Extreme point inequalities and geometry of the rank sparsity ball

Abstract

We investigate geometric features of the unit ball corresponding to the sum of the nuclear norm of a matrix and the l 1 norm of its entries-a common penalty function encouraging joint low rank and high sparsity. As a byproduct of this effort, we develop a calculus (or algebra) of faces for general convex functions, yielding a simple and unified approach for deriving inequalities balancing the various features of the optimization problem at hand, at the extreme points of the solution set.

On Representer Theorems and Convex Regularization

“…Property (19) simply stems from the fact that C K ⊥ and D = ran(L) ∩ B p 1 are in bijection through the operators L and L + .…”

“…The corresponding regularization is sometimes used in order to favor sparse and low-rank matrices. The authors of [19] have described the extreme points of this unit ball. They have proved that the extreme points M of the rank sparsity ball satisfy r(r+1) 2 − |I| ≤ 1, where |I| denotes the number of non-zero entries in M and r denotes its rank.…”

Max-norm optimization for robust matrix recovery

“…In Doan and Vavasis (2013), a similar combination of the two norms (denoted as · 1, * := X 1 + θ X * for a given matrix X and a parameter θ) is used to find hidden sparse rank-one matrices in a given matrix. The geometry of the unit · 1, * -norm ball is further analyzed in Drusvyatskiy et al (2015). It is interesting to note that (2.3) is the first time that the sum of the max-norm and nuclear norm is considered in matrix recovery.…”