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Colloquium: Anna Little (University of Utah)

Thursday, February 20
4:00 PM - 5:00 PM
203 TMCB

Title: Constructing Features from Data: Dimension Reduction, Interpretability, and Invariants

Abstract: This talk explores how to construct meaningful features from noisy, high-dimensional data, with a focus on three key objectives: reducing dimensionality, ensuring interpretability, and achieving invariance to nuisance variations. First, we introduce a geometric framework for dimension reduction using a power-weighted path metric, which effectively de-noises high-dimensional data while preserving its intrinsic geometric structure. This framework is particularly useful for analyzing single-cell RNA data and for multi-manifold clustering, and we provide theoretical guarantees for the convergence of the associated graph Laplacian operators. Next, we address the goal of interpretability in the context of linear distance metric learning, presenting a novel convex optimization approach to learn linear maps between metric spaces even in the presence of noise. We also establish sample complexity bounds and propose a method for truncating to low-rank models without compromising accuracy. Finally, we tackle the challenge of constructing features that are invariant to certain group actions, focusing on the multi-reference alignment (MRA) data model. We extend classical MRA to handle both translation and random scale changes, providing a robust framework for signal recovery from noisy observations.

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