Convolutional Dictionary Learning With Grid Refinement

Abstract

Given a continuous-time signal that can be modeled as the superposition of localized, time-shifted events from multiple sources, the goal of Convolutional Dictionary Learning (CDL) is to identify the location of the events-by Convolutional Sparse Coding (CSC)-and learn the template for each source-by Convolutional Dictionary Update (CDU). In practice, because we observe samples of the continuous-time signal on a uniformlysampled grid in discrete time, classical CSC methods can only produce estimates of the times when the events occur on this grid, which degrades the performance of the CDU. We introduce a CDL framework that significantly reduces the errors arising from performing the estimation in discrete time. Specifically, we construct an expanded dictionary that comprises, not only discrete-time shifts of the templates, but also interpolated variants, obtained by bandlimited interpolation, that account for continuous-time shifts. For CSC, we develop a novel computationally efficient CSC algorithm, termed Convolutional Orthogonal Matching Pursuit with interpolated dictionary (COMP-INTERP). We benchmarked COMP-INTERP to Contiunuous Basis Pursuit (CBP), the state-of-the-art CSC algorithm for estimating offthe-grid events, and demonstrate, on simulated data, that 1) COMP-INTERP achieves a similar level of accuracy, and 2) is two orders of magnitude faster. For CDU, we derive a novel procedure to update the templates given sparse codes that can occur both on and off the discrete-time grid. We also show that 3) dictionary update with the overcomplete dictionary yields more accurate templates. Finally, we apply the algorithms to the spike sorting problem on electrophysiology recording and show their competitive performance.

Gaussian Process Convolutional Dictionary Learning

“…is the linear operator that shifts h c by n j c,i samples and N j c is the number of occurrences of h c in y j [6].…”

“…We use alternating minimization to solve Eq. ( 2), where {h c } and {x j c } are minimized by alternating between a convolutional sparse coding (CSC) step (optimization for {x j c }) and a convolutional dictionary update (CDU) step (optimization for {h c }) [3], [6], [7]. For CSC, we use Convolutional Orthogonal Matching Pursuit (COMP) [18], [19], which is a greedy algorithm that iteratively identifies a template and its corresponding code that minimizes the residual.…”

“…We update h c in a cyclic manner until convergence and obtain h c . Note that the role of (X j c ) T is to extract the segments of z j,(t+1) c where h (t) c occurs, and take their weighted average [6]. Since K N , the computational complexity of matrix inversion for Σ c , ∇ 2 hc L(y) ∈ R K×K is negligible compared to other operations in IRLS.…”

“…In recent years, the practice of modeling signals as a combination of a few repeated templates has gained popularity [1], such as in the modeling of point spread functions for molecular [2] and astronomical imaging [3], or action potentials in biological signals [4], [5], [6]. This is referred to as convolutional dictionary learning (CDL), where the goal is to estimate the shape and the locations of the shift-invariant templates [7].…”

“…Another line of work imposes total variation or Tikhonov like penalties [10], [11], [12] on the templates. More recently, smooth templates were obtained by passing learned dictionary through pre-designed lowpass filters [3], [6].…”