Related papers: Approximation Theory of Tree Tensor Networks: Tens…

We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some linear methods of this approximation for univariate functions in the class induced by the convolution…

Tree tensor networks (TTNs) are widely used in low-rank approximation and quantum many-body simulation. In this work, we present a formal analysis of the differential geometry underlying TTNs. Building on this foundation, we develop…

We investigate smooth approximations of functions, with prescribed gradient behavior on a distinguished stratified subset of the domain. As an application, we outline how our results yield important consequences for a recently introduced…

Deep learning based on deep neural networks of various structures and architectures has been powerful in many practical applications, but it lacks enough theoretical verifications. In this paper, we consider a family of deep convolutional…

The paper briefy reviews several recent results on hierarchical architectures for learning from examples, that may formally explain the conditions under which Deep Convolutional Neural Networks perform much better in function approximation…

We introduce a class of quantum non-Markovian processes -- dubbed process trees -- that exhibit polynomially decaying temporal correlations and memory distributed across time scales. This class of processes is described by a tensor network…

In this paper we apply methods originated in Complexity theory to some problems of Approximation. We notice that the construction of Alman and Williams that disproves the rigidity of Walsh-Hadamard matrices, provides good…

Inspired by coarse-graining approaches used in physics, we show how similar algorithms can be adapted for data. The resulting algorithms are based on layered tree tensor networks and scale linearly with both the dimension of the input and…

In this Master Thesis, we study the approximation capabilities of Neural Networks in the context of numerical resolution of elliptic PDEs and Approximation Theory. First of all, in Chapter 1, we introduce the mathematical definition of…

We show that deep networks are better than shallow networks at approximating functions that can be expressed as a composition of functions described by a directed acyclic graph, because the deep networks can be designed to have the same…

In this work, we firstly apply the Train-Tensor (TT) networks to construct a compact representation of the classical Multilayer Perceptron, representing a reduction of up to 95% of the coefficients. A comparative analysis between tensor…

We introduce a change of perspective on tensor network states that is defined by the computational graph of the contraction of an amplitude. The resulting class of states, which we refer to as tensor network functions, inherit the…

Recently, there has been a growing interest in efficient numerical algorithms based on tensor networks and low-rank techniques to approximate high-dimensional functions and solutions to high-dimensional PDEs. In this paper, we propose a new…

We present new results on the classical algorithm of variable elimination, which underlies many algorithms including for probabilistic inference. The results relate to exploiting functional dependencies, allowing one to perform inference…

For the past 30 years or so, machine learning has stimulated a great deal of research in the study of approximation capabilities (expressive power) of a multitude of processes, such as approximation by shallow or deep neural networks,…

We analyze rates of approximation by quantized, tensor-structured representations of functions with isolated point singularities in ${\mathbb R}^3$. We consider functions in countably normed Sobolev spaces with radial weights and analytic-…

Several recent works have shown separation results between deep neural networks, and hypothesis classes with inferior approximation capacity such as shallow networks or kernel classes. On the other hand, the fact that deep networks can…

Tensor networks have found a wide use in a variety of applications in physics and computer science, recently leading to both theoretical insights as well as practical algorithms in machine learning. In this work we explore the connection…

In this paper, a universal approximation theorem (UAT) for shallow neural networks whose inputs belong to a topological vector space (TVS) and whose outputs take values in a Hausdorff locally convex TVS is established. The networks are…

Despite their simple intuition, convolutions are more tedious to analyze than dense layers, which complicates the transfer of theoretical and algorithmic ideas to convolutions. We simplify convolutions by viewing them as tensor networks…