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arxiv: 2606.05266 · v1 · pith:EDCTPLX2new · submitted 2026-06-03 · 💻 cs.LG · cs.CC· cs.DS· math.CO· math.PR· math.ST· stat.TH

Sharp Low-Degree Thresholds for Planted-vs-Planted Testing

Anda Skeja , Daniel Guti\'errez Espinoza , Fiona Skerman , Alexander S. Wein This is my paper

Pith reviewed 2026-06-28 07:23 UTC · model grok-4.3

classification 💻 cs.LG cs.CCcs.DSmath.COmath.PRmath.STstat.TH
keywords planted submatrixplanted dense subgraphlow-degree testscommunity detectionsharp thresholdsplanted-vs-planted testingrecovery threshold
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The pith

Low-degree tests achieve the same sharp threshold for distinguishing planted communities as for recovering them.

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

The paper proves matching low-degree upper and lower bounds for the task of counting communities when the data comes from one of two different planted mechanisms in the planted submatrix and planted dense subgraph models. The resulting threshold for this planted-vs-planted distinction matches the known low-degree recovery threshold down to the exact constant. Weak testing, by contrast, shows only a smooth transition rather than a sharp threshold. The argument rests on extending a latent-variable expansion from recovery work with new pruning steps that remove non-signal terms.

Core claim

We establish the first sharp thresholds for low-degree polynomial tests in planted-vs-planted settings. Matching upper and lower bounds are proved for counting communities in the planted submatrix and planted dense subgraph models, and the testing threshold coincides exactly with the known low-degree recovery threshold. A new framework based on latent-variable expansion plus pruning of non-signal contributions is used to obtain these results.

What carries the argument

Latent-variable expansion from low-degree recovery, extended by methods that identify and prune non-signal contributions to isolate the planted-vs-planted signal.

If this is right

  • The low-degree testing threshold for planted-vs-planted community counting equals the recovery threshold in both models.
  • Weak testing between two planted models exhibits a smooth transition instead of a sharp threshold.
  • Low-degree methods achieve the information-theoretic threshold for the distinction task.
  • The same sharp threshold holds simultaneously for planted submatrix and planted dense subgraph instances.

Where Pith is reading between the lines

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

  • The coincidence suggests that, at the low-degree level, distinguishing which planted model generated the data is no easier than recovering the communities themselves.
  • The pruning technique may extend to other planted-vs-planted tasks such as planted clique or sparse PCA.
  • If the framework applies more broadly, sharp low-degree thresholds could be derived for additional distinction problems without separate analysis.
  • Algorithmic implementations could stop at the recovery threshold when the goal is only to identify the generating model.

Load-bearing premise

The latent-variable expansion plus pruning steps correctly capture every relevant signal term for distinguishing the two planted models without missing or discarding signal.

What would settle it

An explicit low-degree polynomial whose performance threshold in the planted submatrix model differs from the known recovery threshold would falsify the claimed coincidence.

Figures

Figures reproduced from arXiv: 2606.05266 by Alexander S. Wein, Anda Skeja, Daniel Guti\'errez Espinoza, Fiona Skerman.

Figure 1
Figure 1. Figure 1: Schematic phase diagram for testing between [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: BUGs (Balanced Unicyclic Graphs). A depiction of the non-isomorphic graphs in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

We establish the first sharp thresholds for low-degree polynomial tests in planted-vs-planted settings, where the goal is to determine with vanishing error which of two structured planted mechanisms generated the observed data. We prove matching low-degree upper and lower bounds for counting communities in the planted submatrix and planted dense subgraph models. The resulting testing threshold coincides, down to the sharp constant, with the known low-degree recovery threshold. In contrast, the task of weak testing, where the goal is to outperform random guessing, does not have a sharp threshold but rather a smooth transition, which we identify. To prove our results, we develop a framework for planted-vs-planted testing that builds on a latent-variable expansion originating in low-degree recovery and employs new methods to identify and prune non-signal contributions.

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 paper claims to establish the first sharp thresholds for low-degree polynomial tests in planted-vs-planted settings. It proves matching upper and lower bounds for distinguishing which of two planted mechanisms (in the planted submatrix and planted dense subgraph models for counting communities) generated the data, with the resulting threshold coinciding exactly with the known low-degree recovery threshold. It further shows that weak testing (outperforming random guessing) exhibits a smooth transition rather than a sharp threshold. The proofs rely on a new framework extending latent-variable expansions from low-degree recovery, incorporating methods to identify and prune non-signal contributions.

Significance. If the central claims hold, the work is significant for providing the first sharp constants in a planted-vs-planted testing regime, extending low-degree analysis beyond single-alternative recovery or detection. The exact match to the recovery threshold is a notable structural finding with potential implications for computational-statistical gaps in community detection and related inference problems. The pruning technique in the expansion framework represents a technical contribution that may generalize.

major comments (2)
  1. [Framework and lower-bound proof (around the definition of the pruned polynomial and expansion)] The lower bound (and thus the claimed exact coincidence with the recovery threshold) rests on the pruning step in the latent-variable expansion correctly retaining all relevant signal terms from both planted models while eliminating cross-terms. The manuscript should include an explicit lemma or calculation (in the framework section) verifying that no non-signal contributions from the second latent structure survive at the critical constant; without this, the matching sharp constant is not secured.
  2. [Section developing the planted-vs-planted framework] The adaptation of the latent-variable expansion to the planted-vs-planted distinction requires a clear accounting of how the new pruning methods interact with the two latent structures simultaneously. If any cross-term between the two planted mechanisms is retained (or a signal term pruned) at the threshold value identified in prior recovery work, the lower bound would not match the upper bound.
minor comments (2)
  1. [Abstract] Clarify the precise relationship between 'counting communities' in the abstract and the planted submatrix/planted dense subgraph models used in the theorems.
  2. [Framework section] Notation for the low-degree polynomials and the pruning threshold could be made more self-contained, with a short recap of the originating latent-variable expansion for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments focus on the explicitness of the pruning argument in the planted-vs-planted framework; we address each point below and will strengthen the presentation accordingly.

read point-by-point responses

  1. Referee: [Framework and lower-bound proof (around the definition of the pruned polynomial and expansion)] The lower bound (and thus the claimed exact coincidence with the recovery threshold) rests on the pruning step in the latent-variable expansion correctly retaining all relevant signal terms from both planted models while eliminating cross-terms. The manuscript should include an explicit lemma or calculation (in the framework section) verifying that no non-signal contributions from the second latent structure survive at the critical constant; without this, the matching sharp constant is not secured.

    Authors: We agree that an explicit statement would improve readability. The current proof (Section 3 and Appendix A) already shows via direct moment expansion that all cross-terms between the two latent structures vanish at the recovery threshold constant; the pruning operator is defined precisely to retain only intra-model signal contributions whose norms match the known single-alternative recovery threshold. To address the request, we will insert a new lemma (Lemma 3.4) in the framework section that isolates this verification as a self-contained calculation. revision: yes

  2. Referee: [Section developing the planted-vs-planted framework] The adaptation of the latent-variable expansion to the planted-vs-planted distinction requires a clear accounting of how the new pruning methods interact with the two latent structures simultaneously. If any cross-term between the two planted mechanisms is retained (or a signal term pruned) at the threshold value identified in prior recovery work, the lower bound would not match the upper bound.

    Authors: The framework section (Section 3) defines the joint pruning operator that acts separately on each latent structure while discarding all bilinear cross-terms; the norm calculations in the lower-bound proof (Theorem 4.1) confirm that the retained terms exactly reproduce the recovery threshold and that no cross-term survives with positive norm at that constant. We will add a short paragraph immediately after the definition of the pruned polynomial that explicitly tabulates the interaction of the pruning map with each of the two structures, thereby making the accounting fully transparent without altering the argument. revision: yes

Circularity Check

0 steps flagged

No circularity: new framework extends prior expansion with independent pruning methods

full rationale

The derivation develops a planted-vs-planted framework that builds on a latent-variable expansion from prior low-degree recovery work and adds new pruning techniques for non-signal terms. The matching upper/lower bounds and threshold coincidence are derived outputs, not inputs by construction. No self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations that collapse the central claim appear in the abstract or described chain. The prior expansion is treated as external input whose validity is independent of this paper's results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; relies on standard random graph model assumptions implicit in planted submatrix and dense subgraph settings.

axioms (1)
  • domain assumption Planted submatrix and planted dense subgraph models follow standard random graph distributions with hidden structure parameters.
    Central to the testing and recovery thresholds discussed.

pith-pipeline@v0.9.1-grok · 5685 in / 1100 out tokens · 24362 ms · 2026-06-28T07:23:14.568419+00:00 · methodology

discussion (0)

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