Chordal Decomposition in Rank Minimized Semidefinite Programs with Applications to Subspace Clustering

Semidefinite programs (SDPs) often arise in relaxations of some NP-hard problems, and if the solution of the SDP obeys certain rank constraints, the relaxation will be tight. Decomposition methods based on chordal sparsity have already been applied to speed up the solution of sparse SDPs, but methods for dealing with rank constraints are underdeveloped. This paper leverages a minimum rank completion result to decompose the rank constraint on a single large matrix into multiple rank constraints on a set of smaller matrices. The re-weighted heuristic is used as a proxy for rank, and the specific form of the heuristic preserves the sparsity pattern between iterations. Implementations of rank-minimized SDPs through interior-point and first-order algorithms are discussed. The problem of subspace clustering is used to demonstrate the computational improvement of the proposed method.Comment: 6 pages, 6 figure

Superstabilizing Control of Discrete-Time ARX Models under Error in Variables

This paper applies a polynomial optimization based framework towards the superstabilizing control of an Autoregressive with Exogenous Input (ARX) model given noisy data observations. The recorded input and output values are corrupted with L-infinity bounded noise where the bounds are known. This is an instance of Error in Variables (EIV) in which true internal state of the ARX system remains unknown. The consistency set of ARX models compatible with noisy data has a bilinearity between unknown plant parameters and unknown noise terms. The requirement for a dynamic compensator to superstabilize all consistent plants is expressed using polynomial nonnegativity constraints, and solved using sum-of-squares (SOS) methods in a converging hierarchy of semidefinite programs in increasing size. The computational complexity of this method may be reduced by applying a Theorem of Alternatives to eliminate the noise terms. Effectiveness of this method is demonstrated on control of example ARX models.Comment: 12 pages, 0 figures, 5 table

Data-Driven Stabilizing and Robust Control of Discrete-Time Linear Systems with Error in Variables

This work presents a sum-of-squares (SOS) based framework to perform data-driven stabilization and robust control tasks on discrete-time linear systems where the full-state observations are corrupted by L-infinity bounded input, measurement, and process noise (error in variable setting). Certificates of state-feedback superstability or quadratic stability of all plants in a consistency set are provided by solving a feasibility program formed by polynomial nonnegativity constraints. Under mild compactness and data-collection assumptions, SOS tightenings in rising degree will converge to recover the true superstabilizing controller, with slight conservatism introduced for quadratic stabilizability. The performance of this SOS method is improved through the application of a theorem of alternatives while retaining tightness, in which the unknown noise variables are eliminated from the consistency set description. This SOS feasibility method is extended to provide worst-case-optimal robust controllers under H2 control costs. The consistency set description may be broadened to include cases where the data and process are affected by a combination of L-infinity bounded measurement, process, and input noise. Further generalizations include varying noise sets, non-uniform sampling, and switched systems stabilization.Comment: 27 pages, 1 figure, 9 table

Sequential sparsification for change detection

This paper presents a general method for segmenting a vector valued sequence into an unknown number of subsequences where all data points from a subsequence can be represented with the same affine parametric model. The idea is to cluster the data into the minimum number of such subsequences which, as we show, can be cast as a sparse signal recovery problem by exploiting the temporal correlation between consecutive data points. We try to maximize the sparsity (i.e. the number of zero elements) of the first order differences of the sequence of parameter vectors. Each non-zero element in the first order difference sequence corresponds to a change. A weighted l1 norm based convex approximation is adopted to solve the change detection problem. We apply the proposed method to video segmentation and temporal segmentation of dynamic textures. 1