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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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