159 research outputs found
A note on Probably Certifiably Correct algorithms
Many optimization problems of interest are known to be intractable, and while
there are often heuristics that are known to work on typical instances, it is
usually not easy to determine a posteriori whether the optimal solution was
found. In this short note, we discuss algorithms that not only solve the
problem on typical instances, but also provide a posteriori certificates of
optimality, probably certifiably correct (PCC) algorithms. As an illustrative
example, we present a fast PCC algorithm for minimum bisection under the
stochastic block model and briefly discuss other examples
Robustly Learning Mixtures of Arbitrary Gaussians
We give a polynomial-time algorithm for the problem of robustly estimating a
mixture of arbitrary Gaussians in , for any fixed , in the
presence of a constant fraction of arbitrary corruptions. This resolves the
main open problem in several previous works on algorithmic robust statistics,
which addressed the special cases of robustly estimating (a) a single Gaussian,
(b) a mixture of TV-distance separated Gaussians, and (c) a uniform mixture of
two Gaussians. Our main tools are an efficient \emph{partial clustering}
algorithm that relies on the sum-of-squares method, and a novel \emph{tensor
decomposition} algorithm that allows errors in both Frobenius norm and low-rank
terms.Comment: This version extends the previous one to yield 1) robust proper
learning algorithm with poly(eps) error and 2) an information theoretic
argument proving that the same algorithms in fact also yield parameter
recovery guarantees. The updates are included in Sections 7,8, and 9 and the
main result from the previous version (Thm 1.4) is presented and proved in
Section
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