Computational complexity versus statistical performance on sparse recovery problems

Computational Complexity versus Statistical Performance on Sparse Recovery Problems

We show that several classical quantities controlling compressed sensing performance directly match classical parameters controlling algorithmic complexity. We first describe linearly convergent restart schemes on first-order methods solving a broad range of compressed sensing problems, where sharpness at the optimum controls convergence speed. We show that for sparse recovery problems, this sh...

The Computational Complexity of Some Classical Problems from Statistical Physics

On the Performance of Sparse Recovery via

It is known that a high-dimensional sparse vector x∗ in Rn can be recovered from low-dimensional measurements y = Ax∗ where Am×n(m < n) is the measurement matrix. In this paper, with A being a random Gaussian matrix, we investigate the recovering ability of `p-minimization (0 ≤ p ≤ 1) as p varies, where `p-minimization returns a vector with the least `p quasi-norm among all the vectors x satisf...

Sparse Recovery in Inverse Problems

Within this chapter we present recent results on sparse recovery algorithms for inverse and ill-posed problems, i.e. we focus on those inverse problems in which we can assume that the solution has a sparse series expansion with respect to a preassigned basis or frame. The presented approaches to approximate solutions of inverse problems are limited to iterative strategies that essentially rely ...

On sample complexity for computational classification problems

1 In the statistical setting of the classification (pattern recognition) problem the number of examples required to approximate an unknown labelling function is linear in the VC dimension of the target learning class. In this work we consider the question of whether such bounds exist if we restrict our attention to computable classification methods, assuming that the unknown labelling function ...