The Preisach Extremum Stack is a Shannon-Minimal Sufficient Statistic for Rate-Independent Functionals
2026-06-03 • Information Theory
Information TheoryData Structures and Algorithms
AI summaryⓘ
The authors study a specific class of functions, called computable, causal, rate-independent functionals, which means their outputs don’t change if you speed up or slow down the input in a smooth way. They show that every such function’s output depends only on a special structure called the Preisach extremum stack, which summarizes the input sequence. They also prove that this stack contains the minimal necessary information to predict these functions’ outputs, and maintaining it step-by-step is enough to estimate related characteristics efficiently. Their work links concepts from computation, information theory, and estimation methods.
computable functionalscausal functionalsrate-independencePreisach extremum stackinformation theoryShannon minimal sufficient statistictime reparametrizationNNLS estimatorPreisach measureincremental stack process
Authors
Piotr Frydrych
Abstract
Let R denote the class of all computable, causal functionals that are rate-independent in the classical sense (invariant under monotone time reparametrizations), and let Pi_n be the Preisach extremum stack of an input sequence u_{0:n}. We prove a characterization theorem establishing that every F in R satisfies Fu = f(Pi_n) for a computable f, and derive two information-theoretic results. First, under any probability measure on u_{0:n}, the equality I(u_{0:n}; Fu) = I(Pi_n; Fu) holds for every F in R and is an immediate corollary of the characterization theorem. Second, the main result: Pi_n is a Shannon-minimal sufficient statistic in the sense that I(u_{0:n}; Pi_n) <= I(u_{0:n}; S) for every random variable S from which all R-queries are computable. The proof uses the finite indicator family of [Frydrych, 2026] to reconstruct Pi_n from any sufficient S. As a corollary, online maintenance of Pi_n suffices for rate-independent estimation: the NNLS estimator of the Preisach measure mu can be assembled from the incremental stack process (Pi_t)_{t=0}^n in O(k * L^2) memory per step, where k = |Pi_t| and L is the grid resolution.