We study the sample covariance error of centered Gaussians. A remarkable breakthrough [66] established the correct error scaling order and explicitly revealed the critical role of both the effective rank and the true covariance spectrum. In this work, we move beyond scaling characterizations and determine the precise limiting value of the error's spectral norm. To do so, we develop a generic framework based on Random Duality Theory (RDT). Within this framework, we first determine closed-form, explicit RDT-based upper bounds. We then establish complementary lower bounds by introducing a novel bilinear-quadratic RDT lower-bounding mechanism. By combining this mechanism with a two-replica systems bounding strategy, we show that our lower and upper bounds match in large-dimensional contexts. Our theoretical results are supplemented with numerical evaluations and simulations, demonstrating an excellent agreement already for problem sizes on the order of thousands.
This paper derives non-asymptotic quasi-optimality bounds governed by a regularized inverse Christoffel function associated with the gradient field and shows that additional smoothness of the gradient map yields sharper coherence estimates and motivates tractable importance sampling from kernel diagonal measures.
Fabio Nobile, Matteo Raviola, R. Tempone· 0 citations
In this paper, we develop a systematic framework for deriving explicit error bounds for semidefinite feasibility problems without assuming strict feasibility (Slater's condition), a setting in which existing results are limited. Our main technical contribution is the introduction of radial-type H\"{o}lder error bounds,...
Mitsuhiro Nishijima, Guo-Yin Li, Bruno F. Lourenço· 0 citations
We prove explicit upper bounds for the number of eigenvalues of a one-dimensional time-frequency localization operator in an open transition window. An abstract residual-energy principle combines a rank-N approximant with the Hilbert-Schmidt energy of its remainder and retains the strict integer correction imposed by t...
This work proposes PRISM-ZO, a projection-robust framework that samples low-dimensional random tangent subspaces and combines symmetric finite differences with median-of-means or Huber aggregation and establishes the unbiasedness of the correctly rescaled projected direction in expectation over the random subspace.
Yin-Pu Ma, Cunlin Li, Shiyue Zhang· Journal of King Saud Univers...· 0 citations
Restricted eigenvalue (RE) bounds govern stable recovery by norm-regularized estimators. For isotropic sub-Gaussian measurements, the benchmark sample size is $1+w(A)^2$, where $w(A)$ is the Gaussian width of the normalized descent cone. The COLT 2015 open-problem note (Banerjee et al., 2015) asked whether the same law...
Shi Fu, Hui-Bo Xu, Qi-Xin Zhang et al.· 0 citations
Subspace embeddings obtained by two normalized real Walsh transforms, two independent sign diagonals, and uniform coordinate sampling without replacement suffices to preserve all squared norms on each fixed $r$-dimensional subspace within $1\pm\varepsilon$ with probability at least $0.99".
Yu-Ning Yang· 1 citation
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