A representer theorem for deep kernel learning

In this paper we provide a finite-sample and an infinite-sample representer theorem for the concatenation of (linear combinations of) kernel functions of reproducing kernel Hilbert spaces. These results serve as mathematical foundation for the analysis of machine learning algorithms based on compositions of functions. As a direct consequence in the finite-sample case, the corresponding infinite-dimensional minimization problems can be recast into (nonlinear) finite-dimensional minimization problems, which can be tackled with nonlinear optimization algorithms. Moreover, we show how concatenated machine learning problems can be reformulated as neural networks and how our representer theorem applies to a broad class of state-of-the-art deep learning methods

A Generalized Representer Theorem for Hilbert Space - Valued Functions

The necessary and sufficient conditions for existence of a generalized representer theorem are presented for learning Hilbert space - valued functions. Representer theorems involving explicit basis functions and Reproducing Kernels are a common occurrence in various machine learning algorithms like generalized least squares, support vector machines, Gaussian process regression, and kernel-based deep neural networks to name a few. Due to the more general structure of the underlying variational problems, the theory is also relevant to other application areas like optimal control, signal processing and decision making. The following presents a generalized representer theorem using the theory of closed, densely defined linear operators and subspace valued maps as a means to address variational optimization problems in learning and control. The implications of the theorem are presented with examples of multi-input - multi-output problems from kernel-based deep neural networks, stochastic regression and sparsity learning problems

On the composition of neural and kernel layers for machine learning

Deep Learning architectures in which neural layers alternate with mappings to infinitedimensional feature spaces have been proposed in recent years, showing improvements on the results obtained when using either technique separately. However, these new algorithms have been presented without delving into the rich mathematical structure that sustains kernel methods. The main focus of this thesis is not only to review these advances in the field of Deep Learning, but to extend and generalize them by defining a broader family of models that operate under the mathematical framework defined by the composition of a neural layerwith a kernel mapping, all of which operate in reproducing kernel Hilbert spaces thatare then concatenated. Each of these spaces has a specific reproducing kernel that we can characterize. Together all of this defines a regularization-based learning optimization problem, for which we prove that minimizers exist. This strong mathematical background is complemented by the presentation of a new a model, the Kernel Network, which manages to produce successful results on many classification problems