Arithmetic circuit tensor networks, multivariable function representation, and high-dimensional integration
Abstract
Many computational problems can be formulated in terms of high-dimensional functions. Simple representations of such functions and resulting computations with them typically suffer from the "curse of dimensionality," an exponential cost dependence on dimension. Tensor networks provide a way to represent certain classes of high-dimensional functions with polynomial memory. This results in computations where the exponential cost is ameliorated or, in some cases, removed, if the tensor network representation can be obtained. Here, we introduce a direct mapping from the arithmetic circuit of a function to arithmetic circuit tensor networks, avoiding the need to perform any optimization or functional fit. We demonstrate the power of the circuit construction in examples of multivariable integration on the unit hypercube in up to 50 dimensions, where the complexity of integration can be understood from the circuit structure. We find very favorable cost scaling compared with quasi–Monte Carlo integration for these cases and further give an example where efficient quasi–Monte Carlo integration cannot be performed without knowledge of the underlying tensor network circuit structure.
Additional Information
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. R.P. (arithmetic tensor network development) was partially supported by an Eddleman Graduate Fellowship and the US Department of Energy, Office of Science, via Award No. DE-SC0019390. Work by J.G. (Quimb development and support) was supported by the US Department of Energy, Office of Science, via Award No. DE-SC0018140. G.K.-L.C. acknowledges support from the Simons Investigator award.Attached Files
Published - PhysRevResearch.5.013156.pdf
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Additional details
- Eprint ID
- 121334
- Resolver ID
- CaltechAUTHORS:20230509-419283000.1
- Eddleman Graduate Fellowship
- DE-SC0019390
- Department of Energy (DOE)
- DE-SC0018140
- Department of Energy (DOE)
- Simons Foundation
- Created
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2023-05-13Created from EPrint's datestamp field
- Updated
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2023-05-13Created from EPrint's last_modified field