Uncertainty Relations and Sparse Signal Recovery for Pairs of General Signal Sets

We present an uncertainty relation for the representation of signals in two different general (possibly redundant or incomplete) signal sets. This uncertainty relation is relevant for the analysis of signals containing two distinct features each of which can be described sparsely in a suitable general signal set. Furthermore, the new uncertainty relation is shown to lead to improved sparsity thresholds for recovery of signals that are sparse in general dictionaries. Specifically, our results improve on the well-known $(1+1/d)/2$-threshold for dictionaries with coherence $d$ by up to a factor of two. Furthermore, we provide probabilistic recovery guarantees for pairs of general dictionaries that also allow us to understand which parts of a general dictionary one needs to randomize over to "weed out" the sparsity patterns that prohibit breaking the square-root bottleneck.Comment: submitted to IEEE Trans. Inf. Theor

Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions

We develop a robust uncertainty principle for finite signals in C^N which states that for almost all subsets T,W of {0,...,N-1} such that |T|+|W| ~ (log N)^(-1/2) N, there is no sigal f supported on T whose discrete Fourier transform is supported on W. In fact, we can make the above uncertainty principle quantitative in the sense that if f is supported on T, then only a small percentage of the energy (less than half, say) of its Fourier transform is concentrated on W. As an application of this robust uncertainty principle (QRUP), we consider the problem of decomposing a signal into a sparse superposition of spikes and complex sinusoids. We show that if a generic signal f has a decomposition using spike and frequency locations in T and W respectively, and obeying |T| + |W| <= C (\log N)^{-1/2} N, then this is the unique sparsest possible decomposition (all other decompositions have more non-zero terms). In addition, if |T| + |W| <= C (\log N)^{-1} N, then this sparsest decomposition can be found by solving a convex optimization problem.Comment: 25 pages, 9 figure

Uncertainty Relations for Shift-Invariant Analog Signals

The past several years have witnessed a surge of research investigating various aspects of sparse representations and compressed sensing. Most of this work has focused on the finite-dimensional setting in which the goal is to decompose a finite-length vector into a given finite dictionary. Underlying many of these results is the conceptual notion of an uncertainty principle: a signal cannot be sparsely represented in two different bases. Here, we extend these ideas and results to the analog, infinite-dimensional setting by considering signals that lie in a finitely-generated shift-invariant (SI) space. This class of signals is rich enough to include many interesting special cases such as multiband signals and splines. By adapting the notion of coherence defined for finite dictionaries to infinite SI representations, we develop an uncertainty principle similar in spirit to its finite counterpart. We demonstrate tightness of our bound by considering a bandlimited lowpass train that achieves the uncertainty principle. Building upon these results and similar work in the finite setting, we show how to find a sparse decomposition in an overcomplete dictionary by solving a convex optimization problem. The distinguishing feature of our approach is the fact that even though the problem is defined over an infinite domain with infinitely many variables and constraints, under certain conditions on the dictionary spectrum our algorithm can find the sparsest representation by solving a finite-dimensional problem.Comment: Accepted to IEEE Trans. on Inform. Theor

Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information

This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f \in \C^N and a randomly chosen set of frequencies $\Omega$ of mean size $\tau N$. Is it possible to reconstruct $f$ from the partial knowledge of its Fourier coefficients on the set $\Omega$? A typical result of this paper is as follows: for each $M > 0$, suppose that $f$ obeys # \{t, f(t) \neq 0 \} \le \alpha(M) \cdot (\log N)^{-1} \cdot # \Omega, then with probability at least $1-O(N^{-M})$, $f$ can be reconstructed exactly as the solution to the $\ell_1$ minimization problem $$\min_g \sum_{t = 0}^{N-1} |g(t)|, \quad \text{s.t.} \hat g(\omega) = \hat f(\omega) \text{for all} \omega \in \Omega.$$ In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for $\alpha$ which depends on the desired probability of success; except for the logarithmic factor, the condition on the size of the support is sharp. The methodology extends to a variety of other setups and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one or two-dimensional) object from incomplete frequency samples--provided that the number of jumps (discontinuities) obeys the condition above--by minimizing other convex functionals such as the total-variation of $f$

Analysis of Basis Pursuit Via Capacity Sets

Finding the sparsest solution $\alpha$ for an under-determined linear system of equations $D\alpha=s$ is of interest in many applications. This problem is known to be NP-hard. Recent work studied conditions on the support size of $\alpha$ that allow its recovery using L1-minimization, via the Basis Pursuit algorithm. These conditions are often relying on a scalar property of $D$ called the mutual-coherence. In this work we introduce an alternative set of features of an arbitrarily given $D$, called the "capacity sets". We show how those could be used to analyze the performance of the basis pursuit, leading to improved bounds and predictions of performance. Both theoretical and numerical methods are presented, all using the capacity values, and shown to lead to improved assessments of the basis pursuit success in finding the sparest solution of $D\alpha=s$