Topological Clustering on Riemannian Manifold

Abstract

Clustering is one of the essential tools in machine learning used for automatically detecting the relevant groups in unlabelled data sets. This thesis focuses on comparative study and experimentation of unsupervised clustering of data sets on Riemannian manifold, particularly Symmetric Positive Definite (SPD) matrix manifold: i.e., where each data point is defined as SPD matrix. SPD matrices appear in various applications such as computer vision (CV),information retrieval, machine learning, and pattern recognition. 2-D motion segmentation from consecutive frames of a video sequence is one of the applications of the visual recognition task. The representation of images by covariance features leverages the inherent manifold structure of SPD matrices that leads to enhanced performances in various visual recognition tasks. Change in the intensity of the pixels in an image is an essential feature for object detection and recognition. The ultimate goal is to detect the object’s shape by analyzing the object’s motion over time because motion can reveal the object’s shape. We tried to achieve it in two different ways. Firstly, In Euclidean space, we started studying unsupervised clustering techniques such as k-means, Locally Linear Embedding (LLE), Laplacian Eigenmaps(LE), and Hessian Eigen maps (HLLE), followed by their extensions in Riemannian manifold. We tried to construct the SPD matrices data set considering different image features to achieve better results using the existing techniques. In the second approach, we tried to incorporate topology-based clustering methods to consider the topology of the underlying manifold in the data set. We studied the clustering concepts powered by Topological Data Analysis in the ToMATo (Topological Mode Analysis Tool) algorithm, a persistence-based clustering algorithm in Riemannian manifold. We used the distance matrix computed using AIRM (Affine Invariant Riemannian Metric) in the ToMATo clustering algorithm to achieve better results. The effectiveness of the proposed changes in the existing clustering techniques has been discussed comparatively after experimenting with them on real-world data set