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On the robust learning mixtures of linear regressions
In this note, we consider the problem of robust learning mixtures of linear
regressions. We connect mixtures of linear regressions and mixtures of
Gaussians with a simple thresholding, so that a quasi-polynomial time algorithm
can be obtained under some mild separation condition. This algorithm has
significantly better robustness than the previous result
Settling the Robust Learnability of Mixtures of Gaussians
This work represents a natural coalescence of two important lines of work:
learning mixtures of Gaussians and algorithmic robust statistics. In particular
we give the first provably robust algorithm for learning mixtures of any
constant number of Gaussians. We require only mild assumptions on the mixing
weights (bounded fractionality) and that the total variation distance between
components is bounded away from zero. At the heart of our algorithm is a new
method for proving dimension-independent polynomial identifiability through
applying a carefully chosen sequence of differential operations to certain
generating functions that not only encode the parameters we would like to learn
but also the system of polynomial equations we would like to solve. We show how
the symbolic identities we derive can be directly used to analyze a natural
sum-of-squares relaxation
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