Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Learning Schatten--von Neumann Operators   We study the learnability of a class of compact operators known as
Schatten--von Neumann operators. These operators between infinite-dimensional
function spaces play a central role in a variety of applications in learning
theory and inverse problems. We address the question of sample complexity of
learning Schatten-von Neumann operators and provide an upper bound on the
number of measurements required for the empirical risk minimizer to generalize
with arbitrary precision and probability, as a function of class parameter $p$.
Our results give generalization guarantees for regression of
infinite-dimensional signals from infinite-dimensional data. Next, we adapt the
representer theorem of Abernethy \emph{et al.} to show that empirical risk
minimization over an a priori infinite-dimensional, non-compact set, can be
converted to a convex finite dimensional optimization problem over a compact
set. In summary, the class of $p$-Schatten--von Neumann operators is probably
approximately correct (PAC)-learnable via a practical convex program for any $p
< \infty$.

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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: bayes-an
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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ex
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The categories of this paper are: q-alg
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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-th
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The categories of this paper are: nlin
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The categories of this paper are: q-fin
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The categories of this paper are: chao-dyn