Beyond Moments: Robustly Learning Affine Transformations with Asymptotically Optimal Error

We present a polynomial-time algorithm for robustly learning an unknown affine transformation of the standard hypercube from samples, an important and well-studied setting for independent component analysis (ICA). Specifically, given an epsilon-corrupted sample from a distribution d obtained by applying an unknown affine transformation X -> aX + b to the uniform distribution on d-dimensional hypercube [-1, 1](d) our algorithm constructs (A) over cap, (b) over cap such that the total variation distance of the distribution (D) over cap from D is O(epsilon) using poly(d) time and samples. Total variation distance is the information-theoretically strongest possible notion of distance in our setting and our recovery guarantees in this distance are optimal up to the absolute constant factor multiplying epsilon. In particular, if the rows of A are normalized to be unit length, our total variation distance guarantee implies a bound on the sum of the l(2) distances between the row vectors of A and A', Sigma(d)(i =1) ||a((i)) - (A) over cap ((i))||(2) = O(epsilon). In contrast, the strongest known prior results only yield an epsilon(O(1)) (relative) bound on the distance between individual a(i)'s and their estimates and translate into an O(d epsilon(O(1))) bound on the total variation distance. Prior algorithms for this problem rely on implementing standard approaches [12] for ICA based on the classical method of moments [18], [32] combined with robust moment estimators. We prove that any approach that relies on method of moments must provably fail to obtain a dimension independent bound on the total error Sigma(i)||a((i)) - (A) over cap ((i))||(2) (and consequently, also in total variation distance). Our key innovation is a new approach to ICA (even to outlier-free ICA) that circumvents the difficulties in the classical method of moments and instead relies on a new geometric certificate of correctness of an affine transformation. Our algorithm, Robust Gradient Descent, is based on a new method that iteratively improves its estimate of the unknown affine transformation whenever the requirements of the certificate are not met.