[Paper Review] Number-State Preserving Tensor Networks as Classifiers for Supervised Learning

Glen Evenbly|arXiv (Cornell University)|May 15, 2019

Computational Physics and Python Applications4 citations

TL;DR

This paper proposes number-state preserving tensor networks—built from tensors that map number states to number states—as efficient and interpretable classifiers for supervised learning. By preserving classical data structure during transformations and enabling efficient training via causal cone contraction, the method makes previously intractable networks like MERA computationally viable for tasks such as image recognition, achieving strong performance on benchmark datasets with minimal computational overhead.

ABSTRACT

We propose a restricted class of tensor network state, built from number-state preserving tensors, for supervised learning tasks. This class of tensor network is argued to be a natural choice for classifiers as (i) they map classical data to classical data, and thus preserve the interpretability of data under tensor transformations, (ii) they can be efficiently trained to maximize their scalar product against classical data sets, and (iii) they seem to be as powerful as generic (unrestricted) tensor networks in this task. Our proposal is demonstrated using a variety of benchmark classification problems, where number-state preserving versions of commonly used networks (including MPS, TTN and MERA) are trained as effective classifiers. This work opens the path for powerful tensor network methods such as MERA, which were previously computationally intractable as classifiers, to be employed for difficult tasks such as image recognition.

Motivation & Objective

• To address interpretability and computational intractability in using tensor networks as classifiers for classical data.
• To resolve the issue of generic tensor networks mapping classical product states to entangled quantum states, which obscures classical interpretability.
• To enable efficient training of complex tensor networks—especially MERA—by restricting to number-state preserving tensors.
• To demonstrate that this restricted class of tensor networks performs as well as unrestricted networks on benchmark classification tasks.
• To open the door for powerful tensor network architectures like MERA to be used in practical machine learning applications such as image recognition.

Proposed method

• The method uses a restricted class of tensor networks composed of number-state preserving tensors, which map Fock number states on input indices to number states on output indices.
• It leverages a causal cone structure when contracting against number states, enabling efficient computation of scalar products between the network and training data.
• The training algorithm optimizes the network to maximize the scalar product with labeled training data, using efficient environment evaluation via dynamic programming.
• The approach allows for efficient evaluation of single-tensor environments, a key step in gradient-based optimization, by exploiting the causal structure of number-state preserving networks.
• The method is applied to standard architectures like MPS, TTN, and MERA, adapted to preserve number states, and trained using standard supervised learning objectives.
• The computational cost scales favorably—e.g., O(χ⁸) for a 2D number-state preserving MERA versus O(χ¹⁶) for isometric MERA—making larger bond dimensions feasible.

Experimental results

Research questions

• RQ1 Can number-state preserving tensor networks maintain interpretability while enabling efficient training for supervised learning?
• RQ2 Does restricting tensor networks to number-state preserving tensors preserve sufficient expressive power for classification tasks?
• RQ3 Can computationally intractable tensor networks like MERA be made viable as classifiers through this restriction?
• RQ4 How does the performance of number-state preserving networks compare to unrestricted tensor networks on standard benchmarks?
• RQ5 What is the computational scaling advantage of number-state preserving networks over standard isometric tensor networks in training?

Key findings

• Number-state preserving tensor networks preserve classical data structure during transformations, ensuring interpretability as inputs and outputs remain classical number states.
• The causal cone structure in number-state preserving networks enables efficient evaluation of scalar products with training data, making training feasible even for complex networks like MERA.
• The method allows for efficient training of MERA-like networks, which were previously computationally intractable for classification tasks.
• Benchmark results show that number-state preserving versions of MPS, TTN, and MERA achieve strong performance on standard classification problems, comparable to unrestricted networks.
• The computational cost of optimizing number-state preserving networks scales as O(χ⁸) for a 2D MERA, a square-root improvement over the O(χ¹⁶) scaling of isometric MERA, enabling larger bond dimensions under the same computational budget.
• The approach enables parallelization of environment evaluation over training samples, mitigating the cost of large training sets.