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Nature evolves structures like honeycombs at optimized performance with limited material. These efficient structures can be artificially created with the collaboration of structural topology optimization and additive manufacturing. However,…
Generalizing work of K\"unnemann, Paturi, and Schneider [ICALP 2017], we study a wide class of high-dimensional dynamic programming (DP) problems in which one must find the shortest path between two points in a high-dimensional grid given a…
We demonstrate that perturbative expansions for quantum many-body systems can be rephrased in terms of tensor networks, thereby providing a natural framework for interpolating perturbative expansions across a quantum phase transition. This…
Numerical integration is a classical problem emerging in many fields of science. Multivariate integration cannot be approached with classical methods due to the exponential growth of the number of quadrature nodes. We propose a method to…
Tensor networks are an efficient platform to represent interesting quantum states of matter as well as to compute physical observables and information-theoretic quantities. We present a general protocol to construct fixed-point tensor…
Open dynamical systems are mathematical models of machines that take input, change their internal state, and produce output. For example, one may model anything from neurons to robots in this way. Several open dynamical systems can be…
Low rank tensor decompositions are a powerful tool for learning generative models, and uniqueness results give them a significant advantage over matrix decomposition methods. However, tensors pose significant algorithmic challenges and…
In this work, we study the tensor ring decomposition and its associated numerical algorithms. We establish a sharp transition of algorithmic difficulty of the optimization problem as the bond dimension increases: On one hand, we show the…
Thresholding--the pruning of nodes or edges based on their properties or weights--is an essential preprocessing tool for extracting interpretable structure from complex network data, yet existing methods face several key limitations.…
Adaptive transport networks in biological and physical systems exhibit hierarchical organization, characteristic channel spacing, and robust scaling relations. Existing adaptive network models, formulated on a lattice, successfully…
We introduce structured decompositions, category-theoretic structures which simultaneously generalize notions from graph theory (including treewidth, layered treewidth, co-treewidth, graph decomposition width, tree independence number,…
This paper is concerned with the approximation of tensors using tree-based tensor formats, which are tensor networks whose graphs are dimension partition trees. We consider Hilbert tensor spaces of multivariate functions defined on a…
Recent years have witnessed an increased interest in recovering dynamical laws of complex systems in a largely data-driven fashion under meaningful hypotheses. In this work, we propose a scalable and numerically robust method for this task,…
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…
This paper considers a multiple stopping time problem for a Markov chain observed in noise, where a decision maker chooses at most L stopping times to maximize a cumulative objective. We formulate the problem as a Partially Observed Markov…
Tensors or {\em multi-way arrays} are functions of three or more indices $(i,j,k,\cdots)$ -- similar to matrices (two-way arrays), which are functions of two indices $(r,c)$ for (row,column). Tensors have a rich history, stretching over…
We develop fast spectral algorithms for tensor decomposition that match the robustness guarantees of the best known polynomial-time algorithms for this problem based on the sum-of-squares (SOS) semidefinite programming hierarchy. Our…
The widespread use of multi-sensor technology and the emergence of big datasets has highlighted the limitations of standard flat-view matrix models and the necessity to move towards more versatile data analysis tools. We show that…
Tensor train (TT) decomposition, a powerful tool for analyzing multidimensional data, exhibits superior performance in many machine learning tasks. However, existing methods for TT decomposition either suffer from noise overfitting, or…
Tensor networks (TNs) are one of the best available tools to study many-body quantum systems. TNs are particularly suitable for one-dimensional local Hamiltonians, while their performance for generic geometries is mainly limited by two…