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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 vij∈ℂd, 1≤ i≤ l, 1≤ j≤ ki, be independent random vectors with finite support and 𝔼 vijvij*=Ai, where ∑i=1l Ai=Id and Tr(Ai)≥ε. Set k=maxi ki, n=maxirank(Ai), and K=∑i=1l ki. If n>(6+4√2)k and 2√(nk)−k<ε≤ n/2, then, with positive probability, λK(∑i=1l∑j=1kivijvij*) ≥ n−(√k+√(n−ε))2. As a consequence, we obtain a restricted invertibility result selecting exactly ki vectors from the ith 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.
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BibTeX
@misc{chen2026explicit,
title = {{Explicit Bounds for Restricted Invertibility with Prescribed Group Cardinalities}},
author = {Xinyu Chen and Bochen Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15583},
eprint = {2610.15583},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15583},
}
RIS
TY - UNPB
TI - Explicit Bounds for Restricted Invertibility with Prescribed Group Cardinalities
AU - Xinyu Chen
AU - Bochen Liu
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15583
UR - https://arxiv.org/abs/2610.15583
AB - We prove an explicit lower spectral bound for independent random vectors with prescribed group multiplicities. Let v_ij∈ℂ^d, 1≤ i≤ l, 1≤ j≤ k_i, be independent random vectors with finite support and 𝔼 v_ijv_ij^*=A_i, where ∑_i=1^l A_i=I_d and Tr(A_i)≥ε. Set k=max_i k_i, n=max_irank(A_i), and K=∑_i=1^l k_i. If n>(6+4√2)k and 2√(nk)−k<ε≤ n/2, then, with positive probability, λ_K(∑_i=1^l∑_j=1^k_iv_ijv_ij^*) ≥ n−(√k+√(n−ε))^2. As a consequence, we obtain a restricted invertibility result selecting exactly k_i vectors from the ith 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.
ER -