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.
Over the past decade or so, an increasing interest in utilizing tools from algebraic topology to extract insights from high dimensional data has given rise to the field of TDA. The successful applications of TDA have spanned a large number of areas, ranging from computer vision @cite_24 to medical imaging @cite_6 , biochemistry @cite_4 , neuroscience @cite_17 and materials science @cite_19 . A predominant tool in TDA is persistent homology, which tracks the evolution of the topological features in a multi-scale manner to avoid information loss @cite_16 @cite_7 . The multi-scale information is summarized by the persistence diagram (PD), a multiset of points in @math that encodes the lifetime (i.e., persistence) of the features.
@cite_4 @cite_7 @cite_6 @cite_24 @cite_19 @cite_16 @cite_17
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Topological data analysis (TDA) is an emerging mathematical concept for characterizing shapes in complex data. In TDA, persistence diagrams are widely recognized as a useful descriptor of data, and can distinguish robust and noisy topological properties. This paper proposes a kernel method on persistence diagrams to develop a statistical framework in TDA. The proposed kernel satisfies the stability property and provides explicit control on the effect of persistence. Furthermore, the method allows a fast approximation technique. The method is applied into practical data on proteins and oxide glasses, and the results show the advantage of our method compared to other relevant methods on persistence diagrams. Persistent homology probes topological properties from point clouds and functions. By looking at multiple scales simultaneously, one can record the births and deaths of topological features as the scale varies. In this paper we use a statistical technique, the empirical bootstrap, to separate topological signal from topological noise. In particular, we derive condence sets for persistence diagrams and condence bands for persistence landscapes. Many data sets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a data set. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs. This work introduces a new dataset and framework for the exploration of topological data analysis (TDA) techniques applied to time-series data. We examine the end-toend TDA processing pipeline for persistent homology applied to time-delay embeddings of time series – embeddings that capture the underlying system dynamics from which time series data is acquired. In particular, we consider stability with respect to time series length, the approximation accuracy of sparse filtration methods, and the discriminating ability of persistence diagrams as a feature for learning. We explore these properties across a wide range of time-series datasets spanning multiple domains for single source multi-segment signals as well as multi-source single segment signals. Our analysis and dataset captures the entire TDA processing pipeline and includes time-delay embeddings, persistence diagrams, topological distance measures, as well as kernels for similarity learning and classification tasks for a broad set of time-series data sources. We outline the TDA framework and rationale behind the dataset and provide insights into the role of TDA for time-series analysis as well as opportunities for new work.