Recovery of Sparse Translation-Invariant Signals With Continuous Basis Pursuit

Abstract

We consider the problem of decomposing a signal into a linear combination of features, each a continuously translated version of one of a small set of elementary features. Although these constituents are drawn from a continuous family, most current signal decomposition methods rely on a finite dictionary of discrete examples selected from this family (e.g., shifted copies of a set of basic waveforms), and apply sparse optimization methods to select and solve for the relevant coefficients. Here, we generate a dictionary that includes auxiliary interpolation functions that approximate translates of features via adjustment of their coefficients. We formulate a constrained convex optimization problem, in which the full set of dictionary coefficients represents a linear approximation of the signal, the auxiliary coefficients are constrained so as to only represent translated features, and sparsity is imposed on the primary coefficients using an L1 penalty. The basis pursuit denoising (BP) method may be seen as a special case, in which the auxiliary interpolation functions are omitted, and we thus refer to our methodology as continuous basis pursuit (CBP). We develop two implementations of CBP for a one-dimensional translationinvariant source, one using a first-order Taylor approximation, and another using a form of trigonometric spline. We examine the tradeoff between sparsity and signal reconstruction accuracy in these methods, demonstrating empirically that trigonometric CBP substantially outperforms Taylor CBP, which in turn offers substantial gains over ordinary BP. In addition, the CBP bases can generally achieve equally good or better approximations with much coarser sampling than BP, leading to a reduction in dictionary dimensionality.

A unified framework and method for automatic neural spike identification

“…Solving for spike times/amplitudes: CBP-Given the waveforms, solving directly for the spike times/amplitudes is intractable, because the spike times are embedded as continuous arguments within the nonlinear waveform functions. We use a recently developed sparse optimization method known as Continuous Basis Pursuit (CBP) (Ekanadham et al, 2011a), which is designed to handle superpositions of localized waveforms that occur at arbitrary times, coupled with a sparsity-inducing penalty on the amplitudes (Candes, 2008). This reduces the problem to solving a simpler objective function:…”

“…Here, W is a set of basis functions that approximately span the space of continuously timeshifted versions of the waveforms {W n (t − τ)}, and x⃗ is a vector of coefficients, constrained by inequalities, that encode the spike times and amplitudes (Ekanadham et al, 2011a).…”

“…The remaining two coefficients in each group encode the time-shift within the time bin. The convex constraint set, , is defined by a set of inequalities that ensure that the first (amplitude) coefficient is positive, and that the other two coefficients are bounded, such that they are only used when the first coefficient is nonzero, and their effect is to (approximately) time-shift the waveform within the time bin (see Ekanadham et al, 2011a). The second term in Eq.…”

“…Each iteration amounts to a convex optimization problem which can be solved efficiently. Upon convergence, the coefficient groups, {x nij } j=1,2,3 , are converted into spike amplitudes and times (Ekanadham et al, 2011a).…”

“…We use a recently developed method in sparse signal decomposition, known as Continuous Basis Pursuit (Ekanadham et al, 2011a), as the basis for an accurate and efficient approximation of the solution. The resulting method provides a unified procedure for the estimation of continuous-valued spike times that operates correctly in the presence of overlapping spikes, and does not rely on any auxiliary heuristic pre-processing or post-processing such as alignment of spike segments or searching for spike combinations.…”