A scalable implementation of sparse approximation on a field programmable analog array

Abstract

Compressed sensing is an important optimization problem in signal and image processing applications. A Hopfield Network-like analog system is proposed as a solution , using the Locally Competitive Algorithm (LCA) [1] to solve an overcom plete £1 sparse approximation problem. A scalable system archi tecture using sub-threshold currents is described. A 2x3 system is implemented on the RASP 2.9v chip, a Field Programmable Analog Array. The circuit successfully reproduced the outputs of a digital L1LS solver, converging to within 2.5% RMS error, and successfully matching its support vector. The paper concludes by discussing methods for scaling the architecture and including it in compressed sensing systems. I. ANALOG CIRCUITS FOR SPARSE ApPROXIMATIONSparse approximation is an optimization program that seeks to represent a vector (i.e., signal) by using just a few ele ments of a prescribed dictionary. This class of approximation problems plays an important role in producing state-of-the-art results in many signallimage processing and inverse problems, including denoising, restoration, and efficient data acquisition [2], [3]. If the input vector is first linearly compressed, sparse approximation can be used to recover the signal in a Compressed Sensing system [4], [5]. Unfortunately, these optimization problems are computationally expensive, pre venting practical deployment of digital solutions for portable, low-power applications. This paper seeks to demonstrate an implementation of a system for solving this widely used sparse approximation problem via sub-threshold current mode circuits on a Field Programmable Analog Array (FPAA). A. Optimization Problem FormulationSparse approximation is the desire to approximate an input signal y E IR N with a library P = [(PI, ... , ¢ M 1 using as few non-zero coefficients a E IR M as possible. While direct optimization of this objective is intractable, one of the most widely used surrogate objective functions is known as Basis Pursuit De-Noising [6]:The first term in the objective function represents the MSE of the approximation, the second term represents the sparsity of the solution via the £rNorm, Ilalll = L:i la i l, and A is a tradeoff parameter. B. Impact of an Analog ImplementationThe presence of the £l-norm makes (1) non-smooth, pro viding particular challenges for direct gradient descent solvers. Sensory DataLeA on FPAA Sparse Representation Fig. I. The LC A implemented on the FPAA is capable of performing the same sparse encodings as a digital solver, but at a fraction of the power. The Locally Competitive Algorithm (LCA) is a recent algo rithm that uses a modified gradient descent approach [1] to exploit the sparse structure of the solutions. The Hopfield Neural-Network-like architecture of this algorithm makes it amenable to analog circuit implementation, which promises several benefits. In particular, efficient iterative digital algo rithms require O(N2) floating point operations per iteration, whereas the solution time in a parallel architecture scales as O(N) with ...

Analog Sparse Approximation with Applications to Compressed Sensing

“…As mentioned earlier, the development of reconfigurable analog chips [57] have improved many of the issues related to barriers in the design phase of analog integrated circuits. In fact, the reconfigurable platform described in [57] has been used to implement a small version of the LCA for solving BPDN [39]. The preliminary tests of this LCA implementation are on the same order as the simulated solution speeds shown in the present work.…”

“…In an analog system like the LCA, increasing the size of a matrix multiply requires increasing the circuit size and complexity. While this may increase the system time constant in some implementations [39], it does not appear to require any more time constants for the system to settle on a solution. 2…”

Integrated Circuits for Programming Flash Memories in Portable Applications

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“…Channel hot-electron injection is used to precisely set a specific charge on an FG in many programmable analog systems [17]- [18]. In Chapter 2 considerations regarding injection voltages and scaling were discussed.…”

“…17 shows the performance of the memory cell and programmer combination. We programmed the memory cell to 21 different target values (V targ ) spaced evenly from 1.2V to 3.2V,…”