arXiv subjects

Piotr Frydrych

Publications and source records attributed to Piotr Frydrych.

5 recordsLinked to original sources

arXiv2026

The Preisach Extremum Stack is a Shannon-Minimal Sufficient Statistic for Rate-Independent Functionals

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.

cs.IT↗

arXiv2026

Worker Utility as Hysteresis: A Preisach Model of Transaction Acceptance in Gig Labour Markets

Worker utility is not observed -- only its consequence is. Each gig transaction produces a single bit: accepted or rejected. We argue this structure points directly to the Preisach hysteresis model as the natural representation of latent worker preferences. The Preisach operator models aggregate output as an integral over a population of binary threshold elements -- precisely the structure that emerges when heterogeneous workers each carry a private acceptance wage. We estimate two latent utility surfaces: acceptance utility U_1(X) and rejection utility U_0(X), via a dual-output neural network (shared layers 256->128, margin loss enforcing U_1 >= U_0). Classification reduces to the Preisach gap U_1(X) - U_0(X), passed into an XGBoost classifier alongside clip-stabilised price-to-threshold encodings. On 36,891 gig transactions, this pipeline achieves Jaccard = 0.827 and ROC AUC = 0.799. The price-to-threshold encoding accounts for +11.0 pp AUC over raw utility features. The model confirms the directional asymmetry hysteresis predicts: price decreases depress completion rates more than equivalent increases raise them. Applied to the full dataset, the model's recommendations simultaneously reduce the total wage bill by 21.3% and increase expected fill rate by 9.7 pp. For 74.2% of transactions, P(accept) already exceeds 0.80; reducing the wage keeps it above threshold (mean post-cut P = 0.972), releasing cost savings (median 31%). For the remaining 25.4%, a median 7% wage increase recovers +43 pp acceptance. A model without an explicit indifference zone cannot execute both moves simultaneously.

cs.LG↗

arXiv2026

The Extremum Stack as Optimal Memory for Rate-Independent Sequence Models: Information-Theoretic Foundations and Online Complexity

Rate-independent sequence models respond to the ordered structure of input extrema rather than to absolute timing or token position. This principle underlies Preisach Attention, a hysteretic alternative to softmax attention in which sequence history is represented by a stack of alternating extrema generated by the classical wiping-out rule. This paper establishes the information-theoretic and online algorithmic foundations of that representation. We prove that the Preisach extremum stack is not merely a convenient implementation detail, but the complete invariant of computable rate-independent sequence functionals: a functional is rate-independent if and only if it factors through the stack. We then show that the stack is minimal in two complementary senses. In the Kolmogorov setting, the shortest exact representation answering all rate-independent queries has complexity equal to that of the stack up to an additive constant independent of sequence length and stack depth. In the Shannon setting, under any input distribution, every sufficient representation contains at least as much mutual information about the input as the stack, with equality only for representations informationally equivalent to it. Finally, we analyse the online maintenance cost of this minimal state. Although the standard stack update is amortised constant time, adversarial inputs can induce linear worst-case latency. We prove a matching output-change lower bound, show that binary search reduces boundary detection but not deletion, and give an exact finger-tree implementation with worst-case (O(\log k)) update time, where (k) is the current stack depth. The results provide a principled foundation for replacing full sequence histories or KV-cache-like memories by extremum-stack states in rate-independent neural architectures, without approximation and with bounded online latency.

cs.DS↗

arXiv2026

Preisach Attention: A Hysteretic Model of Sequential Memory

We introduce the Preisach Attention Layer (PAL), a novel sequence modelling architecture grounded in the classical Preisach hysteresis operator from mathematical physics. PAL replaces the softmax attention mechanism with a binary relay operator parameterised by learned activation and deactivation thresholds, maintaining a stack of local extrema as its internal state. A single-layer PAL-Transformer with O(1) depth is Turing-complete under arbitrary precision arithmetic, achievable through simulation of a two-stack pushdown automaton -- in contrast to the O(log n) depth required by standard hard-attention transformers. Second, we prove that the function classes computable by PAL and by the transformer are incomparable: PAL computes historical range statistics in O(1) layers that require O(log n) layers for transformers, while transformers support random-access retrieval that PAL cannot perform without auxiliary state. The separating property is rate-independence -- PAL responds only to the sequence of local extrema, not to absolute token positions or temporal spacing. Third, we show that the extremum stack constitutes a minimal sufficient statistic of the input history for all rate-independent functionals, providing a formal analogue of the wiping property in classical hysteresis theory. PAL is thus an efficient architecture for tasks with long episodic memory and weak positional dependence, with O(n log n) total inference cost versus O(n^2) for standard attention.

cs.LG↗

arXiv2026

The Extremum Stack is a Minimal Sufficient Statistic for Rate-Independent Functionals: A Kolmogorov Complexity Characterisation

We prove that the extremum stack of a discrete sequence is a minimal sufficient statistic for the class of all computable, causal, rate-independent functionals, in the sense of Kolmogorov complexity. Specifically, we establish K(Pi_n) - O(1) <= K_R(u_{0:n}) <= K(Pi_n) + O(1), where K_R(u_{0:n}) is the length of the shortest program answering every query in the class R, and the O(1) overhead is independent of both the sequence length n and the stack depth k. Sufficiency follows from the classical wiping property of the Preisach hysteresis operator. Minimality is established via a finite indicator family whose rate-independence is verified explicitly. Any compression of a hysteresis-driven stream that preserves the full class R must therefore retain at least K(Pi_n) - O(1) bits; the stack-based compression algorithm implied by the result carries a Kolmogorov optimality guarantee that none of the standard time-series compression methods provide.

cs.IT↗