10 Lectures and 42 Open Problems — Gaussian singular-value monotonicity

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an integer $d\ge1$, let $G_{\mathbb R}^{(d)}$ have independent entries distributed as $\mathcal N(0,1/d)$. Let $G_{\mathbb C}^{(d)}$ have independent entries \[\frac{A_{ij}+iB_{ij}}{\sqrt{2d}},\] where all $A_{ij},B_{ij}$ are independent $\mathcal N(0,1)$ variables, so that each complex entry has mean zero and $\mathbb E|G_{ij}|^2=1/d$. For either field $\mathbb K$, write $\sigma_1(G),\ldots,\sigma_d(G)$ for the nonnegative singular values, the square roots of the eigenvalues of $G^*G$, and define \[\alpha_{\mathbb K}(d)=\frac1d\mathbb E\sum_{j=1}^d\sigma_j\left(G_{\mathbb K}^{(d)}\right).\]

Statement recorded in the supplied source.

Do both inequalities \[\alpha_{\mathbb R}(d+1)\ge\alpha_{\mathbb R}(d),\qquad \alpha_{\mathbb C}(d+1)\le\alpha_{\mathbb C}(d)\] hold for every integer $d\ge1$? [S2]

Short English statement

Does the expected average singular value increase with matrix dimension for normalized real Gaussian matrices and decrease for normalized complex Gaussian matrices?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2800102 (AMR-027-0102). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Afonso S. Bandeira, Christopher Kennedy and Amit Singer, Approximating the Little Grothendieck Problem over the Orthogonal and Unitary Groups (2013; revised 2016), arXiv:1308.5207, Definition 2 and Conjecture 8. https://arxiv.org/pdf/1308.5207.