Robust Sparse Estimation Tasks in High Dimensions
This work addresses robust estimation problems for high-dimensional data analysis, offering incremental algorithmic improvements in the presence of noise.
The paper tackles robust sparse estimation in high dimensions with adversarial corruptions, providing the first efficient algorithms for sparse mean estimation and sparse PCA that achieve non-trivial error guarantees with sample complexities sublinear in the ambient dimension.
In this paper we initiate the study of whether or not sparse estimation tasks can be performed efficiently in high dimensions, in the robust setting where an $\eps$-fraction of samples are corrupted adversarially. We study the natural robust version of two classical sparse estimation problems, namely, sparse mean estimation and sparse PCA in the spiked covariance model. For both of these problems, we provide the first efficient algorithms that provide non-trivial error guarantees in the presence of noise, using only a number of samples which is similar to the number required for these problems without noise. In particular, our sample complexities are sublinear in the ambient dimension $d$. Our work also suggests evidence for new computational-vs-statistical gaps for these problems (similar to those for sparse PCA without noise) which only arise in the presence of noise.