Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information
Emmanuel Candes, Justin Romberg, Terence Tao
TL;DR
This work addresses exact reconstruction of sparse discrete signals from highly incomplete Fourier information by framing the problem as a convex l1-minimization under partial Fourier constraints. The authors prove that, with random frequency sampling Ω of average size τN, a signal supported on a set T with size up to a constant multiple of τN/log N can be recovered exactly with high probability via l1 minimization, and they provide explicit α(M) bounds tied to the desired failure probability. The core method hinges on a duality argument: the existence of a dual polynomial with Fourier support in Ω that interpolates the sign of f on its support and remains strictly below magnitude 1 off the support certifies exact recovery. They develop intricate random-matrix moment bounds and Neumann-series-based constructions to establish invertibility and dual certificates, with extensive numerical experiments validating the theory and extensions to higher dimensions and TV-based penalties for piecewise-constant objects. These results yield a robust, probabilistic uncertainty principle and offer practical pathways for reconstructing signals from highly undersampled frequency data.
Abstract
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 $Ω$ of mean size $τN$. Is it possible to reconstruct $f$ from the partial knowledge of its Fourier coefficients on the set $Ω$? A typical result of this paper is as follows: for each $M > 0$, suppose that $f$ obeys $$ # \{t, f(t) \neq 0 \} \le α(M) \cdot (\log N)^{-1} \cdot # Ω, $$ then with probability at least $1-O(N^{-M})$, $f$ can be reconstructed exactly as the solution to the $\ell_1$ minimization problem $$ \min_g \sum_{t = 0}^{N-1} |g(t)|, \quad \text{s.t.} \hat g(ω) = \hat f(ω) \text{for all} ω\in Ω. $$ In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for $α$ 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 $f$.