Transform-Invariant Non-Parametric Clustering of Covariance Matrices and its Application to Unsupervised Joint Segmentation and Action Discovery

TL;DR

This paper introduces a transform-invariant clustering method for covariance matrices and applies it to unsupervised segmentation and action discovery in complex sequential data, demonstrating effectiveness on real-world datasets and human activity tasks.

Contribution

The paper presents the Spectral Polytope Covariance Matrix similarity, a non-parametric clustering model, and couples it with Bayesian models for joint segmentation and action discovery in time-series data.

Findings

Effective clustering of covariance matrices invariant to transformations

Successful unsupervised segmentation of complex sequential tasks

Validated on real-world datasets and human activity demonstrations

Abstract

In this work, we tackle the problem of transform-invariant unsupervised learning in the space of Covariance matrices and applications thereof. We begin by introducing the Spectral Polytope Covariance Matrix (SPCM) Similarity function; a similarity function for Covariance matrices, invariant to any type of transformation. We then derive the SPCM-CRP mixture model, a transform-invariant non-parametric clustering approach for Covariance matrices that leverages the proposed similarity function, spectral embedding and the distance-dependent Chinese Restaurant Process (dd-CRP) (Blei and Frazier, 2011). The scalability and applicability of these two contributions is extensively validated on real-world Covariance matrix datasets from diverse research fields. Finally, we couple the SPCM-CRP mixture model with the Bayesian non-parametric Indian Buffet Process (IBP) - Hidden Markov Model (HMM) (Fox et al., 2009), to jointly segment and discover transform-invariant action primitives from complex sequential data. Resulting in a topic-modeling inspired hierarchical model for unsupervised time-series data analysis which we call ICSC-HMM (IBP Coupled SPCM-CRP Hidden Markov Model). The ICSC-HMM is validated on kinesthetic demonstrations of uni-manual and bi-manual cooking tasks; achieving unsupervised human-level decomposition of complex sequential tasks.