Improving the Threshold for Finding Rank-1 Matrices in a Subspace

Alexander S. Wein; Jeshu Dastidar; Tait Weicht

Improving the Threshold for Finding Rank-1 Matrices in a Subspace

T0 review · 0 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper proves the JLV algorithm's central identity holds for generic planted subspaces up to $R \le \frac12(m-1)(n-2)$, doubling the threshold, and that the identity fails once $R>mn/\sqrt{2}+1$, pinning the plausible ceiling near…

desk verdict A clean, self-contained improvement of the JLV threshold from mn/4 to mn/2, with a matching upper-bound conjecture and strong computational evidence; worth serious refereeing. read the letter →

arxiv 2504.17947 v1 pith:3VG4GQZ2 submitted 2025-04-24 cs.DS

Jeshu Dastidar , Tait Weicht , Alexander S. Wein This is my paper · ORCID

classification cs.DS MSC 15A6968Q25

keywords rank-1matrixrecoveryplantedsubspaceJLValgorithmNullstellensatzhierarchytensordecompositiongenericidentifiabilitythresholdsymmetricsquare

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks how large a random matrix subspace can be before a polynomial-time algorithm can no longer find planted rank-1 matrices inside it. For the JLV algorithm, based on the first level of the Nullstellensatz hierarchy, the previous guarantee was $R \le (1-o(1))mn/4$. The paper proves the algorithm's central identity holds up to $R \le \frac{1}{2}(m-1)(n-2)$, roughly doubling the guarantee, and proves that the identity must fail when $R > mn/\sqrt{2}+1$. Numerical experiments suggest the true boundary is $R = (1+o(1))mn/\sqrt{2}$. The proven bound already gives order-4 tensor decompositions with roughly twice as many components as before, and the conjectured boundary would make the threshold exactly $mn/\sqrt{2}$.

What carries the argument

The machinery is the symbolic matrix $M$ whose rows are indexed by the $2\times2$-minor constraints $(a,b,c,d)$ and whose columns are indexed by pairs $(i,j)\in\Omega=\{i<j\}\cup\{i=j>s\}$, with entries $M_{abcd,ij}=\langle v_i\vee v_j, E_{abcd}\rangle$; full column rank of $M$ is equivalent to the identity (Goal). The proof constructs a square submatrix $M''$ with the help of a combinatorial injection $\varphi:[R]\to[m-1]\times[k]$ (where $2k+1\le n$) and by zeroing many variables, then shows $M''$ is block upper triangular with $1\times1$ and $2\times2$ diagonal blocks whose determinants are explicit polynomials with distinct monomials; a Zero Lemma forces every off-diagonal block to vanish. That block structure converts a high-dimensional genericity statement into a finite check on polynomial determinants.

What would settle it

Choose $m=n=6$, $s=0$, $R=10$, exactly the claimed threshold $R=\frac{1}{2}(5)(4)$. For a random integer-valued subspace, compute the rank of the matrix $M$ over $\mathbb{Q}$ or a large finite field: Theorem 2.5 predicts full column rank for every generic instance, so a single instance with rank below the column count refutes it. Equivalently, verify symbolically that some maximal minor of $M$ is a nonzero polynomial.

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Extended reading notes

Core claim

The central claim is the identity $S_2(\mathcal{U}) \cap \mathrm{span}\{E_{abcd}\}^{\perp} = \mathrm{span}\{(v_1)^{\otimes 2},\dots,(v_s)^{\otimes 2}\}$ for generically chosen planted subspaces: intersecting the symmetric square of the given subspace with the orthocomplement of the $2\times2$ minor constraints leaves exactly the squared planted matrices. Theorem 2.5 establishes this identity whenever $R \le \frac{1}{2}(m-1)(n-2)$, and Theorem 2.7 shows it fails when $\binom{m}{2}\binom{n}{2} < \binom{R+1}{2}-s$, in particular when $R>mn/\sqrt{2}+1$. Because the left-hand side is computable from an arbitrary basis of $\mathcal{U}$, the positive identity is what lets the JLV algorithm output the planted matrices (after an extraction step), and the negative result shows the first Nullstellensatz level cannot be pushed beyond about $1/\sqrt{2}$ of the ambient dimension.

Load-bearing premise

The paper's own positive theorem is the algebraic identity (Goal); the headline claim that the JLV algorithm 'succeeds' additionally assumes the cited extraction step from the earlier JLV paper — that from the span of the squared planted matrices the vectors are recoverable by simultaneous diagonalization — and the negative result shows only that the identity (Goal) fails, not that no algorithm could recover the planted matrices.

Editorial extensions

If this is right

For any $0\le s\le R$ with $R\le \frac{1}{2}(m-1)(n-2)$, the JLV algorithm recovers all $s$ planted rank-1 matrices from an arbitrary basis of $\mathcal{U}$, up to permutation and scalar multiple (Corollary 2.6).
For $R> mn/\sqrt{2}+1$, the identity (Goal) fails on generic inputs, so the degree-2 Nullstellensatz analysis of the algorithm cannot certify recovery or rank-1-freeness beyond that point (Theorem 2.7).
Generic order-4 $n\times n\times n\times n$ tensors decompose in polynomial time up to rank $R\le \frac{1}{2}(n-2)^2$, roughly twice the previous $R\le \frac{1}{4}(n-1)^2$ (Corollary 7.2).
Generic unbalanced order-3 tensors of shape $n_1\times n_2\times n_3$ with $n_3\ge n_1n_2/2$ decompose up to $R\le \frac{1}{2}(n_1-1)(n_2-2)$ (Corollary 7.1).
If Conjecture 6.1 holds, the threshold is exactly the dimension-count boundary $R\sim mn/\sqrt{2}$, and the tensor-decomposition bounds become $R\le \min\{R_{\max}(n_1,n_2), n_3n_4\}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The block-triangular witness construction suggests that a deterministic family of subspaces achieving full column rank at the conjectured threshold should exist for every $m,n$, which would turn the computer-verified cases into a general proof.
If Conjecture 6.1 is true, the first level of the Nullstellensatz hierarchy is exactly as strong as its dimension count allows: the obstruction is purely linear-algebraic (rows versus columns of $M$) and no degree-2 method can do better.
The observed failure mode just above the threshold, with spurious kernel vectors of dimension exactly $\binom{R+1}{2}-\binom{m}{2}\binom{n}{2}$, suggests the breakdown is abrupt rather than gradual, so a smoothed or noise-robust version of the JLV analysis would need a different mechanism.
The symmetric analogue, if Conjecture 6.10 holds, would raise the symmetric order-4 decomposition bound from $\frac{1}{6}n(n-1)$ to $\sim n^2/\sqrt{6}$, a larger relative gain than the nonsymmetric factor of two; the paper's proofs for $m\le 14$ (all $s$) and $m\le 21$ ($s\in\{0,R\}$) are consistent with that degree count.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper doubles the proven success regime of the JLV first-level Nullstellensatz algorithm for finding planted rank-1 matrices: from R ≤ (1-o(1))mn/4 to R ≤ (1-o(1))mn/2, and proves failure (of the key intersection identity) when R ≥ (1+o(1))mn/√2. The main theorem is Theorem 2.5, and the proof is genuinely different from JLV's: a concrete block-upper-triangular submatrix of the symbolic matrix M, with each 1×1 or 2×2 diagonal block shown to have non-vanishing determinant. That argument is the heart of the paper and it checks out. The negative result is a clean dimension count. The paper is honest about what it proves: the algorithmic conclusions depend on the JLV extraction proposition (2.3), which is imported; the tensor-decomposition corollaries are straightforward applications. I agree with the reader: no load-bearing flaw. The minor issues are all minor: the unstated linear independence of the E_abcd tensors in Theorem 2.7 (true but should be one sentence); a likely typo in Corollary 7.2's flattening bound (should be n3*n4, not n2*n3); and the usual caveat that "failure" in Theorem 2.7 is failure of (Goal), not a formal statement that every recovery method fails—though the numerical evidence supports the conjectured sharp threshold. The finite-field certificates (3703 exact cases, p=997) and the floating-point experiments are real credit; they both support Conjecture 6.1 and demonstrate that the JLV algorithm's actual behavior tracks the proven bound. One point of caution: the conjecture's exact form (5) has no dependence on s to first order, and the paper's evidence for that is the certificates; that is a computational conjecture, not a fitted prediction, so it is fine. This is a within-subfield advance, not a field-changing one, but it is a solid, well-written paper. Who is it for: people working on algebraic algorithms for tensor decomposition, sum-of-squares/Nullstellensatz hierarchies, and the JLV line of work. Deserves a serious referee. I would accept it for review and expect it to be accepted after minor revisions.

Referee Report

0 major / 5 minor

Summary. The paper studies the problem of recovering s planted rank-one m×n matrices from an R-dimensional linear subspace U that is generated generically by those planted matrices together with R−s generic matrices. Its main object is the key identity (Goal), which identifies the intersection S2(U) ∩ span{E_abcd}⊥ with the span of the squared planted matrices. Theorem 2.5 proves that (Goal) holds for generic planted subspaces when R ≤ 1/2 (m−1)(n−2), and Corollary 2.6 turns this into polynomial-time recovery via the cited extraction result Proposition 2.3. Theorem 2.7 gives a dimension-counting negative result: (Goal) fails when C(m,2)C(n,2) < C(R+1,2) − s, in particular when R > mn/√2 + 1. Sections 6 and 7 add a precise conjecture on the threshold, finite-field certificates and floating-point experiments for finitely many small parameter ranges, and corollaries for order-3, order-4, and symmetric tensor decomposition.

Significance. If correct, the paper improves the known algorithmic threshold for the JLV procedure from roughly mn/4 to roughly mn/2 and establishes an upper bound near mn/√2, essentially locating the first-order threshold up to the stated conjecture. The proof of Theorem 2.5 is a self-contained symbolic-determinant argument: a square submatrix M′′ is constructed with a block-upper-triangular structure, and Lemma 4.9 verifies that every diagonal block has nonzero determinant by checking that relevant monomials are distinct. The negative theorem is a clean dimension count. Strengths include the parameter-free nature of the threshold, the reproducible code and finite-field certificates for the computer-assisted claims, and the clear labeling of floating-point experiments as non-rigorous.

minor comments (5)

[§7, Corollary 7.2] The second term in the bound should be n3 n4 rather than n2 n3: after flattening along the first two modes, the columns c_i ⊗ d_i live in R^{n3 n4}, so the linear-independence condition is R ≤ n3 n4. This is also consistent with the final paragraph of §7, which states Rmax(n1,n2) and n3 n4.
[§4.4, Eq. (3)] The determinant formula in Eq. (3) appears to have an incorrect scalar factor: the entries of the 2×2 block already contain the factor 1/2 from Lemma 4.8, so the determinant should carry a factor 1/4 rather than 1/2. The nonvanishing conclusion is unaffected because only the zero/nonzero status matters.
[Abstract and §2, Theorem 2.7] The abstract says the algorithm fails when R > (1+o(1))mn/√2, but Theorem 2.7 formally proves only that the identity (Goal) fails. The paper defines this as failure of a key step of the JLV algorithm, and the numerical experiments support the stronger statement, but the abstract should carry the same qualification to avoid overstating the formal result.
[§4.3, Lemma 4.8] In the Class IV zero-entry part of Lemma 4.8, the condition is written as {b,d} ≠ {g(i),g(i)}; it should be {b,d} ≠ {g(i),g(j)}.
[Throughout] Several typographical issues should be cleaned up: the Section 6.1 heading reads “Conjecure”; “neary” appears in §1.3; and the definition of f(v) just before E_abcd says vab vcd − vad vcd and should say vab vcd − vad vcb.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the core rank threshold proof is self-contained, and cited extraction results are external rather than self-citations.

full rationale

The central claim, Theorem 2.5, is proven by reducing (Goal) to full column rank of a symbolic matrix M (Lemma 3.1), constructing a square submatrix M′ whose determinant is shown non-zero by a block-upper-triangular argument (Lemma 4.9). No parameter of the theorem is fitted to the numerical experiments; the threshold R≤1/2(m−1)(n−2) is used only to ensure the injection φ exists, and det(M′′)≠0 is established as a polynomial identity. The negative result, Theorem 2.7, is a dimension count: assuming (Goal) forces C(m,2)C(n,2) ≥ C(R+1,2)−s. The tensor-decomposition corollaries do import Proposition 2.3 (extraction via simultaneous diagonalization) and Proposition 2.4 (s=0 certificate) from [JLV23], but these are external citations by different authors, not self-citations, and they are not used to define the algebraic threshold. The conjectured exact threshold is presented explicitly as Conjecture 6.1, and the computer-aided proofs in Section 6.3 produce witness subspaces and non-zero maximal minors over F_p—evidence independent of the fitted values. There is no step where an input is renamed as a prediction, no uniqueness theorem from the present authors is invoked to force a choice, and no ansatz is smuggled in via citation. The paper is therefore self-contained with respect to its principal algebraic contribution.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The main theorems are parameter-free: no constants are fitted to data. The proof uses standard genericity, the rank-one variety characterization, and imported JLV propositions for the extraction step. The only hand-set computational parameter is the prime 997 used to certify nonzero minors in the finite-case theorems.

free parameters (1)

finite-field modulus p = 997
Hand-chosen prime for the computer-aided certificates in Section 6.3; large enough that pairwise linearly independent vectors exist over F_p^d. It does not enter the main Theorems 2.5 and 2.7.

assumptions (5)

standard math Genericity definition (Definition 2.1): a property holds generically if it holds for all inputs off a measure-zero algebraic set.
Used throughout to convert nonzero determinant polynomials into statements about almost-all inputs.
standard math Rank-one matrices are exactly the variety cut out by all 2x2 minors.
This is the starting point of the algebraic formulation in Section 2.
domain assumption Proposition 2.3 from JLV23: from span{(v_i)^{⊗2}} with linearly independent v_i, the planted vectors can be extracted in polynomial time by simultaneous diagonalization.
Imported without proof; the paper's Corollary 2.6 and tensor decomposition corollaries depend on it.
domain assumption Proposition 2.4 from JLV23: if S2(U)∩span{E_abcd}^⊥ = {0}, then U contains no nonzero rank-one matrix.
Imported without proof; used for the s=0 certificate interpretation.
standard math The tensors E_abcd are linearly independent, so dim(span{E_abcd}) = C(m,2)C(n,2).
Assumed implicitly in the dimension count of Theorem 2.7; true because the monomial v_ab v_cd appears only in E_abcd.

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Pith. "Pith review of Improving the Threshold for Finding Rank-1 Matrices in a Subspace." pith.science (2026). https://pith.science/paper/3VG4GQZ2

@misc{pith2026250417947,
  author       = {Pith},
  title        = {Pith review of: Improving the Threshold for Finding Rank-1 Matrices in a Subspace},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3VG4GQZ2}},
  note         = {Machine review of arXiv:2504.17947}
}

abstract

We consider a basic computational task of finding $s$ planted rank-1 $m \times n$ matrices in a linear subspace $\mathcal{U} \subseteq \mathbb{R}^{m \times n}$ where $\dim(\mathcal{U}) = R \ge s$. The work of Johnston-Lovitz-Vijayaraghavan (FOCS 2023) gave a polynomial-time algorithm for this task and proved that it succeeds when ${R \le (1-o(1))mn/4}$, under minimal genericity assumptions on the input. Aiming to precisely characterize the performance of this algorithm, we improve the bound to ${R \le (1-o(1))mn/2}$ and also prove that the algorithm fails when ${R \ge (1+o(1))mn/\sqrt{2}}$. Numerical experiments indicate that the true breaking point is $R = (1+o(1))mn/\sqrt{2}$. Our work implies new algorithmic results for tensor decomposition, for instance, decomposing order-4 tensors with twice as many components as before.

Figures

Figures reproduced from arXiv: 2504.17947 by the authors.

**Figure 2.** A visualization of Classes I–V, from top to bottom. A tuple ( [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** The structure of the matrix M′′. Columns are indexed by (i, j) pairs, partitioned into Bins 1–4 and visualized as in [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

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Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Efficient Tensor Decomposition via Moment Matrix Extension
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Generic order-4 symmetric tensors of rank up to 2n+1 are efficiently decomposable via moment matrix extension, with a conjectured extension to O(n^2) rank.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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