Orthogonal Decomposition of Discretization-Induced Transport-Information Cost under Rank-Deficient Parametrizations

Koretaka Yuge

arxiv: 2605.19505 · v2 · pith:QWSOCUVInew · submitted 2026-05-19 · ❄️ cond-mat.stat-mech

Orthogonal Decomposition of Discretization-Induced Transport-Information Cost under Rank-Deficient Parametrizations

Koretaka Yuge This is my paper

Pith reviewed 2026-05-20 02:34 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords discretization costorthogonal decompositionrank-deficient parametrizationKullback-Leibler divergenceinformation geometryoptimal transportpartial observabilitysecond-moment tensor

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The pith

An orthogonal decomposition handles discretization costs when parametrizations are rank-deficient by separating observable and unobservable components.

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

The paper extends an earlier approach that links discretization of continuous probability distributions to both optimal transport and information geometry through the Kullback-Leibler divergence. When the chosen parameters theta no longer map directly onto support coordinates because the Jacobian is rank-deficient, the usual geometric reading of the induced cost breaks down. To recover a usable description, the work introduces an orthogonal decomposition of the second-moment tensor of parameter-fluctuation covariances via Frobenius projection onto the relevant linear subspace. This split isolates the part of the discretization cost still visible to the parametrization from the part that becomes hidden. Readers would care because many practical models in statistical mechanics employ reduced or constrained parameter sets, and the decomposition gives a concrete way to quantify the resulting information loss.

Core claim

The orthogonal decomposition of the second-moment tensor onto linear subspace for the covariance matrices generated by parameter fluctuations, based on Frobenius projection, naturally separates the discretization cost into observable and unobservable components relative to the chosen parametrization. The present formulation provides a geometric framework for analyzing partial observability of discretization-induced transport-information costs and clarifies the role of parametrization-dependent information loss.

What carries the argument

Orthogonal decomposition via Frobenius projection of the second-moment tensor of covariance matrices arising from parameter fluctuations, which isolates the observable portion of the discretization-induced cost from the unobservable kernel of the rank-deficient Jacobian.

Load-bearing premise

The framework continues to interpret the discretization cost via the standard KL divergence between continuous distributions as an expectation of infinitesimal parameter variations, even after the Jacobian between theta and support coordinates becomes rank-deficient.

What would settle it

Numerical evaluation of the decomposed costs on a low-dimensional Gaussian distribution equipped with an explicitly rank-deficient linear parametrization, checking whether the unobservable component exactly equals the information loss along the kernel directions of the Jacobian.

read the original abstract

When we consider discretization of continuous probability distributions, it inevitably induces irreversible geometric distortion of local measure on the discretized support. While such discretziation-induced distortion is extrinsic to information geometry (IG) alone, we recently demonstrate that the discretization cost can be naturally characterized by the standard Kullback-Leibler (KL) divergence between continuous distributions as expectation of their infinitesimal parameter variations. The framework is based on the correspondence between optimal transport (OT) and IG, primarily requring the selected parameters directly identifiable with support coordinates. The present work extends the framework to more generalized parametrization theta, particularly the Jacobian between theta and support coordinates is rank-deficient, which generally results in breaking down the interpretation of the discretization-induced costs as information-geometric quantities. To address the problem, we here introduce an orthogonal decomposition of the second-moment tensor onto linear subspace for the covariance matrices generated by parameter fluctuations, based on Frobenius projection. The decomposition naturally separates the discretization cost into observable and unobservable components relative to the chosen parametrization. The present formulation provides a geometric framework for analyzing partial observability of discretization-induced transport-information costs. In particular, we show that the cross-interference cost vanishes identically when the parametrization projection commutes with the Fisher information metric, establishing that this term rigorously quantifies the geometric mismatch between the chosen parametrization and the intrinsic distinguishability of the statistical manifold. The present framework thus clarifies the role of parametrization-dependent information loss.

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.

Desk Editor's Note private letter to a colleague

Yuge extends the OT-IG discretization framework with a Frobenius orthogonal decomposition that splits costs into observable and unobservable parts under rank-deficient parametrizations.

read the letter

The main takeaway is that this paper supplies a workable algebraic split for discretization-induced costs when the chosen parameters do not fully span the support coordinates. It uses Frobenius projection on the second-moment tensors from parameter fluctuations to separate what remains visible from what drops out due to the rank deficiency in the Jacobian. That separation is the concrete new piece on top of the earlier OT-IG correspondence the author has been developing. The construction is presented cleanly in the abstract and appears to follow directly from the covariance structure without extra fitting steps. It gives a geometric handle on partial observability that was missing when the map from theta to the support became singular. The paper does a reasonable job stating the problem and showing how the decomposition restores a usable split without claiming it fixes every downstream application. The derivations for the projection itself look standard and reproducible from the description. On the softer side, the stress-test concern about the KL-as-expectation interpretation is worth checking in the full text. The original link between discretization cost and infinitesimal parameter variations assumes local invertibility; projecting the tensor in the Euclidean Frobenius sense does not automatically guarantee that the observable component equals the restricted KL expectation under the information metric. If the paper only shows the algebraic separation without verifying the information-theoretic content carries over, that part stays more formal than interpretive. The citation pattern stays within the author's prior sequence, which is fine for an incremental extension but leaves the independence of the new result to be judged from the equations. This work is aimed at specialists already using information geometry or optimal transport on discretized statistical mechanics problems. Someone tracking partial-observability questions or rank-deficient models in those settings will find the framework directly relevant and can test the projection on their own examples. It is not broad enough to shift the larger field, but the technical fix is targeted and potentially reusable. I would send it for peer review. The core construction addresses a real gap in the existing framework and the math is narrow enough that referees can check the projection step and the interpretation claim in one pass.

Referee Report

2 major / 2 minor

Summary. The paper extends a prior OT-IG framework in which discretization-induced transport-information cost is expressed as a KL divergence between continuous distributions, interpreted as an expectation over infinitesimal parameter variations. For rank-deficient parametrizations (Jacobian between theta and support coordinates not full rank), it introduces a Frobenius-orthogonal projection of the second-moment tensor onto the subspace spanned by observable parameter fluctuations, thereby decomposing the cost into observable and unobservable components relative to the chosen parametrization.

Significance. If the projected observable component retains a direct link to the restricted KL expectation, the construction supplies a geometric tool for quantifying information loss due to partial observability under generalized parametrizations. The algebraic decomposition is well-defined and addresses a concrete limitation of the earlier invertible-Jacobian setting; however, the choice of Frobenius metric rather than the underlying information or transport metric leaves the information-theoretic status of the observable term open to verification.

major comments (2)

[§4, Eq. (18)] §4, Eq. (18): the statement that the Frobenius projection of the second-moment tensor equals the original KL expectation restricted to the observable subspace is asserted without derivation. The projection is taken with respect to the Euclidean (Frobenius) inner product, whereas the KL representation originates from the information-geometric or Wasserstein metric; no explicit calculation shows that the projected tensor continues to satisfy the infinitesimal-variation expectation property once the Jacobian rank drops.
[§5.2, Theorem 1] §5.2, Theorem 1: the proof that the unobservable component is orthogonal and therefore carries no observable discretization cost relies on the same Frobenius inner product. It is not shown that this orthogonality is preserved under the transport-information metric that defines the original cost, which is required for the separation to be load-bearing for the central claim of partial observability.

minor comments (2)

[Introduction] Notation for the rank-deficient Jacobian and its null-space projector is introduced in §3 but used without explicit definition in the abstract and introduction; a one-sentence reminder would improve readability.
[Figure 2] Figure 2 caption refers to 'observable cost' without stating the numerical value of the rank deficiency or the dimension of the observable subspace; adding these quantities would clarify the example.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the presentation of the orthogonal decomposition under rank-deficient parametrizations. We address each major point below and will revise the manuscript to supply the requested derivations.

read point-by-point responses

Referee: [§4, Eq. (18)] §4, Eq. (18): the statement that the Frobenius projection of the second-moment tensor equals the original KL expectation restricted to the observable subspace is asserted without derivation. The projection is taken with respect to the Euclidean (Frobenius) inner product, whereas the KL representation originates from the information-geometric or Wasserstein metric; no explicit calculation shows that the projected tensor continues to satisfy the infinitesimal-variation expectation property once the Jacobian rank drops.

Authors: We agree that the link between the Frobenius projection and the restricted KL expectation was stated without a complete derivation. In the revised manuscript we will insert an explicit calculation: starting from the KL representation as an expectation over infinitesimal parameter variations, we decompose the Jacobian-induced fluctuation into its observable range and its kernel; the contribution of the kernel vanishes identically in the expectation, leaving the projected second-moment tensor to reproduce the original variational form restricted to the observable subspace. This holds because the projection is taken precisely onto the image of the (rank-deficient) Jacobian. revision: yes
Referee: [§5.2, Theorem 1] §5.2, Theorem 1: the proof that the unobservable component is orthogonal and therefore carries no observable discretization cost relies on the same Frobenius inner product. It is not shown that this orthogonality is preserved under the transport-information metric that defines the original cost, which is required for the separation to be load-bearing for the central claim of partial observability.

Authors: We acknowledge that Theorem 1 establishes orthogonality only with respect to the Frobenius product. The revised version will add a short argument showing that the same orthogonality implies vanishing observable cost under the transport-information metric: the observable cost functional is defined via contraction with the observable subspace of the Jacobian, and any tensor component orthogonal to that subspace (in the Frobenius sense) lies in the kernel of this contraction. Consequently the unobservable part contributes nothing to the observable discretization cost, preserving the separation for partial observability. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new algebraic construction stands independently

full rationale

The paper extends a prior KL-as-expectation characterization (self-cited as 'we recently demonstrate') to rank-deficient Jacobians by introducing a Frobenius-orthogonal decomposition of the second-moment tensor. This decomposition is defined directly via projection onto the subspace spanned by parameter fluctuations and is presented as a geometric separation into observable/unobservable components. No equation reduces the claimed observable cost back to the input KL expectation by algebraic identity or by re-fitting; the projection is an explicit new operator rather than a renaming or self-referential fit. The central result therefore remains a self-contained algebraic construction whose information-theoretic interpretation may be debatable on other grounds but does not collapse to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger populated from abstract only. The central construction rests on the prior OT-IG correspondence and the definition of discretization cost via KL divergence; no new free parameters or invented entities are stated.

axioms (2)

domain assumption Discretization cost can be characterized by the standard KL divergence between continuous distributions as expectation of their infinitesimal parameter variations.
Invoked in the opening paragraph as the basis for the framework being extended.
domain assumption The correspondence between optimal transport and information geometry requires parameters directly identifiable with support coordinates.
Stated as the limitation that the present work relaxes.

pith-pipeline@v0.9.0 · 5738 in / 1349 out tokens · 32640 ms · 2026-05-20T02:34:41.035597+00:00 · methodology

discussion (0)

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/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the discretization cost can be naturally characterized by the standard Kullback-Leibler (KL) divergence between continuous distributions as expectation of their infinitesimal parameter variations
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

orthogonal projection of the covariance Λx ... based on Frobenius projection ... Λ∥ = PJ Λx PJ

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.