Generalized Canonical Polyadic Tensor Decompositions with General Symmetry

2026-01-08

arXiv.org

AI Summary

This work addresses the limitation of existing generalized CANDECOMP/PARAFAC (CP) decomposition methods, which typically overlook the prevalent symmetry structures in tensors—such as those arising in dynamic graphs—and thus struggle to model symmetric data effectively. To overcome this, we propose Symmetric Generalized CP (SymGCP) decomposition, which, for the first time, incorporates symmetry constraints over arbitrary subsets of tensor modes into the generalized CP framework. While preserving the flexibility of general loss functions, SymGCP derives gradient expressions compatible with existing tensor kernels and introduces an efficient stochastic optimization algorithm. Experimental results demonstrate that SymGCP significantly improves both modeling accuracy and scalability on synthetic and real-world large-scale symmetric tensor data.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization

Abstract

Canonical Polyadic (CP) tensor decomposition is a workhorse algorithm for discovering underlying low-dimensional structure in tensor data. This is accomplished in conventional CP decomposition by fitting a low-rank tensor to data with respect to the least-squares loss. Generalized CP (GCP) decompositions generalize this approach by allowing general loss functions that can be more appropriate, e.g., to model binary and count data or to improve robustness to outliers. However, GCP decompositions do not explicitly account for any symmetry in the tensors, which commonly arises in modern applications. For example, a tensor formed by stacking the adjacency matrices of a dynamic graph over time will naturally exhibit symmetry along the two modes corresponding to the graph nodes. In this paper, we develop a symmetric GCP (SymGCP) decomposition that allows for general forms of symmetry, i.e., symmetry along any subset of the modes. SymGCP accounts for symmetry by enforcing the corresponding symmetry in the decomposition. We derive gradients for SymGCP that enable its efficient computation via all-at-once optimization with existing tensor kernels. The form of the gradients also leads to various stochastic approximations that enable us to develop stochastic SymGCP algorithms that can scale to large tensors. We demonstrate the utility of the proposed SymGCP algorithms with a variety of experiments on both synthetic and real data.

Problem

• tensor decomposition
• symmetry
• Generalized Canonical Polyadic
• dynamic graph
• low-dimensional structure

Innovation

• Symmetric GCP
• Tensor decomposition
• Generalized loss functions
• Symmetry constraints
• Stochastic optimization