K-means on Riemannian Manifolds

Traditional clustering models often fail complex datasets commonly found in advanced applications like medical imaging, 3D shape analysis, and natural language processing where data is highly interrelated.

K-means on manifolds respects the intrinsic geometry of the data, such as curvature and metric.

Introduction
K-means
Clustering data on manifold
     Randomly generated manifold data
     Synthetic clusters with SO(3)
Implementation
     Setup
     Euclidean space
     Hypersphere
References
Appendix        

What you will learn:?How to apply k-means clustering on a Riemann manifold (Hypersphere) using Geomstats, contrasted with its implementation in Euclidean space, using scikit-learn library.

A full article, featuring design principles, detailed implementation, in-depth analysis, and exercises, is available in Insights into k-Means on Riemannian Manifolds

Notes:?

• Environments:?Python ?3.10.10, Geomstats 2.7.0, Scikit-learn 1.4.2,?Matplotlib 3.8.3
• The implementation relies on Geomstats, an open-source, object-oriented library following Scikit-Learn’s API conventions to gain hands-on experience with Geometric Learning. It is described in article Introduction to Geomstats for Geometric Learning
• 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

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. Riemannian geometry enables to describe the geometric structures of manifolds equipped with a metric, which defines the concept of distance and angle on these spaces.

This article is the seventh part of our ongoing series focused on geometric learning. In this installment, we utilize the Geomstats Python library [ref.?5] and explore the ubiquitous K-means clustering algorithm on the hypersphere manifold. The hypersphere which was introduced in a previous piece,?Geometric Learning in Python: Manifolds - Hypersphere ?and is detailed in the Geomstats API [ref.?6].?

I highly recommend watching the comprehensive series of 22 YouTube videos?Tensor Calculus - Eigenchris??to familiarize yourself with fundamental concepts of differential geometry.?

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

There are many benefits for clustering data on a manifold for complex data sets [ref?7]:

• Grouping of dense, continuous?non-linear?data depends on the 'shape' of data
• Projection to Euclidean space may introduce distortion
• Loss, distances are better assessed and computed through geodesics than Euclidean metrics (i.e. sphere).

K-means

Among the array of unsupervised learning algorithms, K-means stands out as one of the most well-known. This algorithm has a straightforward goal: to divide the data space so that data points within the same cluster are as similar as possible (intra-cluster similarity), and data points in different clusters are as dissimilar as possible (inter-cluster similarity). K-means aims to identify a predetermined number of clusters in an unlabeled dataset. It employs an iterative approach to finalize the clustering, which depends on the number of clusters specified by the user (denoted by the variable K).

Given?K,?Ck?clusters each with a centroid?mk,?the input data?xi?is distributed across the cluster so to minimize the reconstruction error:

Clustering data on a manifold

To assess and contrast the k-means model in both Euclidean space and on a hypersphere, it's necessary to create clustered data. This involves a two-step process:

1. Generate a template cluster by employing a?random generator?on the manifold.
2. Generate 4 clusters from the template using a?special orthogonal Lie group?in 3-dimensional space,?SO(3).

Randomly generated manifold data

Let's evaluate and compare the following random generators for data points on the hypersphere we introduced in a previous article [ref?8]

• Uniform distribution
• Uniform distribution with constraints
• von Mises-Fisher distribution

Uniform distribution

We start with the basic random uniform generator over interval [0, 1].

The data points for the 4 clusters are visualized in the following plot.

Constrained uniform random generator

In this scenario, we constrain random values?r?on each of the 3 dimension (or axis) within a sub-interval?[ai, bi].

von mises-Fisher random generator

This approach relies on a generative mixture-model approach to clustering directional data based on the von Mises-Fisher distribution [ref?9].

Given a?d-dimensional unit random vector?x?on a hypersphere of dimension d-1, the d-variate von Moses-Fisher distribution is defined by the following probability density distribution:

Id?is the Bessel function and?Cd?is the normalization factor.

As anticipated, using a pure uniform random generator distributes data evenly across the hypersphere, rendering it ineffective for evaluating KMeans.

Instead, we will employ the von Mises-Fisher distribution and a constrained uniform random generator to more effectively analyze the performance of KMeans on a Riemann manifold.

Synthetic clusters using SO(3)

We leverage the SO(3) Lie group to replicate the randomly generated cluster.

Although the discussion of Lie groups and special orthogonal group in 3-dimensional space is beyond the scope of this article, here is a short summary:

In differential geometry, Lie groups play a crucial role by connecting the concepts of algebra and geometry. A Lie group is a mathematical structure that is both a group and a differentiable manifold. This means that the group operations of multiplication and taking inverses are smooth (differentiable), and it allows the application of calculus within the group structure.

The?Special Orthogonal Lie group?in 3-dimension space SO(3) is simply a group of 3 x 3 orthogonal matrices with determinant = 1.?These represent rotation in n-dimensional space and form a compact Lie group.

Implementation

Setup?

Let's wraps the random generators and k-means training methods in a class,?KMeansOnManifold.

The von Mises-Fisher generator for the data in the initial cluster is initialized with a mean,?_mu?and a?kappa?arbitrary value. The constrained uniform random generator, accepts random values in each dimension x:?[-1, -0.35], y:?[0.3, 1]?and z:?[-1. 0.4].

The SO(3) Lie group is initialized without metric?(equip=False) to generate the 4 synthetic clusters.

Data in Euclidean space

The data class,?KMeansCluster?encapsulates the output (centroid and label) of the training of the k-means algorithm on the synthetic clustered data.

We rely k-means implementation in?scikit-learn?library [ref?10] (class?KMeans)?on the ?to identify the clusters in the Euclidean space, selecting the?elkan?algorithm, and?k-means++?initialization.

Output:

Cluster Center: [ 0.56035023 -0.4030522   0.70054776], Label: 0
Cluster Center: [-0.1997325  -0.38496744  0.8826764 ], Label: 2
Cluster Center: [0.04443849 0.86749237 0.46118632], Label: 3
Cluster Center: [-0.83876485 -0.45621187  0.23570083], Label: 1        

Clearly, this implementation of k-means was not able to identify the proper clusters

Data on hypersphere

The methods to?train k-means on the hypersphere uses the same semantic as its sklearn counterpart. It leverage the?Geomstats,?RiemannianKMeans?class method.

Similar to k-means in Euclidean space, we identify the centroids for 4 clusters using 500 randomly generated samples.

Output:

500 random samples on 4 clusters with von-mises-Fisher distribution
Cluster Center: [ 0.17772496 -0.36363422  0.91443097], Label: 2
Cluster Center: [ 0.44403679  0.06735507 -0.89347335], Label: 0
Cluster Center: [ 0.85407911 -0.50905801  0.10681211], Label: 3
Cluster Center: [ 0.90899637  0.02635062 -0.41597025], Label: 1

500 random samples on 4 clusters with constrained uniform distribution
Cluster Center: [-0.05344069 -0.91613807  0.3972847 ], Label: 1
Cluster Center: [ 0.6796575   0.39400079 -0.61873181], Label: 2
Cluster Center: [ 0.51799972 -0.67116261 -0.530299 ], Label: 0
Cluster Center: [ 0.49290501 -0.45790221 -0.73984473], Label: 3        

Note: The labels are arbitrary indices assigned to each cluster for the purpose of visualization and validation against true labels.

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References

[1]?Tensor Calculus - Eigenchris

[2]?Foundation of Geometric Learning [3]?Differentiable Manifolds for Geometric Learning [4]?Vector and Covector fields in Python

[5]?Introduction to Geometric Learning in Python with Geomstats

[6]?Geomstats API

[7]?Clustering Data That Resides on a Low-Dimensional Manifold in a High-Dimensional Measurement Space

[8]?Geomstats: Hypersphere

[9]?Clustering on the Unit Hypersphere using von Mises-Fisher Distributions

[10]?scikit-learn.org

[11] Insights into k-Means on Riemannian Manifolds

For comprehensive topics on geometric learning, including reviews and exercises, subscribe to Hands-on Geometric Deep Learning

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 Hands-on Geometric Deep Learning newsletter.

Appendix

List of articles related to Geometric Learning on this newsletter:

• Geometric Learning in Python: Basics introduces differential geometry as an applied to machine learning and its basic components.
• Differentiable Manifolds describes manifold?components such as?tangent vectors, geodesics with implementation in Python for Hypersphere using the?Geomstats?library.
• Intrinsic Representation Reviews the various coordinates system using extrinsic and intrinsic representation.
• Vector and Covector Fields describes vector and co-vector fields with Python implementation in 2 and 3-dimension spaces.
• Vector Operators in Python illustrates the differential operators, gradient, divergence, curl and laplacian using?SymPy?library.
• Non-linear Functional Data Analysis describes the key elements of non-linear functional data analysis to analysis?curves, images, or functions in very ?high-dimensional spaces

• Riemann Metric & Connection for Geometric Learning reviews the key components of the Levi-Civita connection and its implementation in Python using Geomstats library.
• Riemann Curvature for Geometric Learning in Python Introduces the concept of curvature tensor in the context of the Levi-Civita connection.

#KMeans #UnsupervisedLearning #RiemannianManifold #GeometricLearning #Geomstats #Clustering