Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Approximation with Tensor Networks. Part II: Approximation Rates for
  Smoothness Classes   We study the approximation by tensor networks (TNs) of functions from
classical smoothness classes. The considered approximation tool combines a
tensorization of functions in $L^p([0,1))$, which allows to identify a
univariate function with a multivariate function (or tensor), and the use of
tree tensor networks (the tensor train format) for exploiting low-rank
structures of multivariate functions. The resulting tool can be interpreted as
a feed-forward neural network, with first layers implementing the
tensorization, interpreted as a particular featuring step, followed by a
sum-product network with sparse architecture. In part I of this work, we
presented several approximation classes associated with different measures of
complexity of tensor networks and studied their properties. In this work (part
II), we show how classical approximation tools, such as polynomials or splines
(with fixed or free knots), can be encoded as a tensor network with controlled
complexity. We use this to derive direct (Jackson) inequalities for the
approximation spaces of tensor networks. This is then utilized to show that
Besov spaces are continuously embedded into these approximation spaces. In
other words, we show that arbitrary Besov functions can be approximated with
optimal or near to optimal rate. We also show that an arbitrary function in the
approximation class possesses no Besov smoothness, unless one limits the depth
of the tensor network.

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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: physics
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The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: hep-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat