering": [[45, null]], "Hypothesis Testing on Equality of Means": [[43, "Hypothesis-Testing-on-Equality-of-Means"]], "Immersed manifolds: the example of a surface defined by its graph": [[41, "Immersed-manifolds:-the-example-of-a-surface-defined-by-its-graph"]], "Immersed manifolds: the example of the 2-sphere": [[41, "Immersed-manifolds:-the-example-of-the-2-sphere"]], "Implement your own Riemannian Geometry": [[41, null]], "Important links for technical writers": [[28, "important-links-for-technical-writers"]], "In the Hyperbolic plane": [[38, "In-the-Hyperbolic-plane"]], "Information geometry": [[40, null]], "Interpolation: Encoding Discrete Curves With Same Number of Points": [[43, "Interpolation:-Encoding-Discrete-Curves-With-Same-Number-of-Points"]], "Intrinsic vs extrinsic coordinates (connection)": [[30, "intrinsic-vs-extrinsic-coordinates-connection"]], "Intrinsic vs extrinsic coordinates (manifold)": [[30, "intrinsic-vs-extrinsic-coordinates-manifold"]], "Intro to Docstrings": [[22, "intro-to-docstrings"]], "Introduction": [[40, "Introduction"], [45, "Introduction"]], "Issue Triaging": [[22, "issue-triaging"]], "Issue tracker": [[22, "issue-tracker"]], "K-Means clustering of normal distributions": [[40, "K-Means-clustering-of-normal-distributions"]], "K-Means clustering on a Riemannian Manifold": [[39, null]], "Labelled or Unlabelled?": [[49, "Labelled-or-Unlabelled?"]], "Labelling and assigning the issue": [[22, "labelling-and-assigning-the-issue"]], "Learn More": [[31, "learn-more"]], "Learn differential geometry": [[24, "learn-differential-geometry"]], "Learning on Tangent Data": [[37, null]], "Learning on spaces of distributions": [[40, "Learning-on-spaces-of-distributions"]], "Logarithms on Manifolds": [[34, "Logarithms-on-Manifolds"]], "Loss function.": [[45, "Loss-function."]], "Making changes": [[22, "making-changes"]], "Mammals Grooming": [[52, "Mammals-Grooming"]], "Measuring project\u2019s success": [[28, "measuring-projects-success"]], "Metrics": [[28, "metrics"]], "Module contents": [[0, "module-geomstats"], [1, "module-geomstats.datasets"], [2, "module-geomstats.distributions"], [3, "module-geomstats.geometry"], [4, "module-geomstats.geometry.stratified"], [5, "module-geomstats.information_geometry"], [6, "module-geomstats.learning"], [7, "module-geomstats.numerics"], [8, "module-geomstats.numerics.optimization"], [9, "module-geomstats.test"], [10, "module-geomstats.test_cases"], [11, "module-contents"], [12, "module-geomstats.test_cases.datasets"], [13, "module-geomstats.test_cases.distributions"], [14, "module-geomstats.test_cases.geometry"], [15, "module-geomstats.test_cases.geometry.stratified"], [16, "module-geomstats.test_cases.information_geometry"], [17, "module-geomstats.test_cases.learning"], [18, "module-geomstats.test_cases.numerics"], [19, "module-geomstats.visualization"]], "Monkey\u2019s optical nerve heads: Data as landmarks in 3D": [[35, "Monkey's-optical-nerve-heads:-Data-as-landmarks-in-3D"]], "Motivation and Dataset": [[52, "Motivation-and-Dataset"], [52, "id1"]], "New structure": [[30, "new-structure"]], "No dynamic behavior": [[30, "no-dynamic-behavior"]], "Numerically optimizing the loss function": [[45, "Numerically-optimizing-the-loss-function"]], "Objectives": [[31, "objectives"]], "On the sphere": [[38, "On-the-sphere"]], "Operate Riemannian K-Means clustering": [[39, "Operate-Riemannian-K-Means-clustering"]], "Optimization of Sao Paulo traffic": [[50, null]], "Packages & Modules": [[20, null]], "Parallel transport": [[48, "Parallel-transport"], [48, "id2"]], "Parameter": [[6, "parameter"]], "Parameters": [[44, "Parameters"]], "Parameters and Initialization": [[45, "Parameters-and-Initialization"]], "Participants": [[28, "participants"]], "Past and future hackathons": [[29, "past-and-future-hackathons"]], "Plotting results": [[45, "Plotting-results"]], "Points shape": [[30, "points-shape"]], "Pose of objects in pictures: data on the Lie group of 3D rotations": [[35, "Pose-of-objects-in-pictures:-data-on-the-Lie-group-of-3D-rotations"]], "Practical methods": [[55, "practical-methods"]], "Pre-processing the dataset": [[50, "Pre-processing-the-dataset"]], "Preprocessing into covariance matrices": [[44, "Preprocessing-into-covariance-matrices"]], "Principal Components": [[51, "Principal-Components"]], "Principal geodesic analysis of normal distributions": [[40, "Principal-geodesic-analysis-of-normal-distributions"]], "Problem Statement": [[28, "problem-statement"]], "Project Description": [[28, "project-description"]], "Proposal Abstract": [[28, "proposal-abstract"]], "Proposed Budget": [[28, "proposed-budget"]], "Pull Request Checklist": [[22, "pull-request-checklist"]], "Pullback via diffeomorphism: Warping of the plane \\mathbb{R}^2": [[41, "Pullback-via-diffeomorphism:-Warping-of-the-plane-\\mathbb{R}^2"]], "Purpose of this document": [[54, "purpose-of-this-document"]], "Real world applications": [[55, "real-world-applications"]], "Reference:": [[35, "Reference:"]], "References": [[37, "References"], [40, "References"], [42, "References"], [45, "References"], [46, "References"], [47, "References"], [48, "References"]], "Regression": [[51, "Regression"]], "Related events": [[29, "related-events"]], "Remarks:": [[42, "Remarks:"]], "Remove duplicate samples in curves": [[43, "Remove-duplicate-samples-in-curves"]], "Reporting bugs and features": [[22, "reporting-bugs-and-features"]], "Reproducing issues": [[22, "reproducing-issues"]], "Riemannian optimization.": [[45, "Riemannian-optimization."]], "Roadmap": [[54, null]], "Roles And Responsibilities": [[27, "roles-and-responsibilities"]], "Run learning algorithms": [[24, "run-learning-algorithms"]], "Run the tests": [[22, "run-the-tests"]], "Running tests": [[22, "running-tests"]], "Set of Graphs: GraphSpace": [[49, "Set-of-Graphs:-GraphSpace"]], "Shape Analysis of Cancer Cells": [[43, null]], "Shape analysis of curves with the Square Root Velocity metric": [[42, null]], "Sign Classification": [[44, "Sign-Classification"]], "Social networks: data on the hyperbolic space": [[35, "Social-networks:-data-on-the-hyperbolic-space"]], "Source control with Git": [[22, "source-control-with-git"]], "Spider": [[49, "Spider"]], "Stalled Pull Requests": [[22, "stalled-pull-requests"]], "Statement of purpose: Geomstats in 2022": [[54, "statement-of-purpose-geomstats-in-2022"]], "Stratified spaces": [[49, null]], "Sub-Riemannian structures as limits of Riemannian structures": [[53, "Sub-Riemannian-structures-as-limits-of-Riemannian-structures"]], "Submodules": [[0, "submodules"], [1, "submodules"], [2, "submodules"], [3, "submodules"], [4, "submodules"], [5, "submodules"], [6, "submodules"], [7, "submodules"], [9, "submodules"], [10, "submodules"], [11, "submodules"], [12, "submodules"], [13, "submodules"], [14, "submodules"], [15, "submodules"], [16, "submodules"], [17, "submodules"], [18, "submodules"], [19, "submodules"]], "Subpackages": [[0, "subpackages"], [3, "subpackages"], [7, "subpackages"], [10, "subpackages"], [14, "subpackages"]], "Summary": [[28, "summary"]], "Summary plots": [[44, "Summary-plots"]], "TDD with pytest": [[22, "tdd-with-pytest"]], "Table of contents:": [[40, "Table-of-contents:"]], "Tangent PCA (at the Fr\u00e9chet mean)": [[38, "Tangent-PCA-(at-the-Fr\u00e9chet-mean)"], [38, "id2"]], "Technical Committee": [[27, "technical-committee"]], "Template of a bug/issue report": [[22, "template-of-a-bug-issue-report"]], "Test Driven Development": [[22, "test-driven-development"]], "Testing": [[22, "testing"]], "The Anatomy of a Docstring": [[22, "the-anatomy-of-a-docstring"]], "The Data": [[37, "The-Data"], [44, "The-Data"]], "The Heisenberg group": [[53, "The-Heisenberg-group"]], "The Manifold": [[37, "The-Manifold"]], "The Transformer": [[37, "The-Transformer"]], "The problem": [[28, "the-problem"]], "The project\u2019s scope": [[28, "the-projects-scope"]], "The space \\Sigma_2^3 of 2D triangles": [[48, "The-space-\\Sigma_2^3-of-2D-triangles"]], "The space \\Sigma_3^3 of 3D triangles": [[48, "The-space-\\Sigma_3^3-of-3D-triangles"]], "Timeline and Results": [[28, "timeline-and-results"]], "Title": [[28, "id4"]], "Tutorial: Sub-Riemannian geometry on the Heisenberg group": [[53, null]], "Tutorials": [[55, null]], "Uniform distribution": [[48, "Uniform-distribution"], [48, "id1"]], "Usecase : optical nerve heads": [[48, "Usecase-:-optical-nerve-heads"]], "Vectorization": [[3, "vectorization"]], "Visualization of Interpolated Dataset of Curves": [[43, "Visualization-of-Interpolated-Dataset-of-Curves"]], "Visualization of the Mean of each Treatment": [[43, "Visualization-of-the-Mean-of-each-Treatment"]], "What is the motivation for analyzing data on manifolds?": [[32, null]], "What will you learn in these tutorials?": [[32, "What-will-you-learn-in-these-tutorials?"]], "Workflow": [[29, "workflow"]], "Workflow of a contribution": [[22, "workflow-of-a-contribution"]], "World cities: data on the sphere": [[35, "World-cities:-data-on-the-sphere"]], "Writing Docstrings": [[22, 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