Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Unified Scalable Equivalent Formulations for Schatten Quasi-Norms   The Schatten quasi-norm can be used to bridge the gap between the nuclear
norm and rank function, and is the tighter approximation to matrix rank.
However, most existing Schatten quasi-norm minimization (SQNM) algorithms, as
well as for nuclear norm minimization, are too slow or even impractical for
large-scale problems, due to the SVD or EVD of the whole matrix in each
iteration. In this paper, we rigorously prove that for any p, p1, p2>0
satisfying 1/p=1/p1+1/p2, the Schatten-p quasi-norm of any matrix is equivalent
to minimizing the product of the Schatten-p1 norm (or quasi-norm) and
Schatten-p2 norm (or quasi-norm) of its two factor matrices. Then we present
and prove the equivalence relationship between the product formula of the
Schatten quasi-norm and its weighted sum formula for the two cases of p1 and
p2: p1=p2 and p1\neq p2. In particular, when p>1/2, there is an equivalence
between the Schatten-p quasi-norm of any matrix and the Schatten-2p norms of
its two factor matrices, where the widely used equivalent formulation of the
nuclear norm can be viewed as a special case. That is, various SQNM problems
with p>1/2 can be transformed into the one only involving smooth, convex norms
of two factor matrices, which can lead to simpler and more efficient algorithms
than conventional methods.
  We further extend the theoretical results of two factor matrices to the cases
of three and more factor matrices, from which we can see that for any 0<p<1,
the Schatten-p quasi-norm of any matrix is the minimization of the mean of the
Schatten-(p3+1)p norms of all factor matrices, where p3 denotes the largest
integer not exceeding 1/p. In other words, for any 0<p<1, the SQNM problem can
be transformed into an optimization problem only involving the smooth, convex
norms of multiple factor matrices.

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The categories of this paper are: bayes-an
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
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The categories of this paper are: quant-ph
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The categories of this paper are: econ
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: nlin
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The categories of this paper are: nucl-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th