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Libres pensées d'un mathématicien ordinaire Posts

Least singular value of random matrices with independent rows

For every \( {A\in\mathcal{M}_n(\mathbb{C})} \) let us define

\[ s(A):=\min_{\Vert x\Vert_2=1}\Vert Ax\Vert_2 \quad\text{and}\quad \Vert A\Vert:=\max_{\Vert x\Vert_2=1}\Vert Ax\Vert_2. \]

Let \( {X} \) be a random matrix in \( {\mathcal{M}_n(\mathbb{C})} \) with i.i.d. entries of mean \( {m:=\mathbb{E}(X_{11})} \) and unit variance. Fix \( {0<s_-\leq s_+<\infty} \) and let \( {A_1,\ldots,A_n} \) be invertible deterministic matrices in \( {\mathcal{M}_n(\mathbb{C})} \) s.t.

\[ s_- \leq \min_{1\leq k\leq n}s(A_k) \leq \max_{1\leq k\leq n}\Vert A_k\Vert\leq s_+. \]

Let \( {R_1,\ldots,R_n} \) be the rows of \( {X} \) and \( {Y} \) the random matrix with rows \( {R_1A_1,\ldots,R_nA_n} \) .

Conjecture (RV). If \( {X_{11}} \) is sub-Gaussian, i.e. there exists \( {c_0} \) such that for every \( {t\geq0} \),

\[ \mathbb{P}(|X_{11}-m|>t)\leq 2 e^{-c_0t^2} \]

then there exists \( {C>0} \) and \( {c\in(0,1)} \) depending (polynomially) only on \( {m} \), \( {c_0} \), \( {s_{\pm}} \), such that for large enough \( {n} \) and every \( {\varepsilon\geq0} \),

\[ \mathbb{P}(s(Y)\leq \varepsilon) \leq C\varepsilon+c^n. \]

Conjecture (TV). For every \( {a>0} \) there exists \( {b>0} \) depending only on \( {a,c,m,s_{\pm}} \), such that for every deterministic matrix \( {A\in\mathcal{M}_n(\mathbb{C})} \) with \( {\Vert A\Vert=O(n^c)} \) and large enough \( {n} \),

\[ \mathbb{P}(s(Y+A)\leq n^{-b}) \leq n^{-a}. \]

These conjectures involve a transformation of \( {X} \), which leaves invariant the results of Adamczak et al on the smallest singular values of random matrices with i.i.d. centered log-concave rows.

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Louis Antoine

Nous avons en France très peu de docteurs en mathématiques, bien que nos agrégés aient une haute culture. C’est que les Français n’aiment guère le médiocre et, de peur de ne faire qu’assez bien, beaucoup ne font rien qui pourraient faire très bien. Malgré ses brillantes qualités, sans la guerre, M. Antoine n’aurait peut-être jamais fait de travail personnel. Son exemple incite à oser. Henri Lebesgue, in rapport sur la thèse de Louis Antoine, 1921.

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