Related papers: Nonlinear System Identification via Tensor Complet…
We consider the task of low-multilinear-rank functional regression, i.e., learning a low-rank parametric representation of functions from scattered real-valued data. Our first contribution is the development and analysis of an efficient…
We consider the ill-posed inverse problem of identifying a nonlinearity in a time-dependent PDE model. The nonlinearity is approximated by a neural network, and needs to be determined alongside other unknown physical parameters and the…
Originating from condensed matter physics, tensor networks are compact representations of high-dimensional tensors. In this paper, the prowess of tensor networks is demonstrated on the particular task of one-class anomaly detection. We…
We study the approximation of functions by tensor networks (TNs). We show that Lebesgue $L^p$-spaces in one dimension can be identified with tensor product spaces of arbitrary order through tensorization. We use this tensor product…
This work studies the problem of high-dimensional data (referred to as tensors) completion from partially observed samplings. We consider that a tensor is a superposition of multiple low-rank components. In particular, each component can be…
In general, approximating classes of functions defined over high-dimensional input spaces by linear combinations of a fixed set of basis functions or ``features'' is known to be hard. Typically, the worst-case error of the best basis set…
The Recurrent Neural Networks and their variants have shown promising performances in sequence modeling tasks such as Natural Language Processing. These models, however, turn out to be impractical and difficult to train when exposed to very…
Factorization machines and polynomial networks are supervised polynomial models based on an efficient low-rank decomposition. We extend these models to the multi-output setting, i.e., for learning vector-valued functions, with application…
Neural networks hold great potential to act as approximate models of nonlinear dynamical systems, with the resulting neural approximations enabling verification and control of such systems. However, in safety-critical contexts, the use of…
We prove an inverse approximation theorem for the approximation of nonlinear sequence-to-sequence relationships using recurrent neural networks (RNNs). This is a so-called Bernstein-type result in approximation theory, which deduces…
A highly recurrent traditional bottleneck in applied mathematics, for which the most popular codes (Mathematica and Matlab) do not offer a solution, is to find all the real solutions of a system of N nonlinear equations in a certain finite…
Motivated by neuronal models from neuroscience, we consider the system identification of simple feedback structures whose behaviors include nonlinear phenomena such as excitability, limit-cycles and chaos. We show that output feedback is…
We study the properties of alignment, a form of implicit regularization, in linear neural networks under gradient descent. We define alignment for fully connected networks with multidimensional outputs and show that it is a natural…
A basic problem in machine learning is to find a mapping $f$ from a low dimensional latent space $\mathcal{Y}$ to a high dimensional observation space $\mathcal{X}$. Modern tools such as deep neural networks are capable to represent general…
The success of Neural networks in providing miraculous results when applied to a wide variety of tasks is astonishing. Insight in the working can be obtained by studying the universal approximation property of neural networks. It is proved…
The exploding research interest for neural networks in modeling nonlinear dynamical systems is largely explained by the networks' capacity to model complex input-output relations directly from data. However, they typically need vast…
This paper is concerned with the approximation of high-dimensional functions in a statistical learning setting, by empirical risk minimization over model classes of functions in tree-based tensor format. These are particular classes of…
With new advances in machine learning and in particular powerful learning libraries, we illustrate some of the new possibilities they enable in terms of nonlinear system identification. For a large class of hybrid systems, we explain how…
A central problem in machine learning is often formulated as follows: Given a dataset $\{(x_j, y_j)\}_{j=1}^M$, which is a sample drawn from an unknown probability distribution, the goal is to construct a functional model $f$ such that…
The state-of-the-art deep neural networks (DNNs) have been widely applied for various real-world applications, and achieved significant performance for cognitive problems. However, the increment of DNNs' width and depth in architecture…