v2009.01.01 - Convex Optimization

7.2. SECOND PREVALENT PROBLEM: 519
rankX
g
cenv rankX
Figure 130: Abstraction of convex envelope of rank function. Rank is
a quasiconcave function on the positive semidefinite cone, but its convex
envelope is the smallest convex function enveloping it. Vertical bar labelled g
illustrates a trace/rank gap; id est, rank found exceeds estimate (by 2).
[113] [112] Convex envelope of rank function: for σ i a singular value,
cenv(rankA) on {A∈ R m×n | ‖A‖ 2 ≤κ} = 1 ∑
σ(A) i (1267)
κ
cenv(rankA) on {A∈ S n + | ‖A‖ 2 ≤κ} = 1 tr(A) (1268)
κ
A properly scaled trace thus represents the best convex lower bound on rank
for positive semidefinite matrices. The idea, then, is to substitute the convex
envelope for rank of some variable A∈ S M + (A.6.5)
i
rankA ← cenv(rankA) ∝
trA = ∑ i
σ(A) i = ∑ i
λ(A) i = ‖λ(A)‖ 1 (1269)
which is equivalent to the sum of all eigenvalues or singular values.
[112] Convex envelope of the cardinality function is proportional to the
1-norm:
cenv(cardx) on {x∈ R n | ‖x‖ ∞ ≤κ} = 1 κ ‖x‖ 1 (1270)