Global versus Localized Generative Adversarial Nets

In this paper, we present a novel localized Generative Adversarial Net (GAN) to learn on the manifold of real data. Compared with the classic GAN that {\em globally} parameterizes a manifold, the Localized GAN (LGAN) uses local coordinate charts to parameterize distinct local geometry of how data points can transform at different locations on the manifold. Specifically, around each point there exists a {\em local} generator that can produce data following diverse patterns of transformations on the manifold. The locality nature of LGAN enables local generators to adapt to and directly access the local geometry without need to invert the generator in a global GAN. Furthermore, it can prevent the manifold from being locally collapsed to a dimensionally deficient tangent subspace by imposing an orthonormality prior between tangents. This provides a geometric approach to alleviating mode collapse at least locally on the manifold by imposing independence between data transformations in different tangent directions. We will also demonstrate the LGAN can be applied to train a robust classifier that prefers locally consistent classification decisions on the manifold, and the resultant regularizer is closely related with the Laplace-Beltrami operator. Our experiments show that the proposed LGANs can not only produce diverse image transformations, but also deliver superior classification performances

Geometric Numerical Integration of the Assignment Flow

The assignment flow is a smooth dynamical system that evolves on an elementary statistical manifold and performs contextual data labeling on a graph. We derive and introduce the linear assignment flow that evolves nonlinearly on the manifold, but is governed by a linear ODE on the tangent space. Various numerical schemes adapted to the mathematical structure of these two models are designed and studied, for the geometric numerical integration of both flows: embedded Runge-Kutta-Munthe-Kaas schemes for the nonlinear flow, adaptive Runge-Kutta schemes and exponential integrators for the linear flow. All algorithms are parameter free, except for setting a tolerance value that specifies adaptive step size selection by monitoring the local integration error, or fixing the dimension of the Krylov subspace approximation. These algorithms provide a basis for applying the assignment flow to machine learning scenarios beyond supervised labeling, including unsupervised labeling and learning from controlled assignment flows

Differential geometric regularization for supervised learning of classifiers

We study the problem of supervised learning for both binary and multiclass classification from a unified geometric perspective. In particular, we propose a geometric regularization technique to find the submanifold corresponding to an estimator of the class probability P(y|\vec x). The regularization term measures the volume of this submanifold, based on the intuition that overfitting produces rapid local oscillations and hence large volume of the estimator. This technique can be applied to regularize any classification function that satisfies two requirements: firstly, an estimator of the class probability can be obtained; secondly, first and second derivatives of the class probability estimator can be calculated. In experiments, we apply our regularization technique to standard loss functions for classification, our RBF-based implementation compares favorably to widely used regularization methods for both binary and multiclass classification.http://proceedings.mlr.press/v48/baia16.pdfPublished versio