Directional Statistics in Signal Processing: A Geometry Driven Overview of Models, Learning, and Applications

Many signal-processing objects are defined by periodicity, normalization, sign or global-phase ambiguity, basis invariance, or positive-semidefinite constraints and therefore cannot be modeled faithfully in ordinary Euclidean space. This paper develops a geometry driven framework for directional statistics in signal processing, connecting signal representations to their natural manifolds or quotient spaces and to probability models compatible with their invariances. After reviewing instances of directional statistics in the signal processing literature, we review circular, toroidal, spherical, projective, Stiefel, Grassmann, and covariance-valued distributions, emphasizing their statistical assumptions, computational requirements, and dimensional limitations relevant to signal processing. We describe various modeling and estimation strategies, including analytical and gradient-based maximum likelihood, structured low-rank parameterizations, mixtures and hidden Markov models (HMMs), score-based and contrastive estimation, and geometry-aware K-means. To exemplify the consequences of geometry-aware modeling, we study multivariate phase coherence in synthetic data and functional neuroimaging, comparing toroidal phase vectors to projective, Grassmann, and positive-semidefinite representations. We also study speaker separation in a car environment, showing that directional representations outperform a standard Euclidean model. These examples demonstrate that when the geometric representation, statistical distribution, inference method, and computational regime are considered together, they have the potential to lead to improved models across a broad range of signal processing applications.