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The theory of low-rank tensor-train approximation is well understood when the approximation error is measured in the Frobenius norm. The entrywise maximum norm is equally important but is significantly weaker for large tensors, making the…
We study feedforward neural networks with inputs from a topological vector space (TVS-FNNs). Unlike traditional feedforward neural networks, TVS-FNNs can process a broader range of inputs, including sequences, matrices, functions and more.…
Tree tensor network (TTN) provides an essential theoretical framework for the practical simulation of quantum many-body systems, where the network structure defined by the connectivity of the isometry tensors plays a crucial role in…
Is there any theoretical guarantee for the approximation ability of neural networks? The answer to this question is the "Universal Approximation Theorem for Neural Networks". This theorem states that a neural network is dense in a certain…
Tensor networks are a powerful modeling framework developed for computational many-body physics, which have only recently been applied within machine learning. In this work we utilize a uniform matrix product state (u-MPS) model for…
Belief Propagation is a well-studied message-passing algorithm that runs over graphical models and can be used for approximate inference and approximation of local marginals. The resulting approximations are equivalent to the Bethe-Peierls…
Tensor trains are a versatile tool to compress and work with high-dimensional data and functions. In this work we introduce the Streaming Tensor Train Approximation (STTA), a new class of algorithms for approximating a given tensor…
A topological neural network (TNN), which takes data from a Tychonoff topological space instead of the usual finite dimensional space, is introduced. As a consequence, a distributional neural network (DNN) that takes Borel measures as data…
In this paper, we focus on approximating a natural class of functions that are compositions of smooth functions. Unlike the low-dimensional support assumption on the covariate, we demonstrate that composition functions have an intrinsic…
Progress in the application of machine learning techniques to the prediction of solid-state and molecular materials properties has been greatly facilitated by the development state-of-the-art feature representations and novel deep learning…
Tensor decomposition is one of the fundamental technique for model compression of deep convolution neural networks owing to its ability to reveal the latent relations among complex structures. However, most existing methods compress the…
It is well known that tensor network regression models operate on an exponentially large feature space, but questions remain as to how effectively they are able to utilize this space. Using a polynomial featurization, we propose the…
This work studies approximation based on single-hidden-layer feedforward and recurrent neural networks with randomly generated internal weights. These methods, in which only the last layer of weights and a few hyperparameters are optimized,…
Bivariate matrix functions provide a unified framework for various tasks in numerical linear algebra, including the solution of linear matrix equations and the application of the Fr\'echet derivative. In this work, we propose a novel…
Recurrent neural networks (RNNs) are powerful tools for sequential modeling, but typically require significant overparameterization and regularization to achieve optimal performance. This leads to difficulties in the deployment of large…
The unification of neural and symbolic approaches to artificial intelligence remains a central open challenge. In this work, we introduce a tensor network formalism, which captures sparsity principles originating in the different approaches…
Tensor networks (TNs) enable compact representations of large tensors through shared parameters. Their use in probabilistic modeling is particularly appealing, as probabilistic tensor networks (PTNs) allow for tractable computation of…
Tree tensor networks (TTNs) offer powerful models for image classification. While these TTN image classifiers already show excellent performance on classical hardware, embedding them into quantum neural networks (QNNs) may further improve…
In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only…
Although neural networks traditionally are typically used to approximate functions defined over $\mathbb{R}^n$, the successes of graph neural networks, point-cloud neural networks, and manifold deep learning among other methods have…