Beyond Moments: Robustly Learning Affine Transformations with Asymptotically Optimal Error
This provides a robust solution for ICA with optimal error guarantees, addressing a foundational challenge in machine learning with potential applications in signal processing and data analysis.
The paper tackles the problem of robustly learning an unknown affine transformation from corrupted samples in independent component analysis, achieving an asymptotically optimal total variation distance error of O(ε) with polynomial time and sample complexity, improving prior bounds from O(dε) to O(ε).
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 $ε$-corrupted sample from a distribution $D$ obtained by applying an unknown affine transformation $x \rightarrow Ax+s$ to the uniform distribution on a $d$-dimensional hypercube $[-1,1]^d$, our algorithm constructs $\hat{A}, \hat{s}$ such that the total variation distance of the distribution $\hat{D}$ from $D$ is $O(ε)$ 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 $ε$. In particular, if the columns of $A$ are normalized to be unit length, our total variation distance guarantee implies a bound on the sum of the $\ell_2$ distances between the column vectors of $A$ and $A'$, $\sum_{i =1}^d \|a_i-\hat{a}_i\|_2 = O(ε)$. In contrast, the strongest known prior results only yield a $ε^{O(1)}$ (relative) bound on the distance between individual $a_i$'s and their estimates and translate into an $O(dε)$ bound on the 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 is based on a new method that iteratively improves an estimate of the unknown affine transformation whenever the requirements of the certificate are not met.