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

TL;DR

This work sharpens the threshold for the JLV algorithm, which finds planted rank-1 matrices inside a subspace. By constructing and analyzing a pair of matrices M′′ via a careful combinatorial decomposition, the authors prove a positive result up to R ≤ (1/2)(m−1)(n−2) (and a symmetric analogue), and establish a matching-like negative regime beyond R ≈ mn/√2, supported by extensive numerical evidence and computer-aided certificates. The results yield immediate algorithmic gains for tensor decomposition, enabling roughly a factor-of-two increase in the permissible component count for order-4 and related unbalanced tensors. A systematic combination of exact-symbolic proofs, dimension counting, and numerical experiments underpins the conjectured exact threshold, with symmetric extensions and publicly available code/data enhancing reproducibility and applicability.

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 (3)

• Figure 1: A visualization of Bins 1--4, from top to bottom. An $(i,j)$ pair is visualized as two positions, namely $(f(i),g(i))$ and $(f(j),g(j))$, in the top-left $(m-1) \times k$ block of an $m \times n$ matrix. These two positions are drawn as blue dots. The figures above are representative examples of the various cases that can occur, e.g., $f(i) < f(j)$ vs $f(i) > f(j)$.
• Figure 2: A visualization of Classes I--V, from top to bottom. A tuple $(a,b,c,d)$ is visualized as a $2 \times 2$ submatrix of an $m \times n$ matrix, namely the one with rows $\{a,c\}$ and columns $\{b,d\}$. The 4 positions in the submatrix are drawn as dots, with the blue dots indicating overlap with the corresponding $(i,j)$, visualized as in Figure \ref{['fig:Bin1-4']}.
• Figure 3: The structure of the matrix $M"$. Columns are indexed by $(i,j)$ pairs, partitioned into Bins 1--4 and visualized as in Figure \ref{['fig:Bin1-4']}. In Bin 4, each $(i,j)$ is paired with another $(i',j')$. Rows are indexed by tuples $(a,b,c,d)$, partitioned into Classes I--V and visualized as in Figure \ref{['fig:ClassI-V']}. Classes IV and V are combined in the last block of rows. Each of the first 3 diagonal blocks is a diagonal matrix, with each non-zero entry depicted as a diagram showing how the row and column diagrams overlap. The bottom-right block is a block diagonal matrix with $2 \times 2$ blocks on the diagonal. The blocks with question marks have unknown values that we do not need to control in our proof.

Theorems & Definitions (50)

• Theorem : Informal, see Corollary \ref{['Cor:JLVSuccess']}
• Theorem : Informal, see Theorem \ref{['Thm:JLVFail']}
• Definition 2.1
• Lemma 2.2
• proof
• Proposition 2.3: JLV
• Proposition 2.4: JLV, Observation 11
• Theorem 2.5
• Corollary 2.6
• Theorem 2.7