  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.
