(i) {α - Convex Optimization

Construct H : Random Sensing
Random Sensing: α is s-sparse and given m measurements selected uniformly
at random from an ensemble. If m ≥ Csµ 2 H,Φ log n, then minimizing (P eq)
reconstructs α exactly with overwhelming probability.
Compressible signals/images and l 2 − l 1 instance optimality: (P eq ) solution
recovers the s-largest entries.
Stability to noise y = HΦα + ε: the decoder ∆ Φ,σ (y)
(P σ ) : min ‖α‖ l1
s.t. ‖y − HΦα‖ l2
≤ σ
has l 2 − l 1 instance optimality and a factor of the noise std σ.
Stanford seminar 08-5