Hiding Data Helps: On the Benefits of Masking for Sparse Coding
This addresses a theoretical gap in sparse coding for noisy, over-realized settings, with potential applications in signal processing and computer vision, though it is incremental as it builds on existing self-supervised learning ideas.
The paper tackles the problem of dictionary recovery in sparse coding when the learned dictionary is larger than the ground-truth and data is noisy, showing that standard methods fail, and proposes a masking objective that recovers the ground-truth dictionary optimally as signal increases, with experiments confirming better empirical performance.
Sparse coding, which refers to modeling a signal as sparse linear combinations of the elements of a learned dictionary, has proven to be a successful (and interpretable) approach in applications such as signal processing, computer vision, and medical imaging. While this success has spurred much work on provable guarantees for dictionary recovery when the learned dictionary is the same size as the ground-truth dictionary, work on the setting where the learned dictionary is larger (or over-realized) with respect to the ground truth is comparatively nascent. Existing theoretical results in this setting have been constrained to the case of noise-less data. We show in this work that, in the presence of noise, minimizing the standard dictionary learning objective can fail to recover the elements of the ground-truth dictionary in the over-realized regime, regardless of the magnitude of the signal in the data-generating process. Furthermore, drawing from the growing body of work on self-supervised learning, we propose a novel masking objective for which recovering the ground-truth dictionary is in fact optimal as the signal increases for a large class of data-generating processes. We corroborate our theoretical results with experiments across several parameter regimes showing that our proposed objective also enjoys better empirical performance than the standard reconstruction objective.