On the Identifiability of Sparse Vectors from Modulo Compressed Sensing

On the Identifiability of Sparse Vectors from Modulo Compressed Sensing Measurements Dheeraj Prasanna, Chandrasekhar Sriram and Chandra R. Murthy Abstract—Compressed sensing deals with recovery of sparse pass filtered spikes in a continuous-time sparse signal, and signals from low dimensional projections, but under the as- developed a new sampling theorem and a signal recovery sumption that the measurement setup has infinite dynamic algorithm. In [10], the authors studied the quantization of range. In this paper, we consider a system with finite dynamic range, and to counter the clipping effect, the measurements oversampled signals in the SR-ADC architecture with the crossing the range are folded back into the dynamic range of goal of reducing the overload distortion error. the system through modulo arithmetic. For this setup, we derive A novel HDR imaging system that employs SR-ADCs theoretical results on the minimum number of measurements required for unique recovery of sparse vectors. We also show to overcome limitations due to limited dynamic range was that recovery using the minimum number of measurements is studied in [2], [11]. Mathematically, this involves applying achievable by using a measurement matrix whose entries are an SR-ADC individually to multiple linear measurements of independently drawn from a continuous distribution. Finally, the images, and is termed as modulo compressed sensing we present an algorithm based on convex relaxation and (modulo-CS) [11]. Modulo-CS can also help overcome is- develop a mixed integer linear program (MILP) for recovering sparse signals from the modulo measurements. Our empirical sues introduced by signal clipping in other signal acquisition results demonstrate that the minimum number of measure- systems where compressed measurements of sparse signals ments required for recovery using the MILP algorithm is close are available, such as communication systems [4], [12]. to the theoretical result for signals with low variance. By exploiting the sparsity of the signal, the modulo-CS Index Terms—Modulo compressed sensing, `1 recovery. setup can be used to overcome the losses caused by limited dynamic range. In the somewhat restrictive setting where the modulo-CS measurements are assumed to span at most I. INTRODUCTION two periods, [11] proposes an algorithm and analyzes the The effect of dynamic range in data acquisition systems sample complexity under Gaussian measurement matrices. has been an important research area in signal processing [1]– In [12], a generalized approximate message passing algo- [4]. Systems with low dynamic range lead to signal loss due rithm tailored to modulo-CS was proposed by assuming a to clipping, and high dynamic range systems with finite res- Bernoulli-Gaussian distribution on the sparse signal. The olution sampling are affected by high quantization noise. A results in these papers suggest that sparsity is very useful direction of research in recent years to counter this problem in the recovery of signals from modulo-CS measurements. has been the so-called self-reset analog to digital converters In the context of the above, our contributions in this (SR-ADCs) [5], [6], which fold the amplitudes back into the paper are twofold: (a) we derive necessary and sufficient dynamic range of the ADCs using the modulo arithmetic, conditions on the measurement matrix under which sparse thus mitigating the clipping effect. However, these systems signals are identifiable under modulo-CS measurements, and encounter information loss due to the modulo operation. The (b) we present a novel algorithm for modulo-CS recovery transfer function of the SR-ADC with parameter λ is and derive its theoretical guarantees. To elaborate: s t 1{ 1 1) We derive necessary and sufficient conditions for unique Mλ(t) = 2λ + − ; (1) 2λ 2 2 recovery of sparse signals in the modulo-CS setup. 2) We show that the minimum number of measurements where t , t − btc is the fractional part of t [7]. J K m to uniquely reconstruct every s-sparse signal from In the context of SR-ADCs, an alternative sampling theory modulo measurements is 2s + 1. called the unlimited sampling framework was developed in 3) We also show that m = 2s + 1 is sufficient, and that a [7], [8], which provides sufficient conditions on the sampling measurement matrix with 2s+1 rows and entries drawn rate for guaranteeing the recovery of band-limited signals independently from any continuous distribution satisfies from its folded samples. Extending these results, the work the identifiability conditions with high probability. in [9] considered the inverse problem of recovering K low 4) We present a mixed integer linear program for modulo- CS recovery via convex relaxation. We also identify an The authors are with the Dept. of ECE, Indian Institute of Science, Bengaluru, India. E-mails: [email protected], integer range space property, which guarantees exact [email protected], [email protected]. sparse signal recovery via the relaxed problem. Notation: Bold lowercase and uppercase letters denote Lemma 1 (Necessary and sufficient conditions). Any vector N vectors and matrices, respectively, and script styled letters x satisfying kxk0 ≤ s < 2 is an unique solution to the denote sets. A vector supported on index set S is denoted optimization problem (P0) if and only if any 2s columns of m as xS and AS denotes the sub-matrix with columns of A matrix A are linearly independent of all v 2 Z . corresponding to set S. The ` -norm of a vector, kxk , is the 0 0 Proof. We first prove sufficiency by contradiction. Let z = number of nonzero entries in x. The inner product between Ax , x is an s-sparse vector, and A 2 m×N . Suppose the a and x is denoted by ha; xi. For a set S, jSj and Sc denotes R optimizationJ K problem (P0) returned another s-sparse vector the cardinality and the complement of the set, respectively. # # x (so that kx k0 ≤ s), then II. MODULO COMPRESSED SENSING # # m A(x − x ) = v ) AS (xS − xS ) = v 2 Z ; N Let x 2 R denote an s-sparse vector, i.e., kxk0 ≤ s, where the set S is the union of the supports of x and x#. N with s < 2 . For ease of exposition, instead of SR-ADC Since jSj ≤ 2s, a set of 2s columns of A span an integer transfer function given in (1), we consider an equivalent vector v, which violates the condition in the Lemma. modular arithmetic which returns the fractional part of a real number, i.e., it returns t t−btc. We obtain m projections To prove the necessary part, suppose 9 S such that jSj = , 2s m of x as follows: J K 2s and 0 6= u 2 R such that ASu = v, where v 2 Z . We construct two s-sparse vectors x0; x# 2 RN from u, 0 zi = hai; xi ; i = 1; 2; : : : ; m: (2) where the first s indices of S constitute the support of x J K with the values equal to first s entries of u, and the remaining Usually, m ≤ N in the compressed sensing paradigm, but s entries of S constitute the support of x# with the values we will also present extensions to dense vectors (s ≥ N ) in 2 equal to the corresponding s entries of u. Also, define z = the overdetermined system setup (m > N). Ax0 and y0 = Ax0, so that y0 = z + v0 for some J 0 K m # 0 Stacking the projections hai; xi as a vector y, we can v 2 Z . Then, using ASu = v, we have Ax + Ax = rewrite (2) in a form similar to the CS framework as v which implies y0 − v = −Ax#. Thus, −x# is also a # solution to the optimization problem since kx k0 ≤ s and z = y = Ax ; (3) # 0 J K J K −Ax = y − v = z, which is a contradiction. T m×N J K J K where A = a1 a2 ::: am 2 R is the measurement matrix and · represents the element wise modulo-1 The following corollary presents a similar result for the operation on a vector,J K as before. recovery of dense vectors, which requires m > N. N The non-linearity introduced by the modulo operation Corollary 1. Any vector x satisfying kxk0 ≥ 2 is a unique along with the underdetermined compressive measurements solution to Aw = y with y = Ax + v and v 2 Zm if could lead to an indeterminate system, i.e., it may not have a only if the columns of matrix AJ areK linearly independent unique solution. In this paper, we explore the role of sparsity of all v 2 Zm. Consequently, the minimum number of in uniquely recovering an s-sparse input signal x from the measurements required for unique recovery is m = N + 1. modulo-CS measurements z obtained using (3). Proof. The proof is similar to Lemma 1, with the observation m N # N # P0 formulation: Any real valued vector y 2 R can be that when kxk0 ≥ and kx k0 ≥ , x − x can be any m 2 2 uniquely decomposed as y = z + v; where z 2 [0; 1) and N length real vector. v 2 Zm denote the fractional part and integer part (the floor function) of y, respectively. Using this decomposition, the To compare the modulo-CS problem to the standard CS non-linearity in (3) can be represented using a linear equation problem, we state two necessary conditions for modulo-CS Ax = z + v. Now, consider the optimization problem: recovery below. The proof is immediate from Lemma 1. m arg min kwk0 subject to Aw = z + v; v 2 Z : (P0) Corollary 2. The following two conditions are necessary for w;v recovering any vector x satisfying kxk0 ≤ s as a unique 0 ∗ 0 ∗ Any s -sparse solution x to (P0) satisfies s ≤ s (i.e., x solution of the optimization problem (P0): is s-sparse), since x is s-sparse and satisfies the constraints of (P0).

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