Speaker
Description
Abstract: The shape of complex data -- time series, functional observations, and multimodal measurements -- encodes structural information that classical statistical methods often fail to capture. Topological data analysis addresses this through persistent homology, which tracks the evolution of topological features across a filtration of the data. However, the outcome of any topological analysis depends on the choice of filtration. We present a framework for filtration learning in Exact Multi-Parameter Persistent Homology (EMPH), a method developed specifically for topological time-series analysis. Unlike conventional sliding window approaches, EMPH exploits the geometry of the Liouville torus via Fourier decomposition to derive an exact closed-form formula for the rank invariant, substantially reducing computational complexity while preserving full topological fidelity. Building on this, we formulate filtration selection as an optimization problem and propose an algorithm that learns the filtration minimizing a topological loss -- enabling data-adaptive, statistically grounded topology extraction for complex data. We demonstrate the framework on classification problems for time-series data, where the learned filtration in the multi-parameter persistence space yields improved discriminative performance over fixed filtrations. We further discuss extensions to topological regularization in generative modeling and multimodal learning, where persistence diagrams and persistence images serve as structural targets that constrain the learning objective. These results establish EMPH-based filtration learning as a flexible statistical tool for complex data analysis, providing interpretable topological summaries that are both computationally efficient and optimizable within learning frameworks.