Gilad Lerman

Spectral Curvature Clustering (SCC)

This paper presents novel techniques for improving the performance of a multi-way spectral clustering framework (Govindu in Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), vol. 1, pp. 1150–1157, 2005; Chen and Lerman, 2007, preprint in the supplementary webpage) for segmenting affine subspaces. Specifically, it suggests an iterative sampling procedure to improve the uniform sampling strategy, an automatic scheme of inferring the tuning parameter from the data, a precise initialization procedure for K-means, as well as a simple strategy for isolating outliers. The resulting algorithm, Spectral Curvature Clustering (SCC), requires only linear storage and takes linear running time in the size of the data. It is supported by theory which both justifies its successful performance and guides our practical choices. We compare it with other existing methods on a few artificial instances of affine subspaces. Application of the algorithm to several real-world problems is also discussed.

Hybrid Linear Modeling via Local Best-Fit Flats

We present a simple and fast geometric method for modeling data by a union of affine subspaces. The method begins by forming a collection of local best-fit affine subspaces, i.e., subspaces approximating the data in local neighborhoods. The correct sizes of the local neighborhoods are determined automatically by the Jones’ β2 numbers (we prove under certain geometric conditions that our method finds the optimal local neighborhoods). The collection of subspaces is further processed by a greedy selection procedure or a spectral method to generate the final model. We discuss applications to tracking-based motion segmentation and clustering of faces under different illuminating conditions. We give extensive experimental evidence demonstrating the state of the art accuracy and speed of the suggested algorithms on these problems and also on synthetic hybrid linear data as well as the MNIST handwritten digits data; and we demonstrate how to use our algorithms for fast determination of the number of affine subspaces.

Median K-Flats for hybrid linear modeling with many outliers

We describe the MedianK-flats (MKF) algorithm, a simple online method for hybrid linear modeling, i.e., for approximating data by a mixture of flats. This algorithm simultaneously partitions the data into clusters while finding their corresponding best approximating ℓ<inf>1</inf> d-flats, so that the cumulative ℓ<inf>1</inf> error is minimized. The current implementation restricts d-flats to be d-dimensional linear subspaces. It requires a negligible amount of storage, and its complexity, when modeling data consisting of N points in ℝ^D with K d-dimensional linear subspaces, is of order O(n<inf>s</inf> · K · d · D + n<inf>s</inf> · d² · D), where n<inf>s</inf> is the number of iterations required for convergence (empirically on the order of 10⁴). Since it is an online algorithm, data can be supplied to it incrementally and it can incrementally produce the corresponding output. The performance of the algorithm is carefully evaluated using synthetic and real data.

Robust Computation of Linear Models by Convex Relaxation

Consider a data set of vector-valued observations that consists of noisy inliers, which are explained well by a low-dimensional subspace, along with some number of outliers. This work describes a convex optimization problem, called reaper, that can reliably fit a low-dimensional model to this type of data. This approach parameterizes linear subspaces using orthogonal projectors and uses a relaxation of the set of orthogonal projectors to reach the convex formulation. The paper provides an efficient algorithm for solving the reaper problem, and it documents numerical experiments that confirm that reaper can dependably find linear structure in synthetic and natural data. In addition, when the inliers lie near a low-dimensional subspace, there is a rigorous theory that describes when reaper can approximate this subspace.

Robust recovery of multiple subspaces by geometric lp minimization

We assume i.i.d. data sampled from a mixture distribution with K components along fixed d-dimensional linear subspaces and an additional outlier component. For p>0, we study the simultaneous recovery of the K fixed subspaces by minimizing the l_p-averaged distances of the sampled data points from any K subspaces. Under some conditions, we show that if

Defining functional distance using manifold embeddings of gene ontology annotations

0 1 and p>1, then the underlying subspaces cannot be recovered or even nearly recovered by l_p minimization. The results of this paper partially explain the successes and failures of the basic approach of l_p energy minimization for modeling data by multiple subspaces.

Robust locally linear analysis with applications to image denoising and blind inpainting

Although rigorous measures of similarity for sequence and structure are now well established, the problem of defining functional relationships has been particularly daunting. Here, we present several manifold embedding techniques to compute distances between Gene Ontology (GO) functional annotations and consequently estimate functional distances between protein domains. To evaluate accuracy, we correlate the functional distance to the well established measures of sequence, structural, and phylogenetic similarities. Finally, we show that manual classification of structures into folds and superfamilies is mirrored by proximity in the newly defined function space. We show how functional distances place structure–function relationships in biological context resulting in insight into divergent and convergent evolution. The methods and results in this paper can be readily generalized and applied to a wide array of biologically relevant investigations, such as accuracy of annotation transference, the relationship between sequence, structure, and function, or coherence of expression modules.

Kernel Spectral Curvature Clustering (KSCC)

We study the related problems of denoising images corrupted by impulsive noise and blind inpainting (i.e., inpainting when the deteriorated region is unknown). Our basic approach is to model the set of patches of pixels in an image as a union of low-dimensional subspaces, corrupted by sparse but perhaps large magnitude noise. For this purpose, we develop a robust and iterative method for single subspace modeling and extend it to an iterative algorithm for modeling multiple subspaces. We prove convergence for both algorithms and carefully compare our methods with other recent ideas for such robust modeling. We demonstrate state-of-the-art performance of our method for both imaging problems.

A New Approach to Two-View Motion Segmentation Using Global Dimension Minimization

Multi-manifold modeling is increasingly used in segmentation and data representation tasks in computer vision and related fields. While the general problem, modeling data by mixtures of manifolds, is very challenging, several approaches exist for modeling data by mixtures of affine subspaces (which is often referred to as hybrid linear modeling). We translate some important instances of multi-manifold modeling to hybrid linear modeling in embedded spaces, without explicitly performing the embedding but applying the kernel trick. The resulting algorithm, Kernel Spectral Curvature Clustering, uses kernels at two levels - both as an implicit embedding method to linearize non-flat manifolds and as a principled method to convert a multi-way affinity problem into a spectral clustering one. We demonstrate the effectiveness of the method by comparing it with other state-of-the-art methods on both synthetic data and a real-world problem of segmenting multiple motions from two perspective camera views.

Motion segmentation by SCC on the hopkins 155 database

We present a new approach to rigid-body motion segmentation from two views. We use a previously developed nonlinear embedding of two-view point correspondences into a 9-dimensional space and identify the different motions by segmenting lower-dimensional subspaces. In order to overcome nonuniform distributions along the subspaces, whose dimensions are unknown, we suggest the novel concept of global dimension and its minimization for clustering subspaces with some theoretical motivation. We propose a fast projected gradient algorithm for minimizing global dimension and thus segmenting motions from 2-views. We develop an outlier detection framework around the proposed method, and we present state-of-the-art results on outlier-free and outlier-corrupted two-view data for segmenting motion.