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Abstract Subspace clustering is the unsupervised grouping of points lying near a union of low-dimensional linear subspaces. Algorithms based directly on geometric properties of such data tend to either provide poor empirical performance, lack theoretical guarantees or depend heavily on their initialization. We present a novel geometric approach to the subspace clustering problem that leverages ensembles of the $$K$$-subspace (KSS) algorithm via the evidence accumulation clustering framework. Our algorithm, referred to as ensemble $$K$$-subspaces (EKSSs), forms a co-association matrix whose $(i,j)$th entry is the number of times points $$i$$ and $$j$$ are clustered together by several runs of KSS with random initializations. We prove general recovery guarantees for any algorithm that forms an affinity matrix with entries close to a monotonic transformation of pairwise absolute inner products. We then show that a specific instance of EKSS results in an affinity matrix with entries of this form, and hence our proposed algorithm can provably recover subspaces under similar conditions to state-of-the-art algorithms. The finding is, to the best of our knowledge, the first recovery guarantee for evidence accumulation clustering and for KSS variants. We show on synthetic data that our method performs well in the traditionally challenging settings of subspaces with large intersection, subspaces with small principal angles and noisy data. Finally, we evaluate our algorithm on six common benchmark datasets and show that unlike existing methods, EKSS achieves excellent empirical performance when there are both a small and large number of points per subspace. 

Baseline estimation is a critical task for commercial buildings that participate in demand response programs and need to assess the impact of their strategies. The problem is to predict what the power profile would have been had the demand response event not taken place. This paper explores the use of tensor decomposition in baseline estimation. We apply the method to submetered fan power data from demand response experiments that were run to assess a fast demand response strategy expected to primarily impact the fans. Baselining this fan power data is critical for evaluating the results, but doing so presents new challenges not readily addressed by existing techniques designed primarily for baselining whole building electric loads. We find that tensor decomposition of the fan power data identifies components that capture both dominant daily patterns and demand response events, and that are generally more interpretable than those found by principal component analysis. We conclude by discussing how these components and related techniques can aid in developing new baseline models.