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

This paper proves that the extremum stack serves as a minimal sufficient statistic for all computable, causal, rate-independent functionals by demonstrating that its Kolmogorov complexity is asymptotically equivalent to the shortest program capable of answering any query within this class, thereby establishing a theoretical optimality for stack-based compression of hysteresis-driven streams.

Original authors: Piotr Frydrych

Published 2026-05-20

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Piotr Frydrych

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: The "Memory Filter"

Imagine you are watching a rollercoaster ride. The ride goes up and down, fast and slow. Sometimes it zooms; sometimes it crawls.

Now, imagine you have a special camera that only cares about where the ride turns around (the highest peaks and the lowest valleys). It doesn't care how long it took to get there, or how fast the coaster was moving between the peaks. It only remembers the sequence of "Highs" and "Lows."

The paper calls this special memory the "Extremum Stack."

The author, Piotr Frydrych, proves a very specific and powerful thing about this memory: It is the absolute smallest, most efficient way to remember everything that matters for a specific type of problem.

The Problem: "Rate-Independence"

In the real world, many systems (like magnetic materials, rubber bands, or certain financial models) behave in a way called "rate-independent."

The Analogy: Think of a heavy door with a spring. If you push it open slowly or slam it open quickly, the door ends up in the same spot. The speed of your push doesn't change the result, only the direction and distance you pushed matter.
The Paper's Claim: For any system that works this way, the only thing that actually matters is the list of peaks and valleys (the Extremum Stack). The rest of the data (the speed, the exact timing, the tiny wiggles in between) is just noise.

The Discovery: The "Goldilocks" Memory

The paper asks a question: "Can we compress this data even further? Is there a way to remember less than the list of peaks and valleys?"

The answer is No.

The author uses a mathematical tool called Kolmogorov Complexity (which is basically a way to measure how much information is truly needed to describe something) to prove two things:

It's Enough (Sufficiency): If you have the list of peaks and valleys, you can predict the future behavior of any "rate-independent" system perfectly. You don't need the full history of the rollercoaster ride; the list of turns is enough.
It's Necessary (Minimality): You cannot throw away any part of that list. If you delete even one peak or valley from your memory, you will lose the ability to predict the system correctly.

The Metaphor:
Imagine you are packing for a trip.

The Full Data: You pack your entire house, including every sock, every book, and every dust bunny.
The Extremum Stack: You pack only the essentials: your passport, a toothbrush, and a change of clothes.
The Paper's Proof: The author proves that for "rate-independent" systems, the "essentials" pack is the smallest possible pack that still lets you survive. You can't pack less than that without getting lost.

Why This Matters (According to the Paper)

The paper claims that previous methods of compressing this data were slightly inefficient. They thought you needed a little bit of extra "overhead" (extra space) to make the math work, perhaps growing as the data got longer.

This paper proves that the overhead is actually constant. It's like saying:

"Whether you are packing for a 1-day trip or a 100-year trip, the extra space you need for the 'Essentials Pack' is always just the size of a single coin."

This makes the "Extremum Stack" the perfectly optimal way to store this kind of data.

The "Indicator" Test

To prove that you can't throw away any data, the author created a "test" using a family of simple questions (called an "indicator family").

The Test: Imagine asking, "Did the rollercoaster ever go above 50 feet and then drop below 10 feet?"
The Result: The paper shows that if you don't have the full list of peaks and valleys, you cannot answer all possible versions of this question correctly. If you miss one piece of the stack, you might get the answer wrong for a specific scenario. Therefore, the whole stack is required.

Summary

What is it? A mathematical proof that the "list of peaks and valleys" (Extremum Stack) is the smallest possible memory needed to understand systems that ignore speed and timing.
The Analogy: It's the "Essentials Pack" for data. You can't pack less without losing the ability to function.
The Result: This method is mathematically proven to be the most efficient way to compress this specific type of data, with no wasted space.

Note: The paper focuses strictly on the mathematical proof of this efficiency. It mentions that this applies to things like magnetic materials and financial models, but it does not claim to solve specific medical or engineering problems in this text; it only proves the data structure is optimal.

Technical Summary: The Extremum Stack as a Minimal Sufficient Statistic for Rate-Independent Functionals

Problem Statement
The paper addresses the fundamental question of data compression for sequences governed by rate-independent functionals. A functional $F$ is rate-independent if its output depends solely on the sequence of local extrema of the input, rather than on the temporal spacing, speed of change, or absolute values at non-extremal points. This property is central to modeling phenomena such as ferromagnetic hysteresis, elastoplastic systems, and financial threshold models.

While it is established that the extremum stack ( $\Pi_n$ )—an alternating sequence of local maxima and minima maintained in a specific order—is a sufficient statistic for Preisach functionals, it remained an open question whether this representation is minimal. Specifically, is there a strictly shorter representation that can answer all queries for the class $\mathcal{R}$ of computable, causal, rate-independent functionals? Furthermore, previous informal claims suggested the overhead of using the stack was $O(\log n)$ or $O(\log K(\Pi_n))$ , but the precise Kolmogorov complexity bounds had not been rigorously established.

Methodology
The author employs Kolmogorov complexity theory to quantify the information content required to represent a sequence $u_{0:n}$ such that any functional in the class $\mathcal{R}$ can be computed.

Definitions: The paper defines $K_R(u_{0:n})$ as the length of the shortest program that, given any $F \in \mathcal{R}$ , computes $F[u](n)$ . The extremum stack $\Pi_n$ is defined as a sequence of pairs $(M_i, m_i)$ representing surviving local extrema, updated via a "wiping" mechanism where new extrema erase smaller, older ones.
Sufficiency Proof: The upper bound is derived from the classical wiping property of the Preisach hysteresis operator. Since the output of any rate-independent functional depends only on $\Pi_n$ , a fixed computable function can map $\Pi_n$ to the functional output.
Minimality Proof: To prove that no shorter representation exists, the author constructs a finite indicator family of functionals. For every possible pair of grid values $(M, m)$ , an indicator functional $F_{(M,m)}$ is defined which returns 1 if $(M, m)$ is present in the stack $\Pi_n$ and 0 otherwise. The paper demonstrates that any representation capable of answering all queries in $\mathcal{R}$ must, by necessity, allow for the reconstruction of the full stack $\Pi_n$ via these indicators.

Key Contributions

Minimal Sufficiency: The paper proves that the extremum stack $\Pi_n$ is a minimal sufficient statistic for the class $\mathcal{R}$ . It establishes that any lossless compression preserving the ability to answer all rate-independent queries must retain at least $K(\Pi_n) - O(1)$ bits.
Tight Complexity Bounds: The authors establish the precise Kolmogorov complexity relationship:
$K(\Pi_n) - O(1) \le K_R(u_{0:n}) \le K(\Pi_n) + O(1)$
This corrects and tightens previous informal bounds of $O(\log n)$ or $O(\log K(\Pi_n))$ . The $O(1)$ overhead is shown to be independent of the sequence length $n$ and the stack depth $k$ .
Theoretical Justification for Compression: The result provides a formal optimality guarantee for stack-based compression algorithms (referred to as PSTACK). Unlike standard time-series compression methods (e.g., PAA, SAX, PLR) which offer heuristic or $L_p$ -approximation guarantees, the stack-based approach is asymptotically optimal for the specific class of rate-independent functionals.

Results

Upper Bound: The length of the shortest program answering all $\mathcal{R}$ -queries is bounded above by $K(\Pi_n) + O(1)$ . This follows from the prefix-free invariance theorem, as the mapping from $\Pi_n$ to any $F[u](n)$ is a fixed computable function.
Lower Bound: The length is bounded below by $K(\Pi_n) - O(1)$ . This is proven by showing that the optimal representation $R(u_{0:n})$ allows for the reconstruction of $\Pi_n$ via a computable function $\Phi$ applied to the indicator family. Since $\Pi_n$ can be recovered from the representation, the representation cannot be significantly shorter than the complexity of the stack itself.
Algorithmic Efficiency: The paper notes that the stack can be maintained in amortized $O(1)$ time per step with $O(k)$ space, where $k$ is the current stack depth.

Significance and Claims
The paper claims that the extremum stack is not merely a convenient data structure but is information-theoretically necessary for rate-independent processing.

Optimality: The stack-based representation is asymptotically optimal. Any attempt to compress a hysteresis-driven stream further while preserving the full class of rate-independent queries would result in information loss.
Independence from Sequence Length: The $O(1)$ overhead implies that for signals with long monotone runs (small $k$ relative to $n$ ), the compression ratio $n/k$ can be arbitrarily large without increasing the optimality gap.
Scope: The results apply specifically to discrete sequences over a finite grid $G_L$ . The paper explicitly notes that over the continuous reals $\mathbb{R}$ , the indicator family would be uncountable, and the Kolmogorov framework would not apply directly.

The work concludes that the "wiping" mechanism inherent in the Preisach operator is not just a physical or mathematical curiosity but represents the fundamental limit of information retention for rate-independent systems.