Alternating cross interpolation performs elementwise operations on tensor trains in O(χ³) time with error control, improving on the standard O(χ⁴) scaling when output ranks are controlled.
Compressing multivariate func- tions with tree tensor networks
4 Pith papers cite this work, alongside 2 external citations. Polarity classification is still indexing.
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abstract
Tensor networks are a compressed format for multi-dimensional data. One dimensional tensor networks -- often referred to as tensor trains (TT) or matrix product states (MPS) -- are increasingly being used as a numerical ansatz for continuum functions by ``quantizing'' the inputs into discrete binary digits. Here we demonstrate the power of more general tree tensor networks (TTNs) for this purpose. We provide direct constructions of a number of elementary functions as generic tree tensor networks and interpolative constructions for more complicated functions via a generalization of the tensor cross interpolation algorithm. For a range of multi-dimensional functions we show how more structured tree tensor networks offer a significantly more efficient ansatz than the commonly used tensor train. Finally, we demonstrate how the methods introduced in this work can be used to realize a TTN-based solver for multi-dimensional, non-linear Fredholm equations.
verdicts
UNVERDICTED 4representative citing papers
TN-SHAP-G trains a graph-aligned tensor network multilinear surrogate to enable exact, sampling-free computation of Shapley values and higher-order interactions on graph inputs.
Tailoring tensor network algorithms to the scale hierarchy in quantics representation produces faster, more robust solvers for high-dimensional linear and eigenvalue PDE problems.
Integral decimation builds spectral tensor train representations of integrands via quantum gate sequences to achieve polynomial-time evaluation of high-dimensional integrals for statistical mechanics and quantum dynamics.
citing papers explorer
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Fast elementwise operations on tensor trains with alternating cross interpolation
Alternating cross interpolation performs elementwise operations on tensor trains in O(χ³) time with error control, improving on the standard O(χ⁴) scaling when output ranks are controlled.
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TN-SHAP-G: Graph-Structured Tensor Network Surrogates for Shapley Values and Interactions
TN-SHAP-G trains a graph-aligned tensor network multilinear surrogate to enable exact, sampling-free computation of Shapley values and higher-order interactions on graph inputs.
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Tailoring tensor network techniques to the quantics representation for highly inhomogeneous problems and few body problems
Tailoring tensor network algorithms to the scale hierarchy in quantics representation produces faster, more robust solvers for high-dimensional linear and eigenvalue PDE problems.
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The Integral Decimation Method for Quantum Dynamics and Statistical Mechanics
Integral decimation builds spectral tensor train representations of integrands via quantum gate sequences to achieve polynomial-time evaluation of high-dimensional integrals for statistical mechanics and quantum dynamics.