GOSUS: Grassmannian Online Subspace Updates with Structured-Sparsity

Abstract

We study the problem of online subspace learning in the context of sequential observations involving structured perturbations. In online subspace learning, the observations are an unknown mixture of two components presented to the model sequentially -the main effect which pertains to the subspace and a residual/error term. If no additional requirement is imposed on the residual, it often corresponds to noise terms in the signal which were unaccounted for by the main effect. To remedy this, one may impose 'structural' contiguity, which has the intended effect of leveraging the secondary terms as a covariate that helps the estimation of the subspace itself, instead of merely serving as a noise residual. We show that the corresponding online estimation procedure can be written as an approximate optimization process on a Grassmannian. We propose an efficient numerical solution, GOSUS, Grassmannian Online Subspace Updates with Structured-sparsity, for this problem. GOSUS is expressive enough in modeling both homogeneous perturbations of the subspace and structural contiguities of outliers, and after certain manipulations, solvable via an alternating direction method of multipliers (ADMM). We evaluate the empirical performance of this algorithm on two problems of interest: online background subtraction and online multiple face tracking, and demonstrate that it achieves competitive performance with the state-of-the-art in near real time.

An online background subtraction algorithm deployed on a NAO humanoid robot based monitoring system

“…Third, low rank model can be integrated with other models, combining the advantages of different models. Typical examples are some PCP-like online algorithms [17,18,19,20]. These algorithms incorporate representative back-85 ground model [6,10,11,12] into low rank model [13,14], achieving excellent performance.…”

“…Generally, the framework of background subtraction algorithms includes two components: a background model and a foreground model. The background model estimates the potential background in image sequences, and the foreground model detects foreground re- 20 gions by comparing between captured frames and the estimated background. Currently, although a large number of algorithms have been proposed for background subtraction [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16], some problems remain open for a background subtrac-25 tion algorithm designed for robots.…”

“…However, 90 some problems still exist and restrict the application of low rank based background subtraction algorithms in real robotic systems. First, it has been shown that the contiguous foreground prior contributes significantly to the accuracy of foreground detection [14,20], but using this prior 95 also induces heavy computational burden. It influences the speed of algorithm, even in an online scheme [20].…”

“…Furthermore, to speed up these PCP-based algorithms, a large number of PCP-like [17,18,19,20,21, 22] online algorithms have been proposed. Some of these PCP-like online algorithms have achieved a real time speed [19,21,22].…”

“…The representative algorithms are Principal Component Pursuit (PCP) based al-50 corithms [13,14] and PCP-like online algorithms [17,18,19,20,21,22]. PCP [13] assumes the potential background images of an image sequence lie in a low rank subspace, and the moving objects (foreground) is spatially sparse.…”