  We present a general framework for designing efficient algorithms for
unsupervised learning problems, such as mixtures of Gaussians and subspace
clustering. Our framework is based on a meta algorithm that learns arithmetic
circuits in the presence of noise, using lower bounds. This builds upon the
recent work of Garg, Kayal and Saha (FOCS 20), who designed such a framework
for learning arithmetic circuits without any noise. A key ingredient of our
meta algorithm is an efficient algorithm for a novel problem called Robust
Vector Space Decomposition. We show that our meta algorithm works well when
certain matrices have sufficiently large smallest non-zero singular values. We
conjecture that this condition holds for smoothed instances of our problems,
and thus our framework would yield efficient algorithms for these problems in
the smoothed setting.
