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Suppose x is any exactly k-sparse vector in R^n. We present a class of sparsematrices A, and a corresponding algorithm that we call SHO-FA (for Short andFast) that, with high probability over A, can reconstruct x from Ax. The SHO-FAalgorithm is related to the Invertible Bloom Lookup Tables recently introducedby Goodrich et al., with two important distinctions - SHO-FA relies on linearmeasurements, and is robust to noise. The SHO-FA algorithm is the first tosimultaneously have the following properties: (a) it requires only O(k)measurements, (b) the bit-precision of each measurement and each arithmeticoperation is O (log(n) + P) (here 2^{-P} is the desired relative error in thereconstruction of x), (c) the decoding complexity is O(k) arithmetic operationsand encoding complexity is O(n) arithmetic operations, and (d) if thereconstruction goal is simply to recover a single component of x instead of allof x, with significant probability over A this can be done in constant time.All constants above are independent of all problem parameters other than thedesired success probability. For a wide range of parameters these propertiesare information-theoretically order-optimal. In addition, our SHO-FA algorithmworks over fairly general ensembles of "sparse random matrices", is robust torandom noise, and (random) approximate sparsity for a large range of k. Inparticular, suppose the measured vector equals A(x+z)+e, where z and ecorrespond respectively to the source tail and measurement noise. Underreasonable statistical assumptions on z and e our decoding algorithmreconstructs x with an estimation error of O(||z||_2 +||e||_2). The SHO-FAalgorithm works with high probability over A, z, and e, and still requires onlyO(k) steps and O(k) measurements over O(log n)-bit numbers. This is in contrastto the worst-case z model, where it is known O(k log n/k) measurements arenecessary.

SHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits