On the simplicity and conditioning of low rank semidefinite programs

Abstract

Low rank matrix recovery problems appear widely in statistics, combinatorics, and imaging. One celebrated method for solving these problems is to formulate and solve a semidefinite program (SDP). It is often known that the exact solution to the SDP with perfect data recovers the solution to the original low rank matrix recovery problem. It is more challenging to show that an approximate solution to the SDP formulated with noisy problem data acceptably solves the original problem; arguments are usually ad hoc for each problem setting, and can be complex.In this note, we identify a set of conditions that we call regularity that limit the error due to noisy problem data or incomplete convergence. In this sense, regular SDPs are robust: regular SDPs can be (approximately) solved efficiently at scale; and the resulting approximate solutions, even with noisy data, can be trusted. Moreover, we show that regularity holds generically, and also for many structured low rank matrix recovery problems, including the stochastic block model, Z2 synchronization, and matrix completion. Formally, we call an SDP regular if it has a surjective constraint map, admits a unique primal and dual solution pair, and satisfies strong duality and strict complementarity.However, regularity is not a panacea: we show the Burer-Monteiro formulation of the SDP may have spurious second-order critical points, even for a regular SDP with a rank 1 solution.

Revisiting Spectral Bundle Methods: Primal-dual (Sub)linear Convergence Rates

“…The condition r ≥ r d Modern applications [11,12,33,15] of (P) actually shows that X is unique, admits rank r : = rank(X ) n, and satisfies strict complementarity under certain structural probabilistic assumptions. If in addition, dual solution is unique, then we only need r ≥ r d = r .…”

“…If (P) and (D) admit such pair, we say the pair (P) and (D) (or simply (P)) satisfies strict complementarity. Such condition is satisfied for a generic SDP [3] and also many structural SDPs [15].…”

A Strict Complementarity Approach to Error Bound and Sensitivity of Solution of Conic Programs

“…The error bound and sensitivity of the solution can be regarded as condition numbers for Problem (P). They guarantee that the output of iterative algorithms to solve (P) is still useful despite optimization error (of the algorithm) and measurement error (of the problem data) [Lew14,DU20].…”

“…It is worth noting that the standard notion of strict complementarity (SC) for SDP [AHO97, Definition 4] and LP, both defined algebraically, are equivalent to the geometric notion of DSC here. 5 SC always holds for LP [GT56], holds "generically" for SDP as shown in [AHO97], and even holds for some structured instances of SDP [DU20]. Due to the equivalence, DSC also holds under the same conditions for LP and SDP.…”

“…Consider an SDP with C = −11 , where 1 ∈ R n is the all one vector, A = diag(•), and b = 1. This is a simplification of the SDP for Z 2 synchronization [Ban18,DU20]. For this SDP, it is easily verified that the unique optimal solution is 11 and dual strict complementarity holds with dual optimal slack S = −11 + nI.…”