Adaptive Tensor Networks Decomposition

Grey Ballard, Tamara G. Kolda

Meraj Hashemizadeh, Meilin Liu, Jacob Miller +1 more

Tensor decomposition techniques have shown great successes in machine learning and data science by extending classical algorithms based on matrix factorization to multi-modal and multi-way data. However, there exist many tensor decomposition models~(CP, Tucker, Tensor Train, etc.) and the rank of such a decomposition is typically a collection of integers rather than a unique number, making model and hyper-parameter selection a tedious and costly task. At the same time, tensor network methods are powerful tools developed in the physics community which have recently shown their potential for machine learning applications and offer a unifying view of the various tensor decomposition models. In this paper, we leverage the tensor network formalism to develop a generic and efficient adaptive algorithm for tensor learning. Our method is based on a simple greedy approach optimizing a differentiable loss function starting from a rank one tensor and successively identifying the most promising tensor network edges for small rank increments. Our algorithm can adaptively identify tensor network structures with small number of parameters that effectively optimize the objective function from data. The framework we introduce is very broad and encompasses many common tensor optimization problems. Experiments on tensor decomposition and tensor completion tasks with both synthetic and real world data demonstrate the effectiveness of the proposed algorithm.

Samuel Desrosiers, Glen Evenbly, Thomas E. Baker

Nicole DeGregorio, Srinivasan S. Iyengar

We present an approach to reduce the computational complexity and storage pertaining to quantum nuclear wave functions and potential energy surfaces. The method utilizes tensor networks implemented through sequential singular value decompositions. Two specific forms of tensor networks are considered to adaptively compress the data in multidimensional quantum nuclear wave functions and potential energy surfaces. In one case the well-known matrix product state approximation is used whereas in another case the wave function and potential energy surface space is initially partitioned into "system" and "bath" degrees of freedom through singular value decomposition, following which the individual system and bath tensors (wave functions and potentials) are in turn decomposed as matrix product states. We postulate that this leads to a mean-field version of the well-known projectionally entangled pair state known in the tensor networks community. Both formulations appear as special cases of more general higher order singular value decompositions known in the mathematics literature as Tucker decomposition. The networks are then used to study the hydrogen transfer step in the oxidation of isoprene by peroxy and hydroxy radicals. We find that both networks are extremely efficient in accurately representing quantum nuclear eigenstates and potential energy surfaces and in computing inner products between quantum nuclear eigenstates and a final-state basis to yield product side probabilities. We also present formal protocols that will be useful to perform explicit quantum nuclear dynamics.

Jincheng Zhang

Quantum machine learning represents a paradigm shift in computational capabilities, offering the potential to solve complex problems currently intractable for classical algorithms. This paper explores the application of adaptive tensor decomposition as a novel method to enhance quantum machine learning models. We hypothesize that dynamically adjusting the weight matrices within a quantum neural network through adaptive tensor decomposition can significantly improve model accuracy and efficiency. The core mechanism focuses on a novel algorithm that adjusts the magnitude of each tensor component based on the training process, leading to a more robust and adaptable learning framework. This work details the theoretical foundations, implementation of the adaptive tensor decomposition algorithm, and preliminary experimental results demonstrating its potential to surpass existing classical machine learning approaches in certain benchmark datasets. The findings suggest a significant advancement in quantum machine learning model design and training.

Timo Felser, Simone Montangero