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Sharp Time--Data Tradeoffs for Linear Inverse Problems
In this paper we characterize sharp time-data tradeoffs for optimization
problems used for solving linear inverse problems. We focus on the minimization
of a least-squares objective subject to a constraint defined as the sub-level
set of a penalty function. We present a unified convergence analysis of the
gradient projection algorithm applied to such problems. We sharply characterize
the convergence rate associated with a wide variety of random measurement
ensembles in terms of the number of measurements and structural complexity of
the signal with respect to the chosen penalty function. The results apply to
both convex and nonconvex constraints, demonstrating that a linear convergence
rate is attainable even though the least squares objective is not strongly
convex in these settings. When specialized to Gaussian measurements our results
show that such linear convergence occurs when the number of measurements is
merely 4 times the minimal number required to recover the desired signal at all
(a.k.a. the phase transition). We also achieve a slower but geometric rate of
convergence precisely above the phase transition point. Extensive numerical
results suggest that the derived rates exactly match the empirical performance
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