v2010.10.26 - Convex Optimization

3.2. PRACTICAL NORM FUNCTIONS, ABSOLUTE VALUE 229k/m10.910.90.80.80.7signed0.7positive0.60.60.50.50.40.40.30.30.20.20.10.100 0.2 0.4 0.6 0.8 100 0.2 0.4 0.6 0.8 1m/n(518)minimize ‖x‖ 1xsubject to Ax = bminimize ‖x‖ 1xsubject to Ax = bx ≽ 0(523)Figure 71: Exact recovery transition: Respectively signed [117] [119] orpositive [124] [122] [123] solutions x to Ax=b with sparsity k below thickcurve are recoverable. For Gaussian random matrix A∈ R m×n , thick curvedemarcates phase transition in ability to find sparsest solution x by linearprogramming. These results were empirically reproduced in [38].f 2 (x) f 3 (x) f 4 (x)xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxFigure 72: Under 1-norm f 2 (x) , histogram (hatched) of residual amplitudesAx −b exhibits predominant accumulation of zero-residuals. Nonnegativelyconstrained 1-norm f 3 (x) from (523) accumulates more zero-residualsthan f 2 (x). Under norm f 4 (x) (not discussed), histogram would exhibitpredominant accumulation of (nonzero) residuals at gradient discontinuities.