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On the Optimal Linear Contraction Order of Tree Tensor Networks, and Beyond

Mihail Stoian, Richard Milbradt, Christian B. Mendl

TL;DR

It is shown that the linear contraction ordering problem for tree tensor networks admits a polynomial-time algorithm, by drawing connections to database join ordering, which relies on the adjacent sequence interchange property of the contraction cost.

Abstract

The contraction cost of a tensor network depends on the contraction order. However, the optimal contraction ordering problem is known to be NP-hard. We show that the linear contraction ordering problem for tree tensor networks admits a polynomial-time algorithm, by drawing connections to database join ordering. The result relies on the adjacent sequence interchange property of the contraction cost, which enables a global decision of the contraction order based on local comparisons. Based on that, we specify a modified version of the IKKBZ database join ordering algorithm to find the optimal tree tensor network linear contraction order. Finally, we extend our algorithm as a heuristic to general contraction orders and arbitrary tensor network topologies.

On the Optimal Linear Contraction Order of Tree Tensor Networks, and Beyond

TL;DR

It is shown that the linear contraction ordering problem for tree tensor networks admits a polynomial-time algorithm, by drawing connections to database join ordering, which relies on the adjacent sequence interchange property of the contraction cost.

Abstract

The contraction cost of a tensor network depends on the contraction order. However, the optimal contraction ordering problem is known to be NP-hard. We show that the linear contraction ordering problem for tree tensor networks admits a polynomial-time algorithm, by drawing connections to database join ordering. The result relies on the adjacent sequence interchange property of the contraction cost, which enables a global decision of the contraction order based on local comparisons. Based on that, we specify a modified version of the IKKBZ database join ordering algorithm to find the optimal tree tensor network linear contraction order. Finally, we extend our algorithm as a heuristic to general contraction orders and arbitrary tensor network topologies.
Paper Structure (38 sections, 2 theorems, 20 equations, 14 figures, 2 algorithms)

This paper contains 38 sections, 2 theorems, 20 equations, 14 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Tensor Contraction
  4. Contraction Order
  5. Contraction Tree
  6. Contraction Cost
  7. Algorithm
  8. Precedence Graph
  9. Adjacent Sequence Interchange Property
  10. ASI Exemplification
  11. Cost Function Reformulation
  12. TensorIKKBZ
  13. Pseudocode
  14. Time Complexity
  15. Generalized Optimization Approaches
  16. ...and 23 more sections

Key Result

Theorem 1

$\mathcal{C}$ satisfies the ASI property for the score function $\sigma \colon \mathbb{S} \to \mathbb{N}^+ \times \mathbb{Z}$ defined as In particular, we define "$\leq$" in Def. def:asi as where we refer to $\sigma_1(S)$ and $\sigma_2(S)$ as the numerator and denominator, respectively, of the symbolic fraction $\sigma(S)$.

Figures (14)

  • Figure 1: A network with three tensors
  • Figure 2: Linear contraction order of a tensor network with four tensors
  • Figure 3: General contraction order of a tensor network with four tensors
  • Figure 4: Contraction tree types
  • Figure 5: Precedence graph of $T^{[4]}$ (right) is obtained by rooting the tree tensor network (left) in $T^{[4]}$
  • ...and 9 more figures

Theorems & Definitions (6)

  • Definition 1: ASI Property Monma1979SequencingWS
  • Theorem 1
  • Theorem 2
  • proof
  • proof
  • proof