Pure Relative Structured Matrix Learning with Square-Root Matvecs

Abstract

Let $\mathcal L\subseteq\mathbb R^{n\times m}$ be any known $q$-dimensional linear matrix family, and let an unknown $A$ be accessible only through products $Ax$ and $A^\top y$. A recent result obtains a nearly optimal approximation from $\mathcal L$ in about $\sqrt q$ queries, but incurs factor three and additive error proportional to $\|A\|_F$. Whether sublinear query complexity can give pure $(1+\epsilon)$ relative error was left open.

We resolve this problem with a polynomial-time, nonadaptive algorithm. For a Frobenius-orthonormal basis $B_1,\ldots,B_q$, form the basis-invariant partial trace $S_L=\sum_iB_iB_i^\top$. The algorithm queries the leading eigenspace of $S_L$ exactly from the left, then fits the remaining directions from a Gaussian right sketch. If $\lambda_j$ are the eigenvalues of $S_L$, its constant-success query complexity is

$$\min_s\left\\\{s+C\left[ \begin{aligned} &\max\\\{1,\lambda_{s+1}\\\}\log^2(2q)\\\ &+\lambda_{s+1}/\epsilon \end{aligned}\right]\right\\\}.$$

The transposed profile is also available. Since $\lambda_{s+1}\le q/(s+1)$, every family admits pure relative error with $\widetilde O(\sqrt{q/\epsilon})$ queries, independent of ambient dimensions and $\|A\|F/\mathrm{OPT}$. The key proof is a partial-trace covariance lemma for a Gaussian matrix-valued design. Exact high-leverage queries make its expected Hessian the identity, while the centered regression offset has variance only $O(\lambda{s+1}\mathrm{OPT}^2/k)$. A median construction amplifies success without estimating residual norms. Finally, a symmetric Wishart lower bound for fixed-sparsity approximation yields $\Omega(\sqrt{q/\epsilon})$ adaptive two-sided queries when $q\epsilon$ is bounded below. Thus the universal rate is optimal up to logarithmic factors in this joint regime. We also show that the transformed profile gives pure relative reducible prediction risk under known input and output covariances. For structured positive-definite precisions, it further controls Gaussian KL error and preconditioned condition number.

Contributions

• We give a nonadaptive, polynomial-time algorithm satisfying for every rectangular linear family. Its query complexity is the better of two basis-invariant spectral profiles, one for each query orientation.
• The profile implies a universal $\widetilde O(\sqrt{q/\epsilon})$ upper bound. It has no additive error and no dependence on the ambient dimensions or the ratio $\|A\|_F/\mathrm{OPT}$.
• We prove a Gaussian covariance lemma controlled by a partial trace. The proof combines a dimension-free Schatten moment estimate, an exact fourth-moment calculation, and matrix concentration for unbounded summands.
• We convert the adaptive symmetric Wishart lower bound of into $\Omega(\sqrt{q/\epsilon})$ for general $q$-dimensional linear families when $q\epsilon=\Omega(1)$. This matches our upper bound up to logarithms. A separate GOE construction shows that $\Omega(1/\sqrt\epsilon)$ queries can be necessary even for a one-dimensional family.
• We lift the theorem to covariance-weighted prediction loss without changing its query count. For positive-definite precision matrices, the same output controls both Gaussian KL divergence and preconditioning quality. Circulant, Toeplitz, and known-graph precision families give explicit dimension-adaptive profiles.

Keywords

pure, relative, structured, matrix, learning, square-root, matvecs, known, dimensional

Files

• main.pdf, supplement.pdf
• main.tex, supplement.tex
• references.bib
• aistats2027.sty, fancyhdr.sty
• main.pdf.ots, README_old_2026-08-19.md.ots OpenTimestamps priority proofs

supplement.pdf is the current version and the one to read. supplement_old_2026-08-19.pdf is the file as first published on 2026-08-10, kept only so that its OpenTimestamps proof stays independently verifiable:

ots verify supplement_old_2026-08-19.pdf.ots

The two differ in one constant in the statement of the fixed-sparsity Wishart lower bound quoted from Amsel et al., where the slack in the dimension range is u rather than 1. Every result is unchanged, since only the order u/eps is ever used.

README_old_2026-08-19.md is likewise the README as first published, kept for the same reason.