7,719 research outputs found
Masking Strategies for Image Manifolds
We consider the problem of selecting an optimal mask for an image manifold,
i.e., choosing a subset of the pixels of the image that preserves the
manifold's geometric structure present in the original data. Such masking
implements a form of compressive sensing through emerging imaging sensor
platforms for which the power expense grows with the number of pixels acquired.
Our goal is for the manifold learned from masked images to resemble its full
image counterpart as closely as possible. More precisely, we show that one can
indeed accurately learn an image manifold without having to consider a large
majority of the image pixels. In doing so, we consider two masking methods that
preserve the local and global geometric structure of the manifold,
respectively. In each case, the process of finding the optimal masking pattern
can be cast as a binary integer program, which is computationally expensive but
can be approximated by a fast greedy algorithm. Numerical experiments show that
the relevant manifold structure is preserved through the data-dependent masking
process, even for modest mask sizes
The Riemannian Geometry of Deep Generative Models
Deep generative models learn a mapping from a low dimensional latent space to
a high-dimensional data space. Under certain regularity conditions, these
models parameterize nonlinear manifolds in the data space. In this paper, we
investigate the Riemannian geometry of these generated manifolds. First, we
develop efficient algorithms for computing geodesic curves, which provide an
intrinsic notion of distance between points on the manifold. Second, we develop
an algorithm for parallel translation of a tangent vector along a path on the
manifold. We show how parallel translation can be used to generate analogies,
i.e., to transport a change in one data point into a semantically similar
change of another data point. Our experiments on real image data show that the
manifolds learned by deep generative models, while nonlinear, are surprisingly
close to zero curvature. The practical implication is that linear paths in the
latent space closely approximate geodesics on the generated manifold. However,
further investigation into this phenomenon is warranted, to identify if there
are other architectures or datasets where curvature plays a more prominent
role. We believe that exploring the Riemannian geometry of deep generative
models, using the tools developed in this paper, will be an important step in
understanding the high-dimensional, nonlinear spaces these models learn.Comment: 9 page
Multiclass latent locally linear support vector machines
Kernelized Support Vector Machines (SVM) have gained the status of off-the-shelf classifiers, able to deliver state of the art performance on almost any problem. Still, their practical use is constrained by their computational and memory complexity, which grows super-linearly with the number of training samples. In order to retain the low training and testing complexity of linear classifiers and the exibility of non linear ones, a growing, promising alternative is represented by methods that learn non-linear classifiers through local combinations of linear ones. In this paper we propose a new multi class local classifier, based on a latent SVM formulation. The proposed classifier makes use of a set of linear models that are linearly combined using sample and class specific weights. Thanks to the latent formulation, the combination coefficients are modeled as latent variables. We allow soft combinations and we provide a closed-form solution for their estimation, resulting in an efficient prediction rule. This novel formulation allows to learn in a principled way the sample specific weights and the linear classifiers, in a unique optimization problem, using a CCCP optimization procedure. Extensive experiments on ten standard UCI machine learning datasets, one large binary dataset, three character and digit recognition databases, and a visual place categorization dataset show the power of the proposed approach
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