arXiv subjects

David Jacobs

Publications and source records attributed to David Jacobs.

40 records · Page 3Linked to original sources

arXiv2016

Efficient Representation of Low-Dimensional Manifolds using Deep Networks

We consider the ability of deep neural networks to represent data that lies near a low-dimensional manifold in a high-dimensional space. We show that deep networks can efficiently extract the intrinsic, low-dimensional coordinates of such data. We first show that the first two layers of a deep network can exactly embed points lying on a monotonic chain, a special type of piecewise linear manifold, mapping them to a low-dimensional Euclidean space. Remarkably, the network can do this using an almost optimal number of parameters. We also show that this network projects nearby points onto the manifold and then embeds them with little error. We then extend these results to more general manifolds.

cs.NE↗

arXiv2015

A Hyperelastic Two-Scale Optimization Model for Shape Matching

We suggest a novel shape matching algorithm for three-dimensional surface meshes of disk or sphere topology. The method is based on the physical theory of nonlinear elasticity and can hence handle large rotations and deformations. Deformation boundary conditions that supplement the underlying equations are usually unknown. Given an initial guess, these are optimized such that the mechanical boundary forces that are responsible for the deformation are of a simple nature. We show a heuristic way to approximate the nonlinear optimization problem by a sequence of convex problems using finite elements. The deformation cost, i.e, the forces, is measured on a coarse scale while ICP-like matching is done on the fine scale. We demonstrate the plausibility of our algorithm on examples taken from different datasets.

cs.CG↗

arXiv2014

Locally Scale-Invariant Convolutional Neural Networks

Convolutional Neural Networks (ConvNets) have shown excellent results on many visual classification tasks. With the exception of ImageNet, these datasets are carefully crafted such that objects are well-aligned at similar scales. Naturally, the feature learning problem gets more challenging as the amount of variation in the data increases, as the models have to learn to be invariant to certain changes in appearance. Recent results on the ImageNet dataset show that given enough data, ConvNets can learn such invariances producing very discriminative features [1]. But could we do more: use less parameters, less data, learn more discriminative features, if certain invariances were built into the learning process? In this paper we present a simple model that allows ConvNets to learn features in a locally scale-invariant manner without increasing the number of model parameters. We show on a modified MNIST dataset that when faced with scale variation, building in scale-invariance allows ConvNets to learn more discriminative features with reduced chances of over-fitting.

cs.CV↗

arXiv2014

Comparing apples to apples in the evaluation of binary coding methods

We discuss methodological issues related to the evaluation of unsupervised binary code construction methods for nearest neighbor search. These issues have been widely ignored in literature. These coding methods attempt to preserve either Euclidean distance or angular (cosine) distance in the binary embedding space. We explain why when comparing a method whose goal is preserving cosine similarity to one designed for preserving Euclidean distance, the original features should be normalized by mapping them to the unit hypersphere before learning the binary mapping functions. To compare a method whose goal is to preserves Euclidean distance to one that preserves cosine similarity, the original feature data must be mapped to a higher dimension by including a bias term in binary mapping functions. These conditions ensure the fair comparison between different binary code methods for the task of nearest neighbor search. Our experiments show under these conditions the very simple methods (e.g. LSH and ITQ) often outperform recent state-of-the-art methods (e.g. MDSH and OK-means).

cs.CV↗