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Recovery of an N-dimensional, K-sparse solution $${\mathbf {x}}$$ x from an M-dimensional vector of measurements $${\mathbf {y}}$$ y for multivariate linear regression can be accomplished by minimizing a suitably penalized least-mean-square cost $$||{\mathbf {y}}-{\mathbf {H}} {\mathbf {x}}||_2^2+\lambda V({\mathbf {x}})$$ | | y - H x | | 2 2 + λ V ( x ) . Here $${\mathbf...
Sparse reconstruction algorithms aim to retrieve high-dimensional sparse signals from a limited number of measurements. A common example is LASSO or Basis Pursuit where sparsity is enforced using an ℓ1-penalty together with a cost function ‖y — Hx‖22. For random design matrices H, a sharp phase transition boundary separates the ‘good’ parameter region where error-free recovery of a sufficiently sparse...
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