Please also see my Google scholar profile; it is sometimes more up to date.
Preprints
Published
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Thesis
Laura Balzano, Handling Missing Data in High-Dimensional Subspace Modeling, May 2012.
I have great news that my AFOSR Young Investigator proposal was accepted for funding. My proposal was focused on time-varying low-rank factorization models, and various ways of solving a variety of related non-convex problem formulations. Read more about it here. I look forward to the contributions we will be able to make with the support of AFOSR.
Yesterday I gave a talk on Subspace Clustering using Ensemble methods at the Simons Institute. See the video here!
This is work with John Lipor, David Hong, and Yan Shuo Tan. Our related paper has been just updated on the arxiv. Our key observation was that, while K-Subspaces (KSS) works poorly overall and depends heavily on initialization, it still seems to give partially good clustering information. We therefore use it as a “weak clusterer” and combine ensembles of KSS (EKSS) by averaging the co-association/affinity matrices. This works extremely well, both in simulation and on real data, and also in theory. We were able to show that EKSS gives correct clustering in a variety of common cases: e.g. for subspaces with bounded affinity, and with noisy data and missing data. Our theory generalizes theory of the Thresholded Subspace Clustering algorithm to show that any algorithm that produces an affinity matrix that is an approximation to a monotonic function of absolute inner products will give correct clustering. This general theory should be broadly applicable to many geometric approaches to subspace clsutering.
John Lipor, Andrew Gitlin, Bioashuai Tao, and I have a new paper, “Improving K-Subspaces via Coherence Pursuit,” to be published in the Journal of Special Topics in Signal Processing issue “Robust Subspace Learning and Tracking: Theory, Algorithms, and Applications.” In it we present a new subspace clustering algorithm, Coherence Pursuit – K-Subspaces (CoP-KSS). Here is the code for CoP-KSS and for our figures. Our paper considers specifically the PCA step in K-Subspaces, where a best-fit subspace estimate is determined from a (possibly incorrect) clustering. When a given cluster has points from multiple low-rank subspaces, PCA is not a robust approach. We replace that step with Coherence Pursuit, a new algorithm for Robust PCA. We prove that Coherence Pursuit indeed can recover the “majority” subspace when data from other low-rank subspaces is contaminating the cluster. In this paper we also prove — to the best of our knowledge, for the first time — that the K-Subspaces problem is NP-hard, and indeed even NP-hard to approximate within any finite factor for large enough subspace rank.
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