10 Lectures and 42 Open Problems — Improvement over Non-commutative Khintchine inequality

Definitions and mathematical statement

Let $d,n\geq1$ be integers, let $A_1,\ldots,A_n\in\mathbb R^{d\times d}$ be symmetric, and let $g_1,\ldots,g_n$ be independent standard real Gaussian variables. Put $X=\sum_{k=1}^ng_kA_k$ and use the Euclidean operator norm. The conventional variance parameters for this question are \[ \sigma=\left\|\sum_{k=1}^n A_k^2\right\|^{1/2},\qquad \sigma_* =\sup_{\|u\|_2=\|v\|_2=1} \left(\sum_{k=1}^n|u^{\mathsf T}A_kv|^2\right)^{1/2}. \] The source displays these symbols without definitions, so the conventional choices are explicit rather than asserted to have been recovered from its blog text.

Proposed inequality. Is there an absolute constant $C<\infty$, independent of $n,d$ and all the matrices, such that \[ \mathbb E\|X\|\leq C\big(\sigma+\sqrt{\log d}\,\sigma_*\big) \] for every such Gaussian matrix series? The notation $\lesssim$ in the source means such a dimension-independent constant.

The supplied research summary reports that this is false, citing Bandeira--Boedihardjo--van Handel, Proposition 8.2, and reports failure even if $\sqrt{\log d}$ is replaced by $(\log d)^\beta$ for any fixed exponent $\beta$. Its reported family consists of tensor sums \[ X_{n,N}=\sum_{k=1}^n I_N\otimes\cdots\otimes G_k^N\otimes\cdots\otimes I_N \] of independent normalized Wigner matrices in distinct tensor factors, of dimension $d=N^n$, with reported scales $\sigma=\sqrt n$, $\sigma_*=O(\sqrt{n/N})$, and $\mathbb E\|X_{n,N}\|\geq(2-o(1))n$. The record does not supply the precise Wigner normalization or limiting quantifiers, so this is retained as the cited counterexample report rather than a reconstructed proof.

Short English statement

Can the logarithmic loss in a Gaussian matrix norm bound be confined to the weak variance parameter? The supplied Proposition 8.2 attribution reports a tensor-Wigner counterexample, including failure of every fixed logarithmic power replacement.

Sources

• Afonso S. Bandeira, 42 Open Problems in Data Science (source list labelled 2016), Open Problem 4.1, author blog https://afonsobandeira.wordpress.com/2015/10/25/10l42concentration/; HTML extraction, accessed 2026-07-29. The registry gives proposed year 2015.
• Bandeira--Boedihardjo--van Handel, arXiv:2108.06312, https://arxiv.org/abs/2108.06312; Inventiones Mathematicae 234 (2023), Proposition 8.2. The supplied summary says that proposition explicitly disproves this open-list conjecture; no article title or page range is supplied.
• Local registry data/unsolvedmath/problems.json, permanent ID $2800401$, source code catalogue entry, 10 Lectures and 42 Open Problems --- Improvement over Non-commutative Khintchine inequality. The record labels this solved and SOLVED-IN-LITERATURE, with the conjecture false; its source-status and rights checks remain marked NEEDS_REVIEW.