Regularized system identification using orthonormal basis functions

Most of existing results on regularized system identification focus on regularized impulse response estimation. Since the impulse response model is a special case of orthonormal basis functions, it is interesting to consider if it is possible to tackle the regularized system identification using more compact orthonormal basis functions. In this paper, we explore two possibilities. First, we construct reproducing kernel Hilbert space of impulse responses by orthonormal basis functions and then use the induced reproducing kernel for the regularized impulse response estimation. Second, we extend the regularization method from impulse response estimation to the more general orthonormal basis functions estimation. For both cases, the poles of the basis functions are treated as hyperparameters and estimated by empirical Bayes method. Then we further show that the former is a special case of the latter, and more specifically, the former is equivalent to ridge regression of the coefficients of the orthonormal basis functions.Comment: 6 pages, final submission of an contribution for European Control Conference 2015, uploaded on March 20, 201

Tensor-Based Algorithms for Image Classification

Interest in machine learning with tensor networks has been growing rapidly in recent years. We show that tensor-based methods developed for learning the governing equations of dynamical systems from data can, in the same way, be used for supervised learning problems and propose two novel approaches for image classification. One is a kernel-based reformulation of the previously introduced multidimensional approximation of nonlinear dynamics (MANDy), the other an alternating ridge regression in the tensor train format. We apply both methods to the MNIST and fashion MNIST data set and show that the approaches are competitive with state-of-the-art neural network-based classifiers

A Lorentzian Quantum Geometry

We propose a formulation of a Lorentzian quantum geometry based on the framework of causal fermion systems. After giving the general definition of causal fermion systems, we deduce space-time as a topological space with an underlying causal structure. Restricting attention to systems of spin dimension two, we derive the objects of our quantum geometry: the spin space, the tangent space endowed with a Lorentzian metric, connection and curvature. In order to get the correspondence to differential geometry, we construct examples of causal fermion systems by regularizing Dirac sea configurations in Minkowski space and on a globally hyperbolic Lorentzian manifold. When removing the regularization, the objects of our quantum geometry reduce precisely to the common objects of Lorentzian spin geometry, up to higher order curvature corrections.Comment: 65 pages, LaTeX, 4 figures, many small improvements (published version

Conformal regularization of Einstein's field equations

To study asymptotic structures, we regularize Einstein's field equations by means of conformal transformations. The conformal factor is chosen so that it carries a dimensional scale that captures crucial asymptotic features. By choosing a conformal orthonormal frame we obtain a coupled system of differential equations for a set of dimensionless variables, associated with the conformal dimensionless metric, where the variables describe ratios with respect to the chosen asymptotic scale structure. As examples, we describe some explicit choices of conformal factors and coordinates appropriate for the situation of a timelike congruence approaching a singularity. One choice is shown to just slightly modify the so-called Hubble-normalized approach, and one leads to dimensionless first order symmetric hyperbolic equations. We also discuss differences and similarities with other conformal approaches in the literature, as regards, e.g., isotropic singularities.Comment: New title plus corrections and text added. To appear in CQ

Probing the cosmological singularity with a particle

We examine the transition of a particle across the singularity of the compactified Milne (CM) space. Quantization of the phase space of a particle and testing the quantum stability of its dynamics are consistent to one another. One type of transition of a quantum particle is described by a quantum state that is continuous at the singularity. It indicates the existence of a deterministic link between the propagation of a particle before and after crossing the singularity. Regularization of the CM space leads to the dynamics similar to the dynamics in the de Sitter space. The CM space is a promising model to describe the cosmological singularity deserving further investigation by making use of strings and membranes.Comment: 19 pages, 7 figures, revtex4, added references, version accepted for publication in Class. Quantum Gra