Kernel Spectral Curvature Clustering (KSCC)

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 nonflat manifolds and as a principled method to convert a multiway 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.Comment: accepted to 2009 ICCV Workshop on Dynamical Visio

High-Dimensional Menger-Type Curvatures - Part I: Geometric Multipoles and Multiscale Inequalities

We define a discrete Menger-type curvature of d+2 points in a real separable Hilbert space H by an appropriate scaling of the squared volume of the corresponding (d+1)-simplex. We then form a continuous curvature of an Ahlfors d-regular measure on H by integrating the discrete curvature according to the product measure. The aim of this work, continued in a subsequent paper, is to estimate multiscale least squares approximations of such measures by the Menger-type curvature. More formally, we show that the continuous d-dimensional Menger-type curvature is comparable to the ``Jones-type flatness''. The latter quantity adds up scaled errors of approximations of a measure by d-planes at different scales and locations, and is commonly used to characterize uniform rectifiability. We thus obtain a characterization of uniform rectifiability by using the Menger-type curvature. In the current paper (part I) we control the continuous Menger-type curvature of an Ahlfors d-regular measure by its Jones-type flatness.Comment: 47 pages, 13 figures. Minor revisions and the inclusion of figure

Foundations of a Multi-way Spectral Clustering Framework for Hybrid Linear Modeling

The problem of Hybrid Linear Modeling (HLM) is to model and segment data using a mixture of affine subspaces. Different strategies have been proposed to solve this problem, however, rigorous analysis justifying their performance is missing. This paper suggests the Theoretical Spectral Curvature Clustering (TSCC) algorithm for solving the HLM problem, and provides careful analysis to justify it. The TSCC algorithm is practically a combination of Govindu's multi-way spectral clustering framework (CVPR 2005) and Ng et al.'s spectral clustering algorithm (NIPS 2001). The main result of this paper states that if the given data is sampled from a mixture of distributions concentrated around affine subspaces, then with high sampling probability the TSCC algorithm segments well the different underlying clusters. The goodness of clustering depends on the within-cluster errors, the between-clusters interaction, and a tuning parameter applied by TSCC. The proof also provides new insights for the analysis of Ng et al. (NIPS 2001).Comment: 40 pages. Minor changes to the previous version (mainly revised Sections 2.2 & 2.3, and added references). Accepted to the Journal of Foundations of Computational Mathematic

Regulation of Unethical Billing Practices: Progress and Prospects

During the last ten years billing fraud by lawyers has been recognized as a serious problem that undermines clients\u27 trust of lawyers and the reputation of the profession as a whole. It used to be thought that lawyers who wanted to steal their clients\u27 money would just take money out of the trust account. In recent years it has become clear that dishonest lawyers\u27 methods of misappropriation are far more diverse than that. The focus of this paper is on billing misconduct by lawyers who contract with their clients to bill by the hour. I will not talk about lawyers who borrow from their trust accounts or manipulate contingent fees

A Model State Act: Remedies for Domestic Abuse

The problem of domestic violence has been the subject of increasing national concern. In response to that concern, states have enacted legislation providing protection to victims of domestic violence, encouraging improved police enforcement of protection law\u27s, and constructing appropriate legal sanctions against abusers. Drafting comprehensive legislation to address this problem is difficult because both civil and criminal remedies are needed, because the needs of battered women are diverse and complex, and because effective protection requires a coordinated response by courts, law enforcement agencies, mental health personnel, and the bar. While some new legislation on wife abuse has been enacted in a majority of states, few states have addressed the fill range of available remedies, or made relief available to all victims of domestic violence. In this Article, Ms. Lerman presents a Model Act that consolidates and addresses remedies needed for domestic violence in one comprehensive statute. Ms. Lerman asserts that the primary goal of any lai\u27 on domestic violence should be to protect the victim. Accordingly, the Model Act facilitates the victim\u27s ability to gain access to the courts and to request protection. The Model Act also acknowledges the needfor improved police response to domestic violence and specifies particular police duties. In addition, the Model Act recognizes that domestic violence may be handled as a civil matter, as a criminal matter, or both. Finally, the Model Act considers the appropriate legal treatment of abusers and ihchudes both punitive and rehabilitative dispositional options