Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Noisy Low-rank Matrix Optimization: Geometry of Local Minima and
  Convergence Rate   This paper is concerned with low-rank matrix optimization, which has found a
wide range of applications in machine learning. This problem in the special
case of matrix sense has been studied extensively through the notion of
Restricted Isometry Property (RIP), leading to a wealth of results on the
geometric landscape of the problem and the convergence rate of common
algorithms. However, the existing results are not able to handle the problem
with a general objective function subject to noisy data. In this paper, we
address this problem by developing a mathematical framework that can deal with
random corruptions to general objective functions, where the noise model is
arbitrary. We prove that as long as the RIP constant of the noiseless objective
is less than $1/3$, any spurious local solution of the noisy optimization
problem must be close to the ground truth solution. By working through the
strict saddle property, we also show that an approximate solution can be found
in polynomial time. We characterize the geometry of the spurious local minima
of the problem in a local region around the ground truth in the case when the
RIP constant is greater than $1/3$. This paper offers the first set of results
on the global and local optimization landscapes of general low-rank
optimization problems under arbitrary random corruptions.

37
The categories of this paper are: supr-con
31
The categories of this paper are: ao-sci
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
33
The categories of this paper are: plasm-ph
17
The categories of this paper are: q-fin