Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Sparse Principal Component of a Constant-rank Matrix   The computation of the sparse principal component of a matrix is equivalent
to the identification of its principal submatrix with the largest maximum
eigenvalue. Finding this optimal submatrix is what renders the problem
${\mathcal{NP}}$-hard. In this work, we prove that, if the matrix is positive
semidefinite and its rank is constant, then its sparse principal component is
polynomially computable. Our proof utilizes the auxiliary unit vector technique
that has been recently developed to identify problems that are polynomially
solvable. Moreover, we use this technique to design an algorithm which, for any
sparsity value, computes the sparse principal component with complexity
${\mathcal O}\left(N^{D+1}\right)$, where $N$ and $D$ are the matrix size and
rank, respectively. Our algorithm is fully parallelizable and memory efficient.

28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
33
The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th