A polynomial-time relaxation of the Gromov-Hausdorff distance

arXiv.org Machine Learning

The Gromov-Hausdorff distance provides a metric on the set of isometry classes of compact metric spaces. Unfortunately, computing this metric directly is believed to be computationally intractable. Motivated by applications in shape matching and point-cloud comparison, we study a semidefinite programming relaxation of the Gromov-Hausdorff metric. This relaxation can be computed in polynomial time, and somewhat surprisingly is itself a pseudometric. We describe the induced topology on the set of compact metric spaces. Finally, we demonstrate the numerical performance of various algorithms for computing the relaxed distance and apply these algorithms to several relevant data sets. In particular we propose a greedy algorithm for finding the best correspondence between finite metric spaces that can handle hundreds of points.


Sparse Code Shrinkage: Denoising by Nonlinear Maximum Likelihood Estimation

Neural Information Processing Systems

Such a representation is closely related to redundancy reductionand independent component analysis, and has some neurophysiological plausibility. In this paper, we show how sparse coding can be used for denoising. Using maximum likelihood estimation of nongaussian variables corrupted by gaussian noise, we show how to apply a shrinkage nonlinearity on the components of sparse coding so as to reduce noise. Furthermore, we show how to choose the optimal sparse coding basis for denoising. Our method is closely related to the method of wavelet shrinkage, but has the important benefit over wavelet methods that both the features and the shrinkage parameters are estimated directly from the data. 1 Introduction A fundamental problem in neural network research is to find a suitable representation forthe data.


A Bayesian Approach to Perceptual 3D Object-Part Decomposition Using Skeleton-Based Representations

AAAI Conferences

We present a probabilistic approach to shape decomposition that creates a skeleton-based shape representation of a 3D object while simultaneously decomposing it into constituent parts. Our approach probabilistically combines two prominent threads from the shape literature: skeleton-based (medial axis) representations of shape, and part-based representations of shape, in which shapes are combinations of primitive parts. Our approach recasts skeleton-based shape representation as a mixture estimation problem, allowing us to apply probabilistic estimation techniques to the problem of 3D shape decomposition, extending earlier work on the 2D case. The estimated 3D shape decompositions approximate human shape decomposition judgments. We present a tractable implementation of the framework, which begins by over-segmenting objects at concavities, and then probabilistically merges them to create a distribution over possible decompositions. This results in a hierarchy of decompositions at different structural scales, again closely matching known properties of human shape representation. The probabilistic estimation procedures that arise naturally in the model allow effective prediction of missing parts. We present results on shapes from a standard database illustrating the effectiveness of the approach.


Bayesian Modeling of Facial Similarity

Neural Information Processing Systems

In previous work [6, 9, 10], we advanced a new technique for direct visual matching of images for the purposes of face recognition and image retrieval, using a probabilistic measure of similarity based primarily on a Bayesian (MAP) analysis of image differences, leadingto a "dual" basis similar to eigenfaces [13]. The performance advantage of this probabilistic matching technique over standard Euclidean nearest-neighbor eigenface matching was recently demonstrated using results from DARPA's 1996 "FERET" face recognition competition, in which this probabilistic matching algorithm was found to be the top performer. We have further developed a simple method of replacing the costly compution of nonlinear (online) Bayesian similarity measures by the relatively inexpensive computation of linear (offline) subspace projections and simple (online) Euclidean norms, thus resulting in a significant computational speedup for implementation with very large image databases as typically encountered in real-world applications.


A Computational Model for Cursive Handwriting Based on the Minimization Principle

Neural Information Processing Systems

We propose a trajectory planning and control theory for continuous movements such as connected cursive handwriting and continuous natural speech. Its hardware is based on our previously proposed forward-inverse-relaxation neural network (Wada & Kawato, 1993). Computationally, its optimization principle is the minimum torquechange criterion.Regarding the representation level, hard constraints satisfied by a trajectory are represented as a set of via-points extracted from a handwritten character. Accordingly, we propose a via-point estimation algorithm that estimates via-points by repeating the trajectory formation of a character and the via-point extraction from the character. In experiments, good quantitative agreement is found between human handwriting data and the trajectories generated by the theory. Finally, we propose a recognition schema based on the movement generation. We show a result in which the recognition schema is applied to the handwritten character recognition and can be extended to the phoneme timing estimation of natural speech. 1 INTRODUCTION In reaching movements, trajectory formation is an ill-posed problem because the hand can move along an infinite number of possible trajectories from the starting to the target point.