Wednesday, June 12, 2024

Principal Geodesic Analysis

Target audience: Advanced
Estimated reading time: 7'

Principal Component Analysis (PCA) is essential for dimensionality and noise reduction, feature extraction, and anomaly detection. However, its effectiveness is limited by the strong assumption of linearity. 
Principal Geodesic Analysis (PGA) addresses this limitation by extending PCA to handle non-linear data that lies on a lower-dimensional manifold.


Table of contents
       Tangent PCA
       Setup
       Euclidean PCA
       Tangent PCA on hypersphere
References
Appendix
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What you will learn: How to implement principal geodesic analysis as extension of principal components analysis on a manifolds tangent space.

Notes: 

  • Environments: Python  3.10.10, Geomstats 2.7.0, Scikit-learn 1.4.2
  • This article assumes that the reader is somewhat familiar with differential and tensor calculus [ref 1]. Please refer to our previous articles related to geometric learning [ref 2, 3, 4].
  • Source code is available at  Github.com/patnicolas/Data_Exploration/manifolds
  • To enhance the readability of the algorithm implementations, we have omitted non-essential code elements like error checking, comments, exceptions, validation of class and method arguments, scoping qualifiers, and import statements.

Introduction

This article is the ninth installments of our ongoing series focused on geometric learning. As with previous articles, we utilize the Geomstats Python library [ref. 5] to implement concepts associated with geometric learning.

Note: Summaries of my earlier articles on this topic can be found in the Appendix

As a reminder, the primary goal of learning Riemannian geometry is to understand and analyze the properties of curved spaces that cannot be described adequately using Euclidean geometry alone. 

This article revisits the widely used unsupervised learning technique, Principal Component Analysis (PCA), and its counterpart in non-Euclidean space, Principal Geodesic Analysis (PGA).
The content of this article is as follows:
  1. Brief recap of PCA
  2. Overview of key components of differential geometry
  3. Introduction to PCA on tangent space using the logarithmic map
  4. Implementation in Python using the Geomstats library

   Principal components

    Principal component analysis

Principal Component Analysis (PCA) is a technique for reducing the dimensionality of a large dataset, simplifying it into a smaller set of components while preserving important patterns and trends. The goal is to reduce the number of variables of a data set, while preserving as much information as possible.

For a n-dimensional data, PCA tries to put maximum possible information in the first component c0, then maximum remaining information in the second c1 and so on, until having something like shown in the scree plot below.
Finally, we select the p << n top first components such as:\[\left \{ c_{i} :\ \ \sum_{i=1}^{p} c_{i} < T \ and \ \ c_{1} \geqslant c_{2} .... \geqslant c_{p} \right \}\]
The principal components are actually the eigenvectors of the Covariance matrix.
The first thing to understand about eigenvectors and eigenvalues is that they always appear in pairs, with each eigenvector corresponding to an eigenvalue. Additionally, the number of these pairs matches the number of dimensions in the data.

For instance, in a 3-dimension space, the eigenvalues are extracted from the following 3x3 symmetric Covariance matrix:\[\begin{bmatrix} cov(x, x)) & cov(x,y) & cov(x, z)\\ cov(x, y) & cov(y,y) & cov(y,z) \\ cov(x, z) & cov(y, z) & cov(z, z) \end{bmatrix}\] as cov(a, b) = cov(b, a).
The eigenvectors of the Covariance matrix (principal components) are the directions of the axes where there is the most variance (most information).
Assuming n normalized data points xi, the first principal component (with most significant eigenvalue) is defined as: \[P_{1}= arg\max_{||p||=1} \sum_{i=1}^{n} \left ( p.x_{i} \right )^{2} \ \ \ (1)\]where '.' is the dot product.


Differential geometry

Extending principal components to differentiable manifolds requires basic knowledge of differential and Riemann geometry introduced in previous articles [ref  2, 3, 4].

Smooth manifold
A smooth (or differentiable) manifold is a topological manifold equipped with a globally defined differential structure. Locally, any topological manifold can be endowed with a differential structure by applying the homeomorphisms from its atlas and the standard differential structure of a vector space.

Tangent space
At every point P on a differentiable manifold, one can associate a tangent space, which is a real vector space that intuitively encompasses all the possible directions in which one can move tangentially through P. The elements within this tangent space at P are referred to as the tangent vectors, tgt_vector at P. 
This concept generalizes the idea of a vector originating from a specific point in Euclidean space. For a connected manifold, the dimension of the tangent space at any point is identical to the dimension of the manifold itself.

Geodesics
Geodesics are curves on a surface that only turn as necessary to remain on the surface, without deviating sideways. They generalize the concept of a "straight line" from a plane to a surface, representing the shortest path between two points on that surface.
Mathematically, a curve c(t) on a surface S is a geodesic if at each point c(t), the acceleration is zero or parallel to the normal vector:\[\frac{d^{2}}{dt^{2}}c(t) = 0 \ \ or \ \ \frac{d^{2}}{dt^{2}}c(t).\vec{n}=\frac{d^{2}}{dt^{2}}c(t)\]
Fig. 1 Illustration of a tangent vector and geodesic on a sphere

Logarithmic map
Given two points P and Q on a manifold, the vector on the tangent space v from P to Q is defined as: \[\left \| log_{P}(Q) \right \| = \left \| v \right \|\]

Tangent PCA

On a manifold, tangent spaces (or plane) are local euclidean space for which PCA can be computed. The purpose of Principal Geodesic Analysis is to project the principal components on the geodesic using the logarithmic map (inverse exponential map).
Given a mean m of n data points x[i] on a manifold with a set of geodesics at m, geod, the first principal component on geodesics is defined as:\[\begin{matrix} P_{1}=arg\max_{\left \| v \right \| = 1}\sum_{i=1}^{n}\left \langle v.log_{m}\pi _{geod}(x_{i}) \right \rangle^{2} \\ \pi_{geod}(x_{i}) = arg\max_{g \in geod} \left \| log_{m}(x_{i})-log_{m}(g) \right \|^2 \end{matrix}\]
Fig. 2 Illustration of a tangent vector and geodesic on a sphere

The mean point on the data point x[I] on the manifold can be defined as either a barycenter or a Frechet Mean [ref 6]. Our implementation in the next section relies on the Frechet mean.

  Implementation

For the sake of simplicity, we illustrate the concept of applying PCA on manifold geodesics using a simple manifold, Hypersphere we introduced in a previous article, Differentiable Manifolds for Geometric Learning: Hypersphere

    Setup

Let's encapsulate the evaluation of principal geodesics analysis in a class HyperspherePCA. We leverage the class HyperphereSpace [ref 7] its implementation of random generation of data points on the sphere.

ff from manifolds.hyperspherespace import HypersphereSpace
I   import numpy as np

  
    class HyperspherePCA(object):
  def __init__(self):
     self.hypersphere_space = HypersphereSpace(equip=True)

  def sample(self, num_samples: int) -> np.array:
     return self.hypersphere_space.sample(num_samples)


    Euclidean PCA

First let's implement the traditional PCA algorithm for 3 dimension (features) data set using scikit-learn library with two methods:
  • euclidean_pca_components to extract the 3 eigenvectors along the 3 eigenvalues
  • euclidean_pca_transform to project the evaluation data onto 3 directions defined by the eigenvectors.
    from sklearn.decomposition import PCA

   
    @staticmethod
    def euclidean_pca_components(data: np.array) -> (np.array, np.array):
   num_components = 3
   pca = PCA(num_components)
   pca.fit(data)
   return (pca.singular_values_, pca.components_)

    @staticmethod
    def euclidean_pca_transform(data: np.array) -> np.array:
   num_components = 3
   pca = PCA(num_components)
 
        return pca.fit_transform(data

We compute the principal components on 256 random 3D data points on the hypersphere.

nu num_samples = 256
pca_hypersphere = HyperspherePCA()

# Random data on Hypersphere
data = pca_hypersphere.sample(num_samples)
eigenvalues, components = HyperspherePCA.euclidean_pca_components(data)
transformed = pca_hypersphere.euclidean_pca_transform(data)        

print(f'\nPrincipal components:\n{components}\nEigen values: {eigenvalues')
print(f'\nTransformed data:\n{transformed}')

Output
Principal components:
[[ 0.63124842 -0.7752775  -0.02168467]
 [ 0.68247094  0.54196625  0.49041411]
 [ 0.36845467  0.32437229 -0.8712197 ]]
Eigen values: [9.84728751 9.0183337  8.65835924]

Important note: The 3 eigenvalues are similar because the input data is random.

    Tangent PCA on hypersphere 

The private method __tangent_pca computes principal components on the tangent plane using the logarithmic map. As detailed in the previous section, this implementation employs the Frechet mean as the base point on the manifold, which is the argument of the Geomstats method fit.

The method tangent_pca_components extracts the principal components computed using the logarithmic map. Finally the method tangent_pca_transform projects input data along the 3 principal components, similarly to its Euclidean counterpart, euclidean_pca_transform .

   from geomstats.learning.pca import TangentPCA
from geomstats.learning.frechet_mean import FrechetMean


def tangent_pca_components(self, data: np.array) -> np.array:
    tgt_pca = self.__tangent_pca(data)
    return tgt_pca.components_

def tangent_pca_transform(self, data: np.array) -> np.array:
    tgt_pca = self.__tangent_pca(data)
    tgt_pca.transform(data)

    
def __tangent_pca(self, data: np.array) -> TangentPCA:
     sphere = self.hypersphere_space.space
     tgt_pca = TangentPCA(sphere)
       
          # We use the Frechet mean as center of data points on the hypersphere
     mean = FrechetMean(sphere, method="default")
     mean.fit(data)
          # Frechet mean estimate
     estimate = mean.estimate_
        
          # Invoke Geomstats fitting method for PCA on tangent plane
          tgt_pca.fit(data, base_point=estimate)
     return tgt_pca

    We evaluate the principal geodesic components on hypersphere using the same 256 points randomly generated on the hypersphere and compared with the components on the Euclidean space.

nu  num_samples = 256
pca_hypersphere = HyperspherePCA()
        
# Random data point generated on Hypersphere
data = pca_hypersphere.sample(num_samples)
        
_, components = HyperspherePCA.euclidean_pca_components(data)
tangent_components = pca_hypersphere.tangent_pca_components(data)
print(f'\nEuclidean PCA components:\n{components}\nTangent Space PCA components:\n{tangent_components}')

Output:
Euclidean PCA components:
[[ 0.63124842 -0.7752775  -0.02168467]
 [ 0.68247094  0.54196625  0.49041411]
 [ 0.36845467  0.32437229 -0.8712197 ]]
Tangent Space PCA components:
[[ 0.8202982  -0.45814582  0.34236423]
 [ 0.45806882  0.16783651 -0.87292832]
 [-0.34246724 -0.87288792 -0.34753831]]

References



--------------------------------------
Patrick Nicolas has over 25 years of experience in software and data engineering, architecture design and end-to-end deployment and support with extensive knowledge in machine learning. 
He has been director of data engineering at Aideo Technologies since 2017 and he is the author of "Scala for Machine Learning", Packt Publishing ISBN 978-1-78712-238-3 and Geometric Learning in Python Newsletter on LinkedIn.

Appendix

Here is the list of published articles related to geometric learning:
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