The recent introduction of indefinite SVM by Luss and dAspremont [15] has effectively demonstrated SVM classification with a non-positive semi-definite kernel (indefinite kernel). This paper studies the properties of the objective function introduced there. In particular, we show that the objective function is continuously differentiable and its gradient can be explicitly computed. Indeed, we further show that its gradient is Lipschitz continuous. The main idea behind our analysis is that the objective function is smoothed by the penalty term, in its saddle (min-max) representation, measuring the distance between the indefinite kernel matrix and the proxy positive semi-definite one.

In this paper we explore a connection between deep networks and learning in reproducing kernel Krein space. Our approach is based on the concept of push-forward - that is, taking a fixed non-linear transform on a linear projection and converting it to a linear projection on the output of a fixed non-linear transform, pushing the weights forward through the non-linearity. Applying this repeatedly from the input to the output of a deep network, the weights can be progressively "pushed" to the output layer, resulting in a flat network that has the form of a fixed non-linear map (whose form is determined by the structure of the deep network) followed by a linear projection determined by the weight matrices - that is, we take a deep network and convert it to an equivalent (indefinite) kernel machine. We then investigate the implications of this transformation for capacity control and uniform convergence, and provide a Rademacher complexity bound on the deep network in terms of Rademacher complexity in reproducing kernel Krein space. Finally, we analyse the sparsity properties of the flat representation, showing that the flat weights are (effectively) Lp-"norm" regularised with 0

The recent introduction of indefinite SVM by Luss and dAspremont [15] has effectively demonstrated SVM classification with a non-positive semi-definite kernel (indefinite kernel). This paper studies the properties of the objective function introduced there. In particular, we show that the objective function is continuously differentiable and its gradient can be explicitly computed. Indeed, we further show that its gradient is Lipschitz continuous. The main idea behind our analysis is that the objective function is smoothed by the penalty term, in its saddle (min-max) representation, measuring the distance between the indefinite kernel matrix and the proxy positive semi-definite one.

The recent introduction of indefinite SVM by Luss and dAspremont [15] has effectively demonstrated SVM classification with a non-positive semi-definite kernel (indefinite kernel). This paper studies the properties of the objective function introduced there. In particular, we show that the objective function is continuously differentiable and its gradient can be explicitly computed. Indeed, we further show that its gradient is Lipschitz continuous. The main idea behind our analysis is that the objective function is smoothed by the penalty term, in its saddle (min-max) representation, measuring the distance between the indefinite kernel matrix and the proxy positive semi-definite one. Our elementary result greatly facilitates the application of gradient-based algorithms. Based on our analysis, we further develop Nesterovs smooth optimization approach [16,17] for indefinite SVM which has an optimal convergence rate for smooth problems. Experiments on various benchmark datasets validate our analysis and demonstrate the efficiency of our proposed algorithms.