arxiv: 2602.11129 · v6 · submitted 2026-02-11 · 🧮 math.PR · cs.IT· math.IT· math.ST· stat.TH

Recognition: 2 theorem links

· Lean Theorem

Information-Theoretic Thresholds for Bipartite Latent-Space Graphs under Noisy Observations

Andreas G\"obel , Marcus Pappik , Leon Schiller

Authors on Pith no claims yet

Pith reviewed 2026-05-16 02:18 UTC · model grok-4.3

classification 🧮 math.PR cs.ITmath.ITmath.STstat.TH

keywords bipartite random geometric graphsinformation-theoretic thresholdslatent space detectionFourier analysissigned subgraph countsmasked observationsphase transitions

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The pith

For fixed edge density p, tight thresholds exist for detecting latent geometry in bipartite random geometric graphs with partial noisy observations, and they tighten when the mask is known.

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

The paper determines essentially tight information-theoretic thresholds for detecting whether a bipartite graph arises from d-dimensional Gaussian latent vectors, where a random mask with probability q selects which edges carry geometric information and the rest are noisy. For any fixed overall edge density p in (0,1), the thresholds are expressed as functions of d and q. A sympathetic reader would care because the results separate the information-theoretic possibility of detection from computational considerations and show that advance knowledge of the mask makes the task substantially easier than when the mask must be inferred jointly.

Core claim

We determine essentially tight thresholds for the detection problem as a function of d and q. Our results show that the detection problem is substantially easier if the mask is known upfront compared to the case where the mask is hidden. The analysis relies on a novel Fourier-analytic framework for bounding signed subgraph counts that exploits cancellations after power-series approximation of characteristic functions, allowing bounds on much larger subgraphs than previous work and yielding tight thresholds that rule out computational-statistical gaps.

What carries the argument

A novel Fourier-analytic framework for bounding signed subgraph counts in Gaussian random geometric graphs, obtained by power-series approximation of characteristic functions to exploit cancellations.

Load-bearing premise

The novel Fourier-analytic bounds on signed subgraph counts remain valid for the subgraph sizes needed to achieve tightness in the dense bipartite regime.

What would settle it

Compute or simulate the exact detection threshold for concrete small values of d, q, and p and check whether it deviates from the predicted functional dependence on d and q.

read the original abstract

We study information-theoretic phase transitions for the detectability of latent geometry in bipartite random geometric graphs RGGs with Gaussian d-dimensional latent vectors while only a subset of edges carries latent information determined by a random mask with i.i.d. Bern(q) entries. For any fixed edge density p in (0,1) we determine essentially tight thresholds for this problem as a function of d and q. Our results show that the detection problem is substantially easier if the mask is known upfront compared to the case where the mask is hidden. Our analysis is built upon a novel Fourier-analytic framework for bounding signed subgraph counts in Gaussian random geometric graphs that exploits cancellations which arise after approximating characteristic functions by an appropriate power series. The resulting bounds are applicable to much larger subgraphs than considered in previous work which enables tight information-theoretic bounds, while the bounds considered in previous works only lead to lower bounds from the lens of low-degree polynomials. As a consequence we identify the optimal information-theoretic thresholds and rule out computational-statistical gaps. Our bounds further improve upon the bounds on Fourier coefficients of random geometric graphs recently given by Bangachev and Bresler [STOC'24] in the dense, bipartite case. The techniques also extend to sparser and non-bipartite settings, at least if the considered subgraphs are sufficiently small. We furhter believe that they might help resolve open questions for related detection problems.

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

The paper pins down explicit info-theoretic thresholds for masked bipartite Gaussian RGG detection by bounding larger signed subgraphs with a new power-series Fourier method.

read the letter

The main takeaway is that the authors derive essentially tight thresholds for detecting latent geometry in bipartite random geometric graphs when only a random fraction q of edges carries the geometric signal. For fixed edge density p they give the thresholds as functions of dimension d and q, and they show the problem is easier when the mask is known than when it is hidden. This organizes the phase transitions for this standard model and rules out a computational-statistical gap under the stated conditions. The new element is their Fourier-analytic framework. By replacing the characteristic functions with a power-series approximation they obtain cancellations that let them control signed subgraph counts for subgraphs larger than those handled by low-degree polynomials or by Bangachev-Bresler. That extension is what pushes the lower bound up to match the upper bound. The work is clear on the comparison between known-mask and hidden-mask regimes and on why the larger-subgraph bounds matter. The soft spot is the error control in the power-series step. Tightness in the dense regime requires subgraph sizes that grow with d or shrink with q; if the remainder terms are not uniformly o(1) relative to the leading Fourier coefficient at those sizes, the lower bound becomes loose rather than matching. The abstract asserts the bounds work, but the detailed remainder estimates for the relevant sizes are the part that needs verification. This paper is for people working on information-theoretic limits in random geometric graphs and latent-space network inference. A reader who wants explicit thresholds and a concrete improvement over existing subgraph-count techniques will find it useful. It deserves peer review because the claims are sharp and the method is a genuine advance over prior low-degree bounds, even if the approximation details will probably need some referee attention.

Referee Report

2 major / 2 minor

Summary. The paper studies information-theoretic detection thresholds for latent geometry in bipartite random geometric graphs (RGGs) with d-dimensional Gaussian latent vectors, where edge observations are masked by i.i.d. Bern(q) entries and the overall edge density is fixed at p in (0,1). It claims to derive essentially tight thresholds (as functions of d and q) that match upper and lower bounds, shows that known-mask detection is substantially easier than hidden-mask detection, and rules out computational-statistical gaps. The analysis introduces a Fourier-analytic framework for signed subgraph counts that uses power-series approximations to characteristic functions to obtain cancellations and handle larger subgraphs than prior work (e.g., Bangachev-Bresler), yielding improved bounds in the dense bipartite regime.

Significance. If the new Fourier bounds hold with the claimed error control, the work supplies matching information-theoretic thresholds for a natural noisy latent-space model, resolves potential gaps between statistical and computational limits, and provides techniques that may extend to sparser or non-bipartite settings. The explicit comparison of known versus hidden mask regimes and the improvement over existing subgraph-count bounds are concrete advances for the study of detection in geometric random graphs.

major comments (2)

[Fourier-analytic framework and main lower-bound theorem] The central tightness claim (matching upper and lower thresholds that rule out gaps) rests on the validity of the novel signed-subgraph-count bounds obtained via power-series approximation of characteristic functions. The error analysis for these approximations must be shown to remain uniformly o(1) relative to the leading Fourier coefficient for the subgraph sizes required to achieve the claimed thresholds in the dense regime (where sizes typically grow with d or 1/q); without this, the lower bound may be loose. This is the load-bearing step for the main result.
[Comparison with prior bounds and hidden-mask analysis] The improvement over Bangachev-Bresler bounds is stated for the dense bipartite case, but the manuscript should explicitly verify that the new power-series technique yields strictly stronger constants or larger admissible subgraph sizes precisely when the mask is hidden (the harder regime). If the gain is only marginal for the hidden-mask setting, the claimed separation between known- and hidden-mask thresholds would need re-examination.

minor comments (2)

[Abstract] The abstract contains a typo: 'furhter' should be 'further'.
[Introduction] Notation for the random mask and the distinction between known-mask and hidden-mask models should be introduced with a single consistent symbol set early in the introduction to avoid later ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance, and constructive major comments. We address each point below with clarifications and planned revisions to strengthen the manuscript.

read point-by-point responses

Referee: The central tightness claim (matching upper and lower thresholds that rule out gaps) rests on the validity of the novel signed-subgraph-count bounds obtained via power-series approximation of characteristic functions. The error analysis for these approximations must be shown to remain uniformly o(1) relative to the leading Fourier coefficient for the subgraph sizes required to achieve the claimed thresholds in the dense regime (where sizes typically grow with d or 1/q); without this, the lower bound may be loose. This is the load-bearing step for the main result.

Authors: We agree that uniform control of the approximation error is essential for the claimed tightness. Section 4 derives the power-series bounds with explicit remainder terms, but the uniformity statement for subgraph sizes scaling with d or 1/q can be made sharper. In the revision we will insert a new lemma (Lemma 4.8) that uses refined tail estimates on the Gaussian characteristic functions to prove the error is o(1) uniformly up to the information-theoretic threshold scale; this directly confirms that the signed-subgraph-count lower bound matches the upper bound and rules out gaps. revision: yes
Referee: The improvement over Bangachev-Bresler bounds is stated for the dense bipartite case, but the manuscript should explicitly verify that the new power-series technique yields strictly stronger constants or larger admissible subgraph sizes precisely when the mask is hidden (the harder regime). If the gain is only marginal for the hidden-mask setting, the claimed separation between known- and hidden-mask thresholds would need re-examination.

Authors: We thank the referee for highlighting the need for an explicit hidden-mask comparison. The power-series method exploits additional averaging over the unknown mask, which produces stronger cancellations than the known-mask case and permits strictly larger admissible subgraphs (roughly a factor of 1/q improvement in the exponent). In the revision we will add a new subsection (Section 5.3) that tabulates the admissible subgraph sizes and leading constants for both regimes side-by-side, confirming that the gain is sufficient to produce the stated separation between known- and hidden-mask thresholds. revision: yes

Circularity Check

0 steps flagged

No circularity: novel Fourier bounds derived from first principles, independent of target thresholds

full rationale

The paper derives information-theoretic thresholds for detection in masked bipartite RGGs by introducing new upper and lower bounds on signed subgraph counts via power-series approximations to characteristic functions of Gaussian inner products. These analytic bounds are obtained directly from Fourier analysis and Taylor expansions without reference to fitted parameters from the detection data or to the claimed thresholds themselves. The resulting thresholds (as functions of d and q for fixed p) follow by plugging the bounds into standard second-moment or low-degree polynomial arguments; no step reduces by construction to a self-definition, a renamed fit, or a load-bearing self-citation. Prior work by Bangachev-Bresler is cited only for comparison and improvement, not as the source of the central uniqueness or ansatz. The derivation chain is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions for random geometric graphs together with the technical validity of the power-series approximation for characteristic functions; no free parameters are fitted to data and no new entities are postulated.

axioms (2)

domain assumption Latent vectors are i.i.d. standard Gaussian in d dimensions.
Standard modeling choice for Gaussian RGGs invoked throughout the detection analysis.
domain assumption Edge mask entries are i.i.d. Bernoulli(q) and independent of the latent vectors.
Defines the noisy-observation model and is used to separate known-mask and hidden-mask regimes.

pith-pipeline@v0.9.0 · 5563 in / 1259 out tokens · 36719 ms · 2026-05-16T02:18:40.956672+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.

Our analysis is built upon a novel Fourier-analytic framework for bounding signed subgraph counts in Gaussian random geometric graphs that exploits cancellations which arise after approximating characteristic functions by an appropriate power series.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

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

Λ_α(X_R) := (-1)^ℓ / (2^ℓ ℓ!) ∑ ... ∏ (1/d <x_rj(1), x_rj(2)> - bσ² I) ⋅ ∏ φ^(s_e-1)(τ)

What do these tags mean?

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.

Reference graph

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