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Tensor decompositions such as the canonical format and the tensor train format have been widely utilized to reduce storage costs and operational complexities for high-dimensional data, achieving linear scaling with the input dimension…

The efficient simulation of complex quantum systems remains a central challenge due to the exponential growth of Hilbert space with system size. Tensor network methods have long been established as powerful approximation schemes, and their…

High-dimensional data arise naturally in many areas of science and engineering, including machine learning, signal processing, computational physics, and statistics. Such data are often represented as tensors, multi-dimensional…

Tensor networks are a popular and computationally efficient approach to simulate general quantum systems on classical computers and, in a broader sense, a framework for dealing with high-dimensional numerical problems. This paper presents a…

To respond to the need of efficient training and inference of deep neural networks, a plethora of domain-specific hardware architectures have been introduced, such as Google Tensor Processing Units and NVIDIA Tensor Cores. A common feature…

In this paper, we propose a type of tensor-neural-network-based machine learning method to compute multi-eigenpairs of high dimensional eigenvalue problems without Monte-Carlo procedure. Solving multi-eigenvalues and their corresponding…

In this paper, we introduce a tensor neural network based machine learning method for solving the elliptic partial differential equations with random coefficients in a bounded physical domain. With the help of tensor product structure, we…

Random quantum circuits have been utilized in the contexts of quantum supremacy demonstrations, variational quantum algorithms for chemistry and machine learning, and blackhole information. The ability of random circuits to approximate any…

Tensor networks represent the state-of-the-art in computational methods across many disciplines, including the classical simulation of quantum many-body systems and quantum circuits. Several applications of current interest give rise to…

Motion planning and control problems are embedded and essential in almost all robotics applications. These problems are often formulated as stochastic optimal control problems and solved using dynamic programming algorithms. Unfortunately,…

We develop new approximation algorithms and data structures for representing and computing with multivariate functions using the functional tensor-train (FT), a continuous extension of the tensor-train (TT) decomposition. The FT represents…

We present a high-accuracy procedure for electronic structure calculations of strongly correlated materials. To address limitations in current electronic structure methods, we employ density functional theory in combination with the…

This paper proposes a novel method for learning highly nonlinear, multivariate functions from examples. Our method takes advantage of the property that continuous functions can be approximated by polynomials, which in turn are representable…

We show that under some widely believed assumptions, there are no higher-order algorithms for basic tasks in computational mathematics such as: Computing integrals with neural network integrands, computing solutions of a Poisson equation…

Tensor network contraction is a fundamental mathematical operation that generalizes the dot product and matrix multiplication. It finds applications in numerous domains, such as database systems, graph theory, machine learning, probability…

Hyperdimensional Computing affords simple, yet powerful operations to create long Hyperdimensional Vectors (hypervectors) that can efficiently encode information, be used for learning, and are dynamic enough to be modified on the fly. In…

A tensor network is a type of decomposition used to express and approximate large arrays of data. A given data-set, quantum state or higher dimensional multi-linear map is factored and approximated by a composition of smaller multi-linear…

Low-rank tensor approximation approaches have become an important tool in the scientific computing community. The aim is to enable the simulation and analysis of high-dimensional problems which cannot be solved using conventional methods…

The curse of dimensionality associated with the Hilbert space of spin systems provides a significant obstruction to the study of condensed matter systems. Tensor networks have proven an important tool in attempting to overcome this…

Neural networks successfully capture the computational power of the human brain for many tasks. Similarly inspired by the brain architecture, Nearest Neighbor (NN) representations is a novel approach of computation. We establish a firmer…