v2009.01.01 - Convex Optimization

7.2. SECOND PREVALENT PROBLEM: 521
compaction of the reconstruction by minimizing total distance. (823) It is best
used where some physical equilibrium implies such an energy minimization;
e.g., [305,5].
For this Problem 2, the trace rank-heuristic arose naturally in the
objective in terms of V . We observe: V (in contrast to VN T ) spreads energy
over all available distances (B.4.2 no.20, confer no.22) although the rank
function itself is insensitive to choice of auxiliary matrix.
7.2.2.4 Rank minimization heuristic beyond convex envelope
Fazel, Hindi, & Boyd [114] [338] [115] propose a rank heuristic more potent
than trace (1269) for problems of rank minimization;
rankY ← log det(Y +εI) (1274)
the concave surrogate function log det in place of quasiconcave rankY
(2.9.2.6.2) when Y ∈ S n + is variable and where ε is a small positive constant.
They propose minimization of the surrogate by substituting a sequence
comprising infima of a linearized surrogate about the current estimate Y i ;
id est, from the first-order Taylor series expansion about Y i on some open
interval of ‖Y ‖ 2 (D.1.7)
log det(Y + εI) ≈ log det(Y i + εI) + tr ( (Y i + εI) −1 (Y − Y i ) ) (1275)
we make the surrogate sequence of infima over bounded convex feasible set C
where, for i = 0...
arg inf rankY ← lim Y i+1 (1276)
Y ∈ C i→∞
Y i+1 = arg inf
Y ∈ C tr( (Y i + εI) −1 Y ) (1277)
Choosing Y 0 =I , the first step becomes equivalent to finding the infimum of
trY ; the trace rank-heuristic (1269). The intuition underlying (1277) is the
new term in the argument of trace; specifically, (Y i + εI) −1 weights Y so that
relatively small eigenvalues of Y found by the infimum are made even smaller.