Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fast global convergence of gradient methods for high-dimensional
  statistical recovery   Many statistical $M$-estimators are based on convex optimization problems
formed by the combination of a data-dependent loss function with a norm-based
regularizer. We analyze the convergence rates of projected gradient and
composite gradient methods for solving such problems, working within a
high-dimensional framework that allows the data dimension $\pdim$ to grow with
(and possibly exceed) the sample size $\numobs$. This high-dimensional
structure precludes the usual global assumptions---namely, strong convexity and
smoothness conditions---that underlie much of classical optimization analysis.
We define appropriately restricted versions of these conditions, and show that
they are satisfied with high probability for various statistical models. Under
these conditions, our theory guarantees that projected gradient descent has a
globally geometric rate of convergence up to the \emph{statistical precision}
of the model, meaning the typical distance between the true unknown parameter
$\theta^*$ and an optimal solution $\hat{\theta}$. This result is substantially
sharper than previous convergence results, which yielded sublinear convergence,
or linear convergence only up to the noise level. Our analysis applies to a
wide range of $M$-estimators and statistical models, including sparse linear
regression using Lasso ($\ell_1$-regularized regression); group Lasso for block
sparsity; log-linear models with regularization; low-rank matrix recovery using
nuclear norm regularization; and matrix decomposition. Overall, our analysis
reveals interesting connections between statistical precision and computational
efficiency in high-dimensional estimation.

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The categories of this paper are: hep-ph
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
13
The categories of this paper are: quant-ph