Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?

Suppose we are given a vector f in a class F ⊂ ℝN, e.g., a class of digital signals or digital images. How many linear measurements do we need to make about f to be able to recover f to within precision ε in the Euclidean (ℓ2) metric? This paper shows that if the objects of interest are sparse in a fixed basis or compressible, then it is possible to reconstruct f to within very high accuracy from a small number of random measurements by solving a simple linear program. More precisely, suppose that the nth largest entry of the vector |f| (or of its coefficients in a fixed basis) obeys |f|(n) ≤ R n^(1-p), where R > 0 and p > 0. Suppose that we take measurements yk = {f,Xk}, k = 1, . . .,K, where the Xk are N-dimensional Gaussian vectors with independent standard normal entries. Then for each f obeying the decay estimate above for some 0 < p < 1 and with overwhelming probability, our reconstruction f#, defined as the solution to the constraints yk = 〈f#, Xk〉 with minimal ℓ1 norm, obeys [equation]. \n \nThere is a sense in which this result is optimal; it is generally impossible to obtain a higher accuracy from any set of K measurements whatsoever. The methodology extends to various other random measurement ensembles; for example, we show that similar results hold if one observes a few randomly sampled Fourier coefficients of f. In fact, the results are quite general and require only two hypotheses on the measurement ensemble which are detailed.