Dictionary Learning for the Almost-Linear Sparsity Regime
This addresses a bottleneck in signal processing and data science by providing the first polynomial-time algorithm for overcomplete dictionaries in linear sparsity, which is incremental but improves upon prior methods limited to orthogonal dictionaries or super-polynomial time.
The paper tackles the problem of dictionary learning in the near-linear sparsity regime, introducing SPORADIC, a spectral method that provably recovers overcomplete dictionaries with restricted isometry property in polynomial time, achieving exact support and sign recovery of sparse vectors with high probability.
Dictionary learning, the problem of recovering a sparsely used matrix $\mathbf{D} \in \mathbb{R}^{M \times K}$ and $N$ $s$-sparse vectors $\mathbf{x}_i \in \mathbb{R}^{K}$ from samples of the form $\mathbf{y}_i = \mathbf{D}\mathbf{x}_i$, is of increasing importance to applications in signal processing and data science. When the dictionary is known, recovery of $\mathbf{x}_i$ is possible even for sparsity linear in dimension $M$, yet to date, the only algorithms which provably succeed in the linear sparsity regime are Riemannian trust-region methods, which are limited to orthogonal dictionaries, and methods based on the sum-of-squares hierarchy, which requires super-polynomial time in order to obtain an error which decays in $M$. In this work, we introduce SPORADIC (SPectral ORAcle DICtionary Learning), an efficient spectral method on family of reweighted covariance matrices. We prove that in high enough dimensions, SPORADIC can recover overcomplete ($K > M$) dictionaries satisfying the well-known restricted isometry property (RIP) even when sparsity is linear in dimension up to logarithmic factors. Moreover, these accuracy guarantees have an ``oracle property" that the support and signs of the unknown sparse vectors $\mathbf{x}_i$ can be recovered exactly with high probability, allowing for arbitrarily close estimation of $\mathbf{D}$ with enough samples in polynomial time. To the author's knowledge, SPORADIC is the first polynomial-time algorithm which provably enjoys such convergence guarantees for overcomplete RIP matrices in the near-linear sparsity regime.