Laura Balzano

Local Convergence of an Algorithm for Subspace Identification from Partial Data

Grassmannian rank-one update subspace estimation (GROUSE) is an iterative algorithm for identifying a linear subspace of

On the sample complexity of subspace clustering with missing data

\mathbb {R}^n

Group-sparse subspace clustering with missing data

Rn from data consisting of partial observations of random vectors from that subspace. This paper examines local convergence properties of GROUSE, under assumptions on the randomness of the observed vectors, the randomness of the subset of elements observed at each iteration, and incoherence of the subspace with the coordinate directions. Convergence at an expected linear rate is demonstrated under certain assumptions. The case in which the full random vector is revealed at each iteration allows for much simpler analysis and is also described. GROUSE is related to incremental SVD methods and to gradient projection algorithms in optimization.

Iterative online subspace learning for robust image alignment

GROUSE (Grassmannian Rank-One Update Subspace Estimation) [1] is an incremental algorithm for identifying a subspace of ℝn from a sequence of vectors in this subspace, where only a subset of components of each vector is revealed at each iteration. Recent analysis [2] has shown that GROUSE converges locally at an expected linear rate, under certain assumptions. GROUSE has a similar flavor to the incremental singular value decomposition algorithm [4], which updates the SVD of a matrix following addition of a single column. In this paper, we modify the incremental SVD approach to handle missing data, and demonstrate that this modified approach is equivalent to GROUSE, for a certain choice of an algorithmic parameter.

Inferring the behavior of distributed energy resources with online learning

Subspace clustering is a useful tool for analyzing large complex data, but in many relevant applications missing data are common. Existing theoretical analysis of this problem shows that subspace clustering from incomplete data is possible, but that analysis requires the number of samples (i.e., partially observed vectors) to be super-polynomial in the dimension d. Such huge sample sizes are unnecessary when no data are missing and uncommon in applications. There are two main contributions in this paper. First, it is shown that if subspaces have rank at most r and the number of partially observed vectors greater than dr+1 (times a poly-logarithmic factor), then with high probability the true subspaces are the only subspaces that agree with the observed data. We may conclude that subspace clustering may be possible without impractically large sample sizes and that we can certify the output of any subspace clustering algorithm by checking its fit to the observed data. The second main contribution is a novel EM-type algorithm for subspace clustering with missing data. We demonstrate and compare it to several other algorithms. Experiments with simulated and real data show that such algorithms work well in practice.

Necessary and sufficient conditions for sketched subspace clustering

We present a family of online algorithms for real-time factorization-based structure from motion, leveraging a relationship between the incremental singular value decomposition and recent work in online matrix completion. Our methods are orders of magnitude faster than previous state of the art, can handle missing data and a variable number of feature points, and are robust to noise and sparse outliers. Experiments show that they perform well in both online and batch settings. We also provide an implementation which is able to produce 3D models in real time using a laptop with a webcam

Mixture regression as subspace clustering

This paper explores algorithms for subspace clustering with missing data. In many high-dimensional data analysis settings, data points Lie in or near a union of subspaces. Subspace clustering is the process of estimating these subspaces and assigning each data point to one of them. However, in many modern applications the data are severely corrupted by missing values. This paper describes two novel methods for subspace clustering with missing data: (a) group-sparse sub-space clustering (GSSC), which is based on group-sparsity and alternating minimization, and (b) mixture subspace clustering (MSC), which models each data point as a convex combination of its projections onto all subspaces in the union. Both of these algorithms are shown to converge to a local minimum, and experimental results show that they outperform the previous state-of-the-art, with GSSC yielding the highest overall clustering accuracy.

Margin-based active subspace clustering

Robust high-dimensional data processing has witnessed an exciting development in recent years, as theoretical results have shown that it is possible using convex programming to optimize data fit to a low-rank component plus a sparse outlier component. This problem is also known as Robust PCA, and it has found application in many areas of computer vision. In image and video processing and face recognition, an exciting opportunity for processing of massive image databases is emerging as people upload photo and video data online in unprecedented volumes. However, the data quality and consistency is not controlled in any way, and the massiveness of the data poses a serious computational challenge. In this paper we present t-GRASTA, or “Transformed GRASTA (Grassmannian Robust Adaptive Subspace Tracking Algorithm)”. t-GRASTA performs incremental gradient descent constrained to the Grassmann manifold of subspaces in order to simultaneously estimate a decomposition of a collection of images into a low-rank subspace, a sparse part of occlusions and foreground objects, and a transformation such as rotation or translation of the image. We show that t-GRASTA is 4× faster than state-of-the-art algorithms, has half the memory requirement, and can achieve alignment for face images as well as jittered camera surveillance images.