Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Optimal Convergence Rates for the Orthogonal Greedy Algorithm   We analyze the orthogonal greedy algorithm when applied to dictionaries
$\mathbb{D}$ whose convex hull has small entropy. We show that if the metric
entropy of the convex hull of $\mathbb{D}$ decays at a rate of
$O(n^{-\frac{1}{2}-\alpha})$ for $\alpha > 0$, then the orthogonal greedy
algorithm converges at the same rate on the variation space of $\mathbb{D}$.
This improves upon the well-known $O(n^{-\frac{1}{2}})$ convergence rate of the
orthogonal greedy algorithm in many cases, most notably for dictionaries
corresponding to shallow neural networks. These results hold under no
additional assumptions on the dictionary beyond the decay rate of the entropy
of its convex hull. In addition, they are robust to noise in the target
function and can be extended to convergence rates on the interpolation spaces
of the variation norm. We show empirically that the predicted rates are
obtained for the dictionary corresponding to shallow neural networks with
Heaviside activation function in two dimensions. Finally, we show that these
improved rates are sharp and prove a negative result showing that the iterates
generated by the orthogonal greedy algorithm cannot in general be bounded in
the variation norm of $\mathbb{D}$.

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The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
19
The categories of this paper are: eess
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
7
The categories of this paper are: hep-th
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
18
The categories of this paper are: econ
9
The categories of this paper are: nlin
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The categories of this paper are: adap-org
15
The categories of this paper are: nucl-ex