Explicit Bounds for Restricted Invertibility with Prescribed Group Cardinalities

Abstract

We prove an explicit lower spectral bound for independent random vectors with prescribed group multiplicities. Let $v_{ij}\in\mathbb C^d$, $1\le i\le l$, $1\le j\le k_i$, be independent random vectors with finite support and $\mathbb E v_{ij}v_{ij}^*=A_i$, where $\sum_{i=1}^l A_i=I_d$ and $\operatorname{Tr}(A_i)\geε$. Set $k=\max_i k_i$, $n=\max_i\operatorname{rank}(A_i)$, and $K=\sum_{i=1}^l k_i$. If $n>(6+4\sqrt2)k$ and $2\sqrt{nk}-k<ε\le n/2$, then, with positive probability, \[ λ_K\left(\sum_{i=1}^l\sum_{j=1}^{k_i}v_{ij}v_{ij}^*\right) \ge n-(\sqrt{k}+\sqrt{n-ε})^2. \] As a consequence, we obtain a restricted invertibility result selecting exactly $k_i$ vectors from the $i$th group of a Parseval frame. The proof combines interlacing families, a reciprocal transformation of mixed characteristic polynomials, and a multivariate barrier estimate that accommodates repeated differentiation in the same variable.