Efficient construction of canonical polyadic approximations of tensor networks

TL;DR

The work tackles the challenge of constructing a CP decomposition for a tensor network, specifically the 4-index Coulomb interaction tensor in quantum chemistry, by exploiting a generalized square-root (SQ) factorization (density fitting or Cholesky) to form a SQ representation that feeds a 4-way CP decomposition (CP4). This approach reduces the dominant gradient/computation cost from $\mathcal{O}(n^4 R)$ to $\mathcal{O}(n^3 R)$ and storage from $\mathcal{O}(n^4)$ to $\mathcal{O}(n^3)$, enabling scalable CP4 factorizations. A CP3-SQ initialization plus ALS optimization yields competitive or superior accuracy compared to CP3-DF and related PS/THC-like methods, with speedups in CP4 ALS iterations and applicability to large tensors (e.g., a 31 GB tensor for a (H2O)$_{12}$ cluster). The study also highlights symmetry considerations: CP4 does not automatically preserve 8-fold permutational symmetry, but a posteriori symmetrization improves accuracy, suggesting symmetry-adapted CP ans"atze as a promising direction. Overall, the results demonstrate a practical, scalable route to accurate CP representations of tensor networks in electronic-structure calculations and point to broader applicability beyond Coulomb integrals.

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 2 order-3 tensors via an approximate generalized square-root (SQ) factorization, such as density fitting or (pivoted) Cholesky. The complexity of constructing a 4-way CP decomposition is reduced from $\mathcal{O}(n^4 R_\text{CP})$ (for the non-approximated Coulomb tensor) to $\mathcal{O}(n^3 R_\text{CP})$ for the SQ-factorized tensor, where $n$ and $R_\text{CP}$ are the basis and CP ranks, respectively. This reduces the cost of constructing the CP approximation of 2-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 4-way CP approximation of the Coulomb interaction tensor is shown to be more accurate than the known approaches utilizing CP-decomposed SQ factors (also obtained at the $\mathcal{O}(n^3 R_\text{CP})$ 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 4-way CP factors.

Figures (5)

• Figure 1: Absolute average element error for approximated $g^{ab}_{ci}$ tensor using an ALS solver thresholdVRG:pierce:2021:JCTC of (a) $\epsilon_\mathrm{ALS}=10^{-3}$ and (b) $\epsilon_\mathrm{ALS}=10^{-5}$. Solid lines represents mean error and dashed lines represent max error. Note that the CP4 rank is in units of a factor times the DF fitting basis, a metric which grows linearly with rank. Data was collected using the S66/5 dataset with a aVDZ/aVDZ-RI basis
• Figure 2: Absolute average element error for approximated $g^{ab}_{cd}$ tensor using an ALS solver thresholdVRG:pierce:2021:JCTC of (a) $\epsilon_\mathrm{ALS}=10^{-3}$ and (b) $\epsilon_\mathrm{ALS}=10^{-5}$. Solid lines represents mean error and dashed lines represent max error. Note that the CP4 rank is in units of a factor times the DF fitting basis, a metric which grows linearly with rank. Data was collected using the S66/5 dataset with a aVDZ/aVDZ-RI basis
• Figure 3: Distributions of the absolute element values for $g^{ab}_{ci}$ [subfigures (a) and (b)] and $g^{ab}_{cd}$ [subfigures (c) and (d)] tensors for representative systems. "Exact" refers to the DF-based tensor free of CP approximations, while "nX" corresponds to the counterpart approximated with CP4 decomposition of rank $nX$. All computations utilized a aVDZ/aVDZ-RI basis and $\epsilon_\mathrm{ALS} = 10^{-3}$.
• Figure 4: The cost per iteration of the exact and DF approximated CP4 ALS decomposition and the total speedup associated with the DF approximation, i.e. $\frac{t_\mathrm{exact}}{t_\mathrm{DF}}$. The error bars represent the maximum and minimum per iteration cost. Data was collected using the S66/5 dataset and the aVDZ/aVDZ-RI basis pair.
• Figure 5: The total speedup of the CP4 ALS optimization of the (a) $g^{ab}_{ci}$ tensor and (c) $g^{ab}_{cd}$ tensor and the per iteration speedup of the (b) $g^{ab}_{ci}$ tensor and (d) $g^{ab}_{cd}$ tensor using a (H2O)$_{2}$ cluster from the S66 dataset and 4 different orbital and DF basis sets