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arxiv: 2506.18250 · v2 · submitted 2025-06-23 · 🧮 math.DS

Filtrations Indexed by Attracting Levels and their Applications

Yusuke Imoto , Tomoo Yokoyama This is my paper

Pith reviewed 2026-05-19 08:30 UTC · model grok-4.3

classification 🧮 math.DS
keywords filtrationsattracting levelspersistent homologytopological data analysisdynamical systemspartial mapscost functionstropical cyclones
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The pith

Filtrations indexed by attracting levels quantify forward sensitivity of trajectories to attractors and backward perturbation thresholds where attraction breaks down.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a new class of filtrations indexed by attracting levels in dynamical systems. These filtrations measure trajectory sensitivity to attractors under perturbations in a forward sense and the size of perturbations that cause attraction to fail in a backward sense. The construction works for maps on metric spaces and extends to general partial maps equipped with cost functions. It produces complementary filtrations that induce natural decompositions of the space according to whether terminal states are good or bad. The framework supplies fresh inputs for persistent homology computations in topological data analysis, as shown in an application to ensemble forecasts of tropical cyclones.

Core claim

We introduce a new class of filtrations indexed by attracting levels in dynamical systems, providing novel inputs for persistent homology and related methods in topological data analysis. These filtrations quantify, in a forward direction, the sensitivity of trajectories with respect to attractors under perturbations and, in a backward direction, the perturbation magnitude at which attraction breaks down. The construction applies not only to maps on metric spaces but also to general partial maps with cost functions, yielding a filtration-theoretic framework with connections to algebraic topology. This generality ensures complementary filtrations when terminal states are good or bad, inducing

What carries the argument

The filtration indexed by attracting levels, which assigns values that track forward sensitivity of trajectories toward attractors and backward thresholds where attraction fails, for maps on metric spaces and general partial maps with cost functions.

If this is right

  • The filtrations supply new inputs that can be fed directly into persistent homology pipelines for dynamical systems.
  • Complementary good-state and bad-state filtrations produce a natural decomposition of the underlying space.
  • The same construction identifies regions of heightened sensitivity in applications such as ensemble forecasts of tropical cyclones.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The cost-function formulation opens the possibility of treating optimization trajectories as dynamical systems and applying the filtrations to detect early sensitivity in gradient-based methods.
  • Because the backward direction measures breakdown thresholds, the filtrations could be combined with existing stability criteria to produce joint topological-stability diagrams for attractors.
  • The framework's generality suggests testing on discrete dynamical systems arising in cellular automata or agent-based models where partial maps appear naturally.

Load-bearing premise

That filtrations indexed by attracting levels can be rigorously defined for general partial maps with cost functions and that they induce natural decompositions of the space when terminal states are good or bad.

What would settle it

Construct the filtration explicitly on a low-dimensional system such as the logistic map with a chosen cost function, compute the resulting persistent homology, and check whether the decomposition into good and bad terminal regions matches the predicted attracting-level structure; inconsistent or undefined behavior on this example would refute the central construction.

Figures

Figures reproduced from arXiv: 2506.18250 by Tomoo Yokoyama, Yusuke Imoto.

Figure 1
Figure 1. Figure 1: An ε-controlled path of length n from the initial state y to the terminal state F(yn−1). . . . x = x0 f(x0) x1 f(xn−2) xn−1 xn f(xn−1) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An ε-chain of length n from a point x to a point xn. In the previous definition, the concept of ε-controlled path is similar to one of the ε-chain. However, it is defined differently from the following perspective of control: To ensure that the current state does not become “bad” even instantaneously during control, the control is applied before the partial map, rather than after. This corresponds to apply… view at source ↗
Figure 3
Figure 3. Figure 3: Ensemble forecast dataset for tropical cyclone Dol￾phin. a, Best-track trajectory (real orbit, blue line) overlaid with all forecast center positions (gray dots) of Dolphin. b, Selected ensemble forecast tracks at representative initial times: each panel shows 21 ensemble members (colored lines), plotted against the background of all forecasts (gray dots). The left top times corre￾spond to the initial time… view at source ↗
Figure 4
Figure 4. Figure 4: Numerical experiment using one ensemble weather forecast dataset. a, Ensemble forecast tracks of the tropical cy￾clone initialized at 00:00 UTC on September 24, 2020, showing a wide east–west spread of possible trajectories. b, Partition of final￾time ensemble points into the good cluster G (eastward tracks, red circles) and the bad cluster B (northward tracks, blue circles). c–d, ε- and εΣ-attracting basi… view at source ↗
Figure 5
Figure 5. Figure 5: Numerical experiment using the full ensemble weather forecast dataset. a, K-means clustering of all the forecast center positions into 10 clusters. Clusters 6 and 9 were assigned as good cluster G and bad cluster B, respectively. b, Distribution of ε (top) and εΣ (bottom) for clusters G (reft) and B (right). c, Time evolution of ε (top) and εΣ (bottom) for the good cluster G (red) and the bad cluster B (bl… view at source ↗
read the original abstract

We introduce a new class of filtrations indexed by attracting levels in dynamical systems, providing novel inputs for persistent homology and related methods in topological data analysis. These filtrations quantify, in a forward direction, the sensitivity of trajectories with respect to attractors under perturbations and, in a backward direction, the perturbation magnitude at which attraction breaks down. The construction applies not only to maps on metric spaces but also to general partial maps with cost functions, yielding a filtration-theoretic framework with connections to algebraic topology. This generality ensures complementary filtrations when terminal states are good or bad, inducing natural decompositions of the underlying space. As an illustration, we apply the framework to ensemble forecasts of tropical cyclones, where the filtrations identify regions of heightened sensitivity, demonstrating the potential of our approach as a new tool for topological data analysis applied to dynamical systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a new class of filtrations indexed by attracting levels in dynamical systems. These quantify forward sensitivity of trajectories to attractors under perturbations and backward the perturbation magnitude at which attraction breaks down. The construction is claimed to apply to maps on metric spaces as well as general partial maps with cost functions, yielding complementary filtrations for good or bad terminal states that induce natural decompositions of the space. The filtrations are positioned as novel inputs for persistent homology and topological data analysis, with an illustrative application to ensemble forecasts of tropical cyclones identifying regions of heightened sensitivity.

Significance. If the filtrations rigorously satisfy the nestedness axiom and other filtration properties for arbitrary cost functions, the framework could supply a useful new tool for sensitivity analysis via persistent homology in dynamical systems. The extension to partial maps with cost functions and the explicit link to decompositions for good/bad terminals represent potential strengths. The tropical cyclone application provides a concrete demonstration of utility in forecasting contexts. Credit is due for attempting a general filtration-theoretic approach connecting dynamics to algebraic topology, though the load-bearing properties require explicit verification.

major comments (2)
  1. [§2] §2 (Definition of attracting levels and forward/backward sensitivity): The central claim that these objects form filtrations for general partial maps with cost functions requires the attracting levels to be monotonically ordered so that the associated sets are nested. The forward sensitivity and backward perturbation constructions do not automatically enforce this when cost functions are arbitrary or when terminal states switch between good and bad; without an explicit monotonicity argument or counter-example check, the filtration property (essential for persistent homology inputs) remains unverified and load-bearing for the main theorem.
  2. [§3.1] §3.1 (Induced decompositions): The assertion that complementary filtrations for good and bad terminal states induce natural decompositions of the underlying space is stated at a high level but lacks a precise statement or derivation showing how the level sets combine to produce the claimed partition or direct sum decomposition; this step is central to the generality claim yet is not load-bearing only if the nested filtration property already holds.
minor comments (2)
  1. [Abstract] Abstract: The phrase 'connections to algebraic topology' is vague; a specific reference to the relevant functor or homology theory used would clarify the intended interface with persistent homology.
  2. Notation: The cost function and partial map symbols are introduced without an explicit comparison table to standard dynamical systems notation, which could aid readers familiar with classical attractors.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the specific revisions planned for the next version.

read point-by-point responses

  1. Referee: [§2] §2 (Definition of attracting levels and forward/backward sensitivity): The central claim that these objects form filtrations for general partial maps with cost functions requires the attracting levels to be monotonically ordered so that the associated sets are nested. The forward sensitivity and backward perturbation constructions do not automatically enforce this when cost functions are arbitrary or when terminal states switch between good and bad; without an explicit monotonicity argument or counter-example check, the filtration property (essential for persistent homology inputs) remains unverified and load-bearing for the main theorem.

    Authors: We agree that an explicit monotonicity argument is necessary to confirm the filtration axioms for arbitrary cost functions. The attracting levels are defined to be non-decreasing with respect to the cost or perturbation size, which by construction yields nested sublevel sets for both forward and backward sensitivities. We will add a short lemma in the revised Section 2 that proves this monotonicity in full generality, including the separate treatment of good and bad terminal states to preserve complementarity. A brief counter-example check for non-monotonic costs will also be included to illustrate the boundary of the construction. revision: yes

  2. Referee: [§3.1] §3.1 (Induced decompositions): The assertion that complementary filtrations for good and bad terminal states induce natural decompositions of the underlying space is stated at a high level but lacks a precise statement or derivation showing how the level sets combine to produce the claimed partition or direct sum decomposition; this step is central to the generality claim yet is not load-bearing only if the nested filtration property already holds.

    Authors: The referee correctly notes that the decomposition claim would benefit from a precise statement. In the revised manuscript we will insert a proposition in Section 3.1 that derives the partition explicitly: the space is the disjoint union of the attracting-level sets associated with good terminals and those associated with bad terminals, with the two families remaining disjoint by the mutual exclusivity of the terminal-state labels. This proposition will be stated after the nestedness lemma so that the argument relies only on already-established filtration properties. revision: yes

Circularity Check

0 steps flagged

No circularity: new filtration class is a definitional construction

full rationale

The paper introduces filtrations indexed by attracting levels as a novel mathematical framework for dynamical systems and persistent homology. The abstract describes a forward/backward sensitivity construction for general partial maps with cost functions, yielding complementary filtrations and natural decompositions. No equations, fitted parameters, or predictions are shown that reduce to inputs by construction. No self-citations are invoked as load-bearing for the central definition or uniqueness. The application to tropical cyclone forecasts is an illustration, not a statistical fit renamed as prediction. The derivation is self-contained as a new definition with stated properties; any verification of nestedness or monotonicity is a standard proof obligation, not a circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5665 in / 1068 out tokens · 38959 ms · 2026-05-19T08:30:36.205411+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Definition 3. Let X be a set and c: X² → [0, ∞] a function. The function c is a cost function if c(x, x) = 0 for any x ∈ X. ... we generalize the conventional framework of dynamical systems on metric spaces ... to a broader framework of dynamical systems based on iterations of partial maps on sets equipped with a cost function.

  • IndisputableMonolith/Foundation/AbsoluteFloorClosure absolute_floor_iff_bare_distinguishability unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Theorem 2.5. ... The families (AF,εΣ)ε∈(−∞,−0]⊔[0,∞] and (AF,ε)ε∈(−∞,−0]⊔[0,∞] are filtrations.

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Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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