Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spectral properties of sample covariance matrices arising from random
  matrices with independent non identically distributed columns   Given a random matrix $X= (x_1,\ldots, x_n)\in \mathcal M_{p,n}$ with
independent columns and satisfying concentration of measure hypotheses and a
parameter $z$ whose distance to the spectrum of $\frac{1}{n} XX^T$ should not
depend on $p,n$, it was previously shown that the functionals
$\text{tr}(AR(z))$, for $R(z) = (\frac{1}{n}XX^T- zI_p)^{-1}$ and $A\in
\mathcal M_{p}$ deterministic, have a standard deviation of order $O(\|A\|_* /
\sqrt n)$. Here, we show that $\|\mathbb E[R(z)] - \tilde R(z)\|_F \leq
O(1/\sqrt n)$, where $\tilde R(z)$ is a deterministic matrix depending only on
$z$ and on the means and covariances of the column vectors $x_1,\ldots, x_n$
(that do not have to be identically distributed). This estimation is key to
providing accurate fluctuation rates of functionals of $X$ of interest (mostly
related to its spectral properties) and is proved thanks to the introduction of
a semi-metric $d_s$ defined on the set $\mathcal D_n(\mathbb H)$ of diagonal
matrices with complex entries and positive imaginary part and satisfying, for
all $D,D' \in \mathcal D_n(\mathbb H)$: $d_s(D,D') = \max_{i\in[n]} |D_i -
D_i'|/ (\Im(D_i) \Im(D_i'))^{1/2}$. Possibly most importantly, the underlying
concentration of measure assumption on the columns of $X$ finds an extremely
natural ground for application in modern statistical machine learning
algorithms where non-linear Lipschitz mappings and high number of classes form
the base ingredients.

34
The categories of this paper are: chem-ph
19
The categories of this paper are: eess
17
The categories of this paper are: q-fin
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
1
The categories of this paper are: cs
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
21
The categories of this paper are: adap-org
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The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph