# Sparse-TDA: Sparse Realization of Topological Data Analysis for Multi-Way Classification

## TL;DR

Sparse-TDA introduces a novel method combining topological data analysis with sparse sampling, efficiently capturing shape features for multi-way classification tasks in high-dimensional data.

## Contribution

It presents a new algorithm that selects sparse samples from persistent topological features using QR pivoting, enhancing classification performance.

## Key findings

- Effective on human posture recognition
- Improves image texture classification
- Demonstrates promising results on benchmark datasets

## Abstract

Topological data analysis (TDA) has emerged as one of the most promising techniques to reconstruct the unknown shapes of high-dimensional spaces from observed data samples. TDA, thus, yields key shape descriptors in the form of persistent topological features that can be used for any supervised or unsupervised learning task, including multi-way classification. Sparse sampling, on the other hand, provides a highly efficient technique to reconstruct signals in the spatial-temporal domain from just a few carefully-chosen samples. Here, we present a new method, referred to as the Sparse-TDA algorithm, that combines favorable aspects of the two techniques. This combination is realized by selecting an optimal set of sparse pixel samples from the persistent features generated by a vector-based TDA algorithm. These sparse samples are selected from a low-rank matrix representation of persistent features using QR pivoting. We show that the Sparse-TDA method demonstrates promising performance on three benchmark problems related to human posture recognition and image texture classification.