Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Adaptivity of deep ReLU network for learning in Besov and mixed smooth
  Besov spaces: optimal rate and curse of dimensionality   Deep learning has shown high performances in various types of tasks from
visual recognition to natural language processing, which indicates superior
flexibility and adaptivity of deep learning. To understand this phenomenon
theoretically, we develop a new approximation and estimation error analysis of
deep learning with the ReLU activation for functions in a Besov space and its
variant with mixed smoothness. The Besov space is a considerably general
function space including the Holder space and Sobolev space, and especially can
capture spatial inhomogeneity of smoothness. Through the analysis in the Besov
space, it is shown that deep learning can achieve the minimax optimal rate and
outperform any non-adaptive (linear) estimator such as kernel ridge regression,
which shows that deep learning has higher adaptivity to the spatial
inhomogeneity of the target function than other estimators such as linear ones.
In addition to this, it is shown that deep learning can avoid the curse of
dimensionality if the target function is in a mixed smooth Besov space. We also
show that the dependency of the convergence rate on the dimensionality is tight
due to its minimax optimality. These results support high adaptivity of deep
learning and its superior ability as a feature extractor.

20
The categories of this paper are: acc-phys
2
The categories of this paper are: math
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc