18 research outputs found
Approximating the Little Grothendieck Problem over the Orthogonal and Unitary Groups
The little Grothendieck problem consists of maximizing
over binary variables , where C is a
positive semidefinite matrix. In this paper we focus on a natural
generalization of this problem, the little Grothendieck problem over the
orthogonal group. Given C a dn x dn positive semidefinite matrix, the objective
is to maximize restricting to take
values in the group of orthogonal matrices, where denotes the (ij)-th
d x d block of C. We propose an approximation algorithm, which we refer to as
Orthogonal-Cut, to solve this problem and show a constant approximation ratio.
Our method is based on semidefinite programming. For a given , we show
a constant approximation ratio of , where is
the expected average singular value of a d x d matrix with random Gaussian
i.i.d. entries. For d=1 we recover the known
approximation guarantee for the classical little Grothendieck problem. Our
algorithm and analysis naturally extends to the complex valued case also
providing a constant approximation ratio for the analogous problem over the
Unitary Group.
Orthogonal-Cut also serves as an approximation algorithm for several
applications, including the Procrustes problem where it improves over the best
previously known approximation ratio of~. The little
Grothendieck problem falls under the class of problems approximated by a recent
algorithm proposed in the context of the non-commutative Grothendieck
inequality. Nonetheless, our approach is simpler and it provides a more
efficient algorithm with better approximation ratios and matching integrality
gaps.
Finally, we also provide an improved approximation algorithm for the more
general little Grothendieck problem over the orthogonal (or unitary) group with
rank constraints.Comment: Updates in version 2: extension to the complex valued (unitary group)
case, sharper lower bounds on the approximation ratios, matching integrality
gap, and a generalized rank constrained version of the problem. Updates in
version 3: Improvement on the expositio
Open problem: Tightness of maximum likelihood semidefinite relaxations
We have observed an interesting, yet unexplained, phenomenon: Semidefinite
programming (SDP) based relaxations of maximum likelihood estimators (MLE) tend
to be tight in recovery problems with noisy data, even when MLE cannot exactly
recover the ground truth. Several results establish tightness of SDP based
relaxations in the regime where exact recovery from MLE is possible. However,
to the best of our knowledge, their tightness is not understood beyond this
regime. As an illustrative example, we focus on the generalized Procrustes
problem
Disentangling Orthogonal Matrices
Motivated by a certain molecular reconstruction methodology in cryo-electron
microscopy, we consider the problem of solving a linear system with two unknown
orthogonal matrices, which is a generalization of the well-known orthogonal
Procrustes problem. We propose an algorithm based on a semi-definite
programming (SDP) relaxation, and give a theoretical guarantee for its
performance. Both theoretically and empirically, the proposed algorithm
performs better than the na\"{i}ve approach of solving the linear system
directly without the orthogonal constraints. We also consider the
generalization to linear systems with more than two unknown orthogonal
matrices