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arXiv:2208.09992 · Journal of Chemical Theory and Computation

Efficient Construction of Canonical Polyadic Approximations of Tensor Networks

Karl Pierce, Edward F. Valeev

Published Dec 9, 2022• 11 citations
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Abstract

We consider the problem of constructing a canonical polyadic (CP) decomposition for a tensor network, rather than a single tensor. We illustrate how it is possible to reduce the complexity of constructing an approximate CP representation of the network by leveraging its structure in the course of the CP factor optimization. The utility of this technique is demonstrated for the order-4 Coulomb interaction tensor approximated by two order-3 tensors via an approximate generalized square-root (SQ) factorization, such as density fitting or (pivoted) Cholesky. The complexity of constructing a four-way CP decomposition is reduced from O ( n 4 R CP ) (for the nonapproximated Coulomb tensor) to O ( n 3 R CP ) (for the SQ-factorized Coulomb tensor), where n and R CP are the basis and CP ranks, respectively. This reduces the cost of constructing the CP approximation of two-body interaction tensors of relevance to accurate many-body electronic structure by up to 2 orders of magnitude for systems with up to 36 atoms studied here. The full four-way CP approximation of the Coulomb interaction tensor is shown to be more accurate than the known approaches which utilize CP-factorizations of the SQ factors (which are also constructed with an O ( n 3 R CP ) cost), such as the algebraic pseudospectral and tensor hypercontraction approaches. The CP-decomposed SQ factors can also serve as a robust initial guess for the four-way CP factors.

DOI
10.1021/acs.jctc.2c00861

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We consider the problem of constructing a canonical polyadic (CP) decomposition for a tensor network, rather than a single tensor. We illustrate how it is possible to reduce the complexity of constructing an approximate CP representation of the network by leveraging its structure in the course of the CP factor optimization. The utility of this technique is demonstrated for the order-4 Coulomb interaction tensor approximated by two order-3 tensors via an approximate generalized square-root (SQ) factorization, such as density fitting or (pivoted) Cholesky. The complexity of constructing a four-way CP decomposition is reduced from O ( n 4 R CP ) (for the nonapproximated Coulomb tensor) to O ( n 3 R CP ) (for the SQ-factorized Coulomb tensor), where n and R CP are the basis and CP ranks, respectively. This reduces the cost of constructing the CP approximation of two-body interaction tensors of relevance to accurate many-body electronic structure by up to 2 orders of magnitude for systems with up to 36 atoms studied here. The full four-way CP approximation of the Coulomb interaction tensor is shown to be more accurate than the known approaches which utilize CP-factorizations of the SQ factors (which are also constructed with an O ( n 3 R CP ) cost), such as the algebraic pseudospectral and tensor hypercontraction approaches. The CP-decomposed SQ factors can also serve as a robust initial guess for the four-way CP factors.
Approximation of a tensor network by approximating (e.g., factorizing) one or more of its constituent tensors can be improved by canceling the leading-order error due to the constituents’ approximation. The utility of such robust approximation is demonstrated for robust canonical polyadic (CP) approximation of a (density-fitting) factorized two-particle Coulomb interaction tensor. The resulting algebraic (grid-free) approximation for the Coulomb tensor, closely related to the factorization appearing in pseudospectral and tensor hypercontraction approaches, is efficient and accurate, with significantly reduced rank compared to the naive (nonrobust) approximation. Application of the robust approximation to the particle–particle ladder term in the coupled-cluster singles and doubles reduces the size complexity from O ( N 6 ) to O ( N 5 ) with robustness ensuring negligible errors in chemically relevant energy differences using CP ranks approximately equal to the size of the density-fitting basis.
We investigate a novel approach to approximate tensor-network contraction via the exact, matrix-free decomposition of full tensor-networks. We study this method as a means to eliminate the propagation of error in the approximation of tensor-networks. Importantly, this decomposition-based approach is generic, i.e., it does not depend on a specific tensor-network, the tensor index (physical) ordering, or the choice of tensor decomposition. Careful consideration should be made to determine the best decomposition strategy. Furthermore, this method does not rely on robust cancellation of errors (i.e., the Taylor expansion). As a means to study the effectiveness of the approach, we replace the exact contraction of the particle–particle ladder (PPL) tensor diagram in the popular coupled-cluster with single and double excitation (CCSD) method with a low-rank tensor decomposition, namely the canonical polyadic decomposition (CPD). With this approach, we replace an O ( N 6 ) tensor contractions with a potentially reduced-scaling O ( N 4 R ) optimization problem, where R is the CP rank, and we reduce the computational storage of the PPL tensor from O ( N 4 ) to O ( N R ), although we do not take advantage of this compression in this study. To minimize the cost of the CPD optimization, we utilize the iterative structure of CCSD to efficiently initialize the CPD optimization. We show that accurate chemically relevant energy values can be computed with an error of less than 1 kcal/mol using a relatively low CP rank.
We investigate a novel approach to approximate tensor-network contraction via the exact, matrix-free decomposition of full tensor-networks. We study this method as a means to eliminate the propagation of error in the approximation of tensor-networks. Importantly, this decomposition-based approach is generic, i.e. it does not depend on a specific tensor-network, the tensor index (physical) ordering, or the choice of tensor decomposition. Careful consideration should be made to determine the best decomposition strategy. Furthermore, this method does not rely on robust cancellation of errors (i.e. the Taylor expansion). As a means to study the effectiveness of the approach, we replace the exact contraction of the particle particle ladder (PPL) tensor diagram in the popular coupled-cluster with single and double excitation (CCSD) method with a low-rank tensor decomposition, namely the canonical polyadic decomposition (CPD). With this approach, we replace an O(N6)\mathcal{O}(N^6)O(N6) tensor contractions with a potentially reduced-scaling O(N4R)\mathcal{O}(N^4R)O(N4R) optimization problem, where RRR is the CP rank, and we reduce the computational storage of the PPL tensor from O(N4)\mathcal{O}(N^4)O(N4) to O(NR)\mathcal{O}(NR)O(NR), although we do not take advantage of this compression in this study. To minimize the cost of the CPD optimization, we utilize the iterative structure of CCSD to efficiently initialize the CPD optimization. We show that accurate chemically-relevant energy values can be computed with an error of less than 1 kcal/mol using a relatively low CP rank.
We investigate the efficient combination of the canonical polyadic decomposition (CPD) and tensor hyper-contraction (THC) approaches. We first present a novel low-cost CPD solver that leverages a precomputed THC factorization of an order-4 tensor to efficiently optimize the order-4 CPD with O ( N R 2 ) scaling. With the matrix-free THC-based optimization strategy in hand, we can efficiently generate CPD factorizations of the order-4 two-electron integral tensors and develop novel electronic structure methods that take advantage of both the THC and CPD approximations. Next, we investigate the application of a combined CPD and THC approximation of the Laplace transform (LT) second-order Mo̷ller-Plesset (MP2) method. We exploit the ability to switch efficiently between the THC and CPD factorizations of the two-electron integrals to reduce the computational complexity of the LT MP2 method while preserving the accuracy of the approach. Furthermore, we take advantage of the robust fitting approximation to eliminate the leading-order error in the CPD approximated tensor networks. Finally, we show that modest values of THC and CPD rank preserve the accuracy of the LT MP2 method and that this CPD + THC LT MP2 strategy realizes a performance advantage over canonical LT MP2 in both computational wall-times and memory resource requirements.
In this thesis the application of novel tensor decomposition and tensor representation techniques in highly accurate post Hartree-Fock methods is evaluated. These representation techniques can help to overcome the steep scaling behaviour of high level ab-initio calculations with increasing system size and therefore break the "curse of dimensionality". After a comparison of various tensor formats the application of the "canonical polyadic" format (CP) is described in detail. There, especially the casting of a normal, index based tensor into the CP format (tensor decomposition) and a method for a low rank approximation (rank reduction) of the two-electron integrals in the AO basis are investigated. The decisive quantity for the applicability of the CP format is the scaling of the rank with increasing system and basis set size. The memory requirements and the computational effort for tensor manipulations in the CP format are only linear in the number of dimensions but still depend on the expansion length (rank) of the approximation. Furthermore, the AO-MO transformation and a MP2 algorithm with decomposed tensors in the CP format is evaluated and the scaling with increasing system and basis set size is investigated. Finally, a Coupled-Cluster algorithm based only on low-rank CP representation of the MO integrals is developed. There, especially the successive tensor contraction during the iterative solution of the amplitude equations and the error propagation upon multiple application of the reduction procedure are discussed. In conclusion the overall complexity of a Coupled-Cluster procedure with tensors in CP format is evaluated and some possibilities for improvements of the rank reduction procedure tailored to the needs in electronic structure calculations are shown.
We investigate the efficient combination of the canonical polyadic decomposition (CPD) and tensor hyper-contraction (THC) approaches. We first present a novel low-cost CPD solver which leverages a precomputed THC factorization of an order-444 tensor to efficiently optimize the order-444 CPD with O(NR2)\mathcal{O}(NR^2)O(NR2) scaling. With the matrix-free THC-based optimization strategy in hand we can: efficiently generate CPD factorizations of the order-4 two-electron integral tensors; and develop novel electronic structure methods which take advantage of both the THC and CPD approximations. Next, we investigate the application of a combined CPD and THC approximation of the Laplace transform (LT) second-order Møller-Plesset (MP2) method. We exploit the ability to switch efficiently between the THC and CPD factorizations of the two electron integrals to reduce the computational complexity of the LT MP2 method while preserving the accuracy of the approach. Furthermore we take advantage of the robust fitting approximation to eliminate leading order error in the CPD approximated tensor networks. Finally, we show that modest values of THC and CPD rank preserve the accuracy of the LT MP2 method and that this CPD+THC LT MP2 strategy realizes a performance advantage over canonical LT MP2 in both computational wall-times and memory resource requirements.