v2010.10.26 - Convex Optimization

572 CHAPTER 7. PROXIMITY PROBLEMS[135] [134] Convex envelope of rank function: for σ i a singular value,(1566)cenv(rankA) on {A∈ R m×n | ‖A‖ 2 ≤κ} = 1 κ 1T σ(A) = 1 κ tr √ A T A (1362)cenv(rankA) on {A normal | ‖A‖ 2 ≤κ} = 1 κ ‖λ(A)‖ 1 = 1 κ tr √ A T A (1363)cenv(rankA) on {A∈ S n + | ‖A‖ 2 ≤κ} = 1 κ 1T λ(A) = 1 tr(A) (1364)κA properly scaled trace thus represents the best convex lower bound on rankfor positive semidefinite matrices. The idea, then, is to substitute convexenvelope for rank of some variable A∈ S M + (A.6.5)rankA ← cenv(rankA) ∝trA = ∑ iσ(A) i = ∑ iλ(A) i (1365)which is equivalent to the sum of all eigenvalues or singular values.[134] Convex envelope of the cardinality function is proportional to the1-norm:cenv(cardx) on {x∈ R n | ‖x‖ ∞ ≤κ} = 1 κ ‖x‖ 1 (1366)cenv(cardx) on {x∈ R n + | ‖x‖ ∞ ≤κ} = 1 κ 1T x (1367)7.2.2.2 Applying trace rank-heuristic to Problem 2Substituting rank envelope for rank function in Problem 2, for D ∈ EDM N(confer (1042))cenv rank(−V T NDV N ) = cenv rank(−V DV ) ∝ − tr(V DV ) (1368)and for desired affine dimension ρ ≤ N −1 and nonnegative H [sic] we geta convex optimization problemminimize ‖ ◦√ D − H‖ 2 FDsubject to − tr(V DV ) ≤ κρ(1369)D ∈ EDM N