Sketching Algorithms for Sparse Dictionary Learning: PTAS and Turnstile Streaming
This work addresses the challenge of scalable and space-efficient algorithms for machine learning tasks like clustering and dictionary learning, which is incremental in extending sketching techniques to these problems.
The paper tackles the problem of designing efficient algorithms for sparse dictionary learning and Euclidean k-means clustering, particularly in settings requiring output assignments for all input points, and achieves results including a PTAS for sparse dictionary learning and space upper bounds like O~(nr/ε² + dk/ε) for turnstile streams, along with matching lower bounds.
Sketching algorithms have recently proven to be a powerful approach both for designing low-space streaming algorithms as well as fast polynomial time approximation schemes (PTAS). In this work, we develop new techniques to extend the applicability of sketching-based approaches to the sparse dictionary learning and the Euclidean $k$-means clustering problems. In particular, we initiate the study of the challenging setting where the dictionary/clustering assignment for each of the $n$ input points must be output, which has surprisingly received little attention in prior work. On the fast algorithms front, we obtain a new approach for designing PTAS's for the $k$-means clustering problem, which generalizes to the first PTAS for the sparse dictionary learning problem. On the streaming algorithms front, we obtain new upper bounds and lower bounds for dictionary learning and $k$-means clustering. In particular, given a design matrix $\mathbf A\in\mathbb R^{n\times d}$ in a turnstile stream, we show an $\tilde O(nr/ε^2 + dk/ε)$ space upper bound for $r$-sparse dictionary learning of size $k$, an $\tilde O(n/ε^2 + dk/ε)$ space upper bound for $k$-means clustering, as well as an $\tilde O(n)$ space upper bound for $k$-means clustering on random order row insertion streams with a natural "bounded sensitivity" assumption. On the lower bounds side, we obtain a general $\tildeΩ(n/ε+ dk/ε)$ lower bound for $k$-means clustering, as well as an $\tildeΩ(n/ε^2)$ lower bound for algorithms which can estimate the cost of a single fixed set of candidate centers.