Shu Wen | Search arXiv

arXiv2026

Universality of smallest singular values of random row submatrices and phase retrieval problems

We establish universal logarithmic laws for extreme least singular values of random row submatrices and for the global stability of real phase retrieval near its injectivity threshold. Let $A_m\in\mathbb{R}^{N_m\times m}$ have i.i.d. entries distributed as a centered, variance-one real subgaussian random variable with a bounded Lebesgue density, and suppose $N_m/m\toγ>1$. For integers $0\le r_m\le N_m-m$, let $M_{m,r_m}(A_m)$ be the minimum least singular value over all row submatrices of $A_m$ with $m+r_m$ rows. Writing $k_m=r_m+1$, we prove that whenever $k_m\log m=o(m)$, $(k_m/m)\log M_{m,r_m}(A_m)$ converges in probability to $-h(γ)$, where $h(γ)=γ\logγ-(γ-1)\log(γ-1)$. This covers both fixed and moderately growing surplus. The limiting rate is universal within the stated class of entry distributions. For real phase retrieval from $N_m=2m-1+r_m$ amplitude measurements, let $L_{A_m}$ denote the optimal lower Lipschitz constant of the map $x\mapsto|A_mx|$, with signals identified up to sign, and let $κ_{\mathrm{ph}}(A_m)=\|A_m\|_{\mathrm{op}}/L_{A_m}$ be its scale-invariant global condition number. In the same surplus regime, $(k_m/m)\log L_{A_m}$ and $(k_m/m)\logκ_{\mathrm{ph}}(A_m)$ converge in probability to $-\log 4$ and $\log 4$, respectively. Although these measurements determine every signal up to sign almost surely, the condition number grows exponentially in dimension for fixed surplus and exceeds every fixed power of $m$ throughout the stated regime, with probability tending to one. These conclusions quantify the distinction between uniqueness and uniform stability under measurement noise. All stated limits also hold almost surely on any common probability space, without requiring independence across dimensions.

math.PR↗

arXiv2026

Extreme least singular values of random row submatrices with bounded-density subgaussian entries

Let $ξ$ be a centered real subgaussian random variable with positive variance and a bounded Lebesgue density, and let $A_m\in\mathbb{R}^{N_m\times m}$ have independent entries distributed as $ξ$, where $N_m/m\toγ>1$. For each set $I\subset[N_m]$ with $|I|=m$, let $(A_m)_I$ denote the row submatrix indexed by $I$, and define $M_m(A_m):=\min_{I\subset[N_m],\,|I|=m}σ_{\min}((A_m)_I)$. We determine its exponential scale: $\frac{1}{m}\log M_m(A_m)\xrightarrow{\mathbb{P}}-h(γ)$, where $h(γ):=γ\logγ-(γ-1)\log(γ-1)$. This extends the corresponding real Gaussian result. The main new ingredient is an upper-tail argument that avoids uniform control over exponentially many random hyperplanes. We combine a density-level local central limit theorem for delocalized directions, an averaged delocalization estimate for hyperplane normals, an exponential bound for nearly parallel pairs, and amplification using a linear number of independent probe rows. For every fixed $\varepsilon\in(0,h(γ))$, the probability of an $\varepsilon$-deviation is at most $C\exp(-c\sqrt{m})$ for all sufficiently large $m$. Under the canonical coupling induced by a single infinite i.i.d. array, this summable deviation estimate yields a uniform almost-sure exponential law over every compact range of aspect ratios. In particular, at the real phase-retrieval threshold $N_m=2m-1$, the Balan--Wang stability parameter has exponential base $1/4$ in probability and, under this coupling, almost surely.

math.PR↗