Maximal Submatrix Bounds Without Unconditionality

Definitions and mathematical statement

Let $n,N\geq 1$ be integers, and let $X_1,\ldots,X_n$ be independent random vectors in ${\mathbb{R}}^N$, without an assumption that they are identically distributed. Each vector is assumed to be isotropic, meaning \[ {\mathbb{E}} X_i=0,\qquad {\mathbb{E}}[X_iX_i^{\mathsf T}]=I_N, \] and log-concave, meaning that its distribution has a density $f_i$ satisfying \[ f_i((1-t)x+ty)\geq f_i(x)^{1-t}f_i(y)^t \quad(x,y\in{\mathbb{R}}^N,\ 0<t<1). \] A vector $X_i$ is unconditional if, for every $\varepsilon\in\{-1,1\}^N$, the vectors $X_i$ and $(\varepsilon_1X_{i1},\ldots,\varepsilon_NX_{iN})$ have the same distribution. Unconditionality is the hypothesis whose removal is being asked about.

Let $A\in{\mathbb{R}}^{n\times N}$ have rows $X_i^{\mathsf T}$. For integers $1\leq k\leq n$ and $1\leq m\leq N$, and index sets $J\subseteq\{1,\ldots,n\}$, $I\subseteq\{1,\ldots,N\}$ of sizes $k$ and $m$, respectively, let $A(J,I)$ be the corresponding submatrix. Define \[ A_{k,m}=\max_{\substack{J\subseteq\{1,\ldots,n\},\ |J|=k\\ I\subseteq\{1,\ldots,N\},\ |I|=m}} \|A(J,I)\|_{2\to2}, \qquad \|B\|_{2\to2}=\sup_{\|u\|_2=1}\|Bu\|_2. \] Source question. Without assuming unconditionality, does the following estimate hold with high probability? \[ A_{k,m}\leq C\left(\sqrt{m\log\frac{3N}{m}}+\sqrt{k\log\frac{3n}{k}}\right). \] Here $\log$ denotes the natural logarithm, and $C$ represents the implicit multiplicative constant in the supplied notation $\lesssim$. The source record does not specify the failure probability, the permitted dependencies of $C$, or whether one probability event must give the estimate simultaneously for all admissible $(k,m)$. Consequently, the displayed question preserves the supplied target but does not determine a complete quantitative probability assertion; these qualifications require clarification from the underlying source.

Short English statement

The question asks whether independent isotropic log-concave rows satisfy the stated bound on the largest Euclidean operator norm of a submatrix, even without invariance under coordinate sign changes. The supplied record leaves the meaning of "with high probability," the constant dependencies, and simultaneous validity over submatrix sizes unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $30001781$, source problem code catalogue entry, title Maximal Submatrix Bounds Without Unconditionality; supplied complete source record associated with data/unsolvedmath/problems.json.
• Random Matrices, Geometric Functional Analysis and Algorithms (2011), Oberwolfach Reports, pp. 1287--1360, DOI 10.4171/owr/2011/24. Supplied source URL and literature reference labeled "original OWR report": https://doi.org/10.4171/owr/2011/24. The record attributes the problem to workshop open-problem discussions documented in this report and supplies no named proposer.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open, says that the report leaves removal of unconditionality as a conjecture, and describes available work as retaining hypotheses or providing related special cases. This is supplied dated curation, not an independently verified literature assessment.
• The supplied statement status is corrected_verified; its verification note reports that the official report was checked and that the random-matrix setup, maximal submatrix norm, target estimate, and unconditionality question were restored. This is a supplied verification claim, not verification performed for this draft.