Random embeddings with an almost Gaussian distortion.(English)Zbl 1501.46010
Authors’ abstract: Let \(X\) be a symmetric, isotropic random vector in \(\mathbb{R}^m\) and let \(X_1 \dots, X_n\) be independent copies of \(X\). We show that under mild assumptions on \(\| X \|_2\) (a suitable thin-shell bound) and on the tail-decay of the marginals \(\langle X, u \rangle\), the random matrix \(A\) whose columns are \(X_i / \sqrt{ m}\) exhibits a Gaussian-like behaviour in the following sense: for an arbitrary subset of \(T \subset \mathbb{R}^n\), the distortion \(\sup_{t \in T} | \| A t \|_2^2 - \| t \|_2^2 |\) is almost the same as if \(A\) were a Gaussian matrix.
A simple outcome of our result is that if \(X\) is a symmetric, isotropic, log-concave random vector and \(n \leq m \leq c_1(\alpha) n^\alpha\) for some \(\alpha > 1\), then with high probability, the extremal singular values of \(A\) satisfy the optimal estimate: \(1 - c_2(\alpha) \sqrt{ n / m} \leq \lambda_{\min} \leq \lambda_{\max} \leq 1 + c_2(\alpha) \sqrt{ n / m}\).
A simple outcome of our result is that if \(X\) is a symmetric, isotropic, log-concave random vector and \(n \leq m \leq c_1(\alpha) n^\alpha\) for some \(\alpha > 1\), then with high probability, the extremal singular values of \(A\) satisfy the optimal estimate: \(1 - c_2(\alpha) \sqrt{ n / m} \leq \lambda_{\min} \leq \lambda_{\max} \leq 1 + c_2(\alpha) \sqrt{ n / m}\).
Reviewer: Dmitry Zaporozhets (Sankt-Peterburg)
MSC:
| 46B09 | Probabilistic methods in Banach space theory |
| 60B20 | Random matrices (probabilistic aspects) |
| 46N30 | Applications of functional analysis in probability theory and statistics |
| 52A23 | Asymptotic theory of convex bodies |
| 60G99 | Stochastic processes |
Keywords:
random embeddings;almost Gaussian distortion;log-concave random vectors;extremal singular values of random matricesCite
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