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Non-asymptotic bounds for the average singular value of a complex Gaussian matrix

  • Luis Daniel Abreu
  • Pratik Patil
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Abstract

Let Gd be a d × d matrix with independent standard complex Gaussian entries, let αℂ(d) be the expected average singular value of Gd/√d, and set Δd := αℂ(d)−αℂ(d+1). The statistic αℂ(d) admits a variational representation as an expected normalized maximum over the unitary group and governs approximation guarantees for the little Grothendieck problem over the unitary group and related unitary registration problems. We obtain a strictly positive lower bound and an upper bound for Δd, both valid in every dimension, together with corresponding bounds for αℂ(d) around the Marchenko--Pastur limit. These bounds match the sharp leading behavior of complete asymptotic expansions for both quantities, whose coefficients are explicitly computable. The proof combines a three-term recurrence for Yd = d3/2αℂ(d), obtained from its continuous dual Hahn representation, along with singularity analysis of the underlying Laguerre moment generating function.

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Cite

BibTeXRIS
BibTeX
@misc{abreu2026nonasymptotic,
  title = {{Non-asymptotic bounds for the average singular value of a complex Gaussian matrix}},
  author = {Luis Daniel Abreu and Pratik Patil},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07802},
  eprint = {2609.07802},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07802},
}
RIS
TY  - UNPB
TI  - Non-asymptotic bounds for the average singular value of a complex Gaussian matrix
AU  - Luis Daniel Abreu
AU  - Pratik Patil
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07802
UR  - https://arxiv.org/abs/2609.07802
AB  - Let G_d be a d × d matrix with independent standard complex Gaussian entries, let α_ℂ(d) be the expected average singular value of G_d/√d, and set Δ_d := α_ℂ(d)−α_ℂ(d+1). The statistic α_ℂ(d) admits a variational representation as an expected normalized maximum over the unitary group and governs approximation guarantees for the little Grothendieck problem over the unitary group and related unitary registration problems. We obtain a strictly positive lower bound and an upper bound for Δ_d, both valid in every dimension, together with corresponding bounds for α_ℂ(d) around the Marchenko--Pastur limit. These bounds match the sharp leading behavior of complete asymptotic expansions for both quantities, whose coefficients are explicitly computable. The proof combines a three-term recurrence for Y_d = d^3/2α_ℂ(d), obtained from its continuous dual Hahn representation, along with singularity analysis of the underlying Laguerre moment generating function.
ER  -