Fusion Subspace Clustering: Full and Incomplete Data

Inferring low-dimensional structures that explain high-dimensional data has become a cornerstone of discovery in virtually all fields of science. Principal component analysis (PCA), which identifies the low-dimensional linear subspace that best explains a dataset, is arguably the most prominent technique for this purpose. However, in many applications -- computer vision, image processing, bioinformatics, linguistics, networks analysis, and more [1-10] -- data is often composed of a mixture of several classes, each of which can be explained with a different subspace. Clustering and inferring subspaces that explain data is an important unsupervised learning problem that has received tremendous 1 attention in recent years, producing theory and algorithms to handle outliers, noisy measurements, privacy concerns, and data constraints, among other difficulties [11-22]. However, one major challenge in contemporary problems is that data is often incomplete. For example, in image inpainting, the values of some pixels are missing due to faulty sensors and image contamination [23]; in computer vision features are often missing due to occlusions and tracking algorithms malfunctions [24]; in recommender systems each user only rates a limited number of items [25]; in a network, most nodes communicate in subsets, producing only a handful of all the possible measurements [7].