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We consider estimating a piecewise-constant image, or a gradient-sparsesignal on a general graph, from noisy linear measurements. We propose and studyan iterative algorithm to minimize a penalized least-squares objective, with apenalty given by the "l_0-norm" of the signal's discrete graph gradient. Themethod proceeds by approximate proximal descent, applying the alpha-expansionprocedure to minimize a proximal gradient in each iteration, and using ageometric decay of the penalty parameter across iterations. Under acut-restricted isometry property for the measurement design, we prove globalrecovery guarantees for the estimated signal. For standard Gaussian designs,the required number of measurements is independent of the graph structure, andimproves upon worst-case guarantees for total-variation (TV) compressed sensingon the 1-D and 2-D lattice graphs by polynomial and logarithmic factors,respectively. The method empirically yields lower mean-squared recovery errorcompared with TV regularization in regimes of moderate undersampling andmoderate to high signal-to-noise, for several examples of changepoint signalsand gradient-sparse phantom images.

Iterative Alpha Expansion for estimating gradient-sparse signals from linear measurements