Jacob Miller

Raphaelle Tihon

Alejandro Perdomo-ortiz

2023-10-31

ArXiv (prépublication)

doi.org

Quantum Tensor Networks, Stochastic Processes, and Weighted Automata

Siddarth Srinivasan

Sandesh M. Adhikary

Byron Boots

Modeling joint probability distributions over sequences has been studied from many perspectives. The physics community developed matrix prod… (voir plus)uct states, a tensor-train decomposition for probabilistic modeling, motivated by the need to tractably model many-body systems. But similar models have also been studied in the stochastic processes and weighted automata literature, with little work on how these bodies of work relate to each other. We address this gap by showing how stationary or uniform versions of popular quantum tensor network models have equivalent representations in the stochastic processes and weighted automata literature, in the limit of infinitely long sequences. We demonstrate several equivalence results between models used in these three communities: (i) uniform variants of matrix product states, Born machines and locally purified states from the quantum tensor networks literature, (ii) predictive state representations, hidden Markov models, norm-observable operator models and hidden quantum Markov models from the stochastic process literature,and (iii) stochastic weighted automata, probabilistic automata and quadratic automata from the formal languages literature. Such connections may open the door for results and methods developed in one area to be applied in another.

2021-03-18

Proceedings of The 24th International Conference on Artificial Intelligence and Statistics (publié)

proceedings.mlr.press

Towards a Trace-Preserving Tensor Network Representation of Quantum Channels

Siddarth Srinivasan

Sandesh M. Adhikary

Bibek Pokharel

Byron Boots

The problem of characterizing quantum channels arises in a number of contexts such as quantum process tomography and quantum error correctio… (voir plus)n. However, direct approaches to parameterizing and optimizing the Choi matrix representation of quantum channels face a curse of dimensionality: the number of parameters scales exponentially in the number of qubits. Recently, Torlai et al. [2020] proposed using locally puriﬁed density operators (LPDOs), a tensor network representation of Choi matrices, to overcome the unfavourable scaling in parameters. While the LPDO structure allows it to satisfy a ‘complete positivity’ (CP) constraint required of physically valid quantum channels, it makes no guarantees about a similarly required ‘trace preservation’ (TP) constraint. In practice, the TP constraint is violated, and the learned quantum channel may even be trace-increasing, which is non-physical. In this work, we present the problem of optimizing over TP LPDOs, discuss two approaches to characterizing the TP constraints on LPDOs, and outline the next steps for developing an optimization scheme.

Quantum Tensor Networks, Stochastic Processes, and Weighted Automata

Siddarth Srinivasan

Sandesh M. Adhikary

Byron Boots

Modeling joint probability distributions over sequences has been studied from many perspectives. The physics community developed matrix prod… (voir plus)uct states, a tensor-train decomposition for probabilistic modeling, motivated by the need to tractably model many-body systems. But similar models have also been studied in the stochastic processes and weighted automata literature, with little work on how these bodies of work relate to each other. We address this gap by showing how stationary or uniform versions of popular quantum tensor network models have equivalent representations in the stochastic processes and weighted automata literature, in the limit of infinitely long sequences. We demonstrate several equivalence results between models used in these three communities: (i) uniform variants of matrix product states, Born machines and locally purified states from the quantum tensor networks literature, (ii) predictive state representations, hidden Markov models, norm-observable operator models and hidden quantum Markov models from the stochastic process literature,and (iii) stochastic weighted automata, probabilistic automata and quadratic automata from the formal languages literature. Such connections may open the door for results and methods developed in one area to be applied in another.

2020-10-20

ArXiv (preprint)

A Fully Tensorized Recurrent Neural Network

Charles Onu

Doina Precup

2020-10-08

ArXiv (prépublication)

Adaptive Learning of Tensor Network Structures

Tensor Networks (TN) offer a powerful framework to efficiently represent very high-dimensional objects. TN have recently shown their potenti… (voir plus)al for machine learning applications and offer a unifying view of common tensor decomposition models such as Tucker, tensor train (TT) and tensor ring (TR). However, identifying the best tensor network structure from data for a given task is challenging. In this work, we leverage the TN formalism to develop a generic and efficient adaptive algorithm to jointly learn the structure and the parameters of a TN from data. Our method is based on a simple greedy approach starting from a rank one tensor and successively identifying the most promising tensor network edges for small rank increments. Our algorithm can adaptively identify TN structures with small number of parameters that effectively optimize any differentiable objective function. Experiments on tensor decomposition, tensor completion and model compression tasks demonstrate the effectiveness of the proposed algorithm. In particular, our method outperforms the state-of-the-art evolutionary topology search [Li and Sun, 2020] for tensor decomposition of images (while being orders of magnitude faster) and finds efficient tensor network structures to compress neural networks outperforming popular TT based approaches [Novikov et al., 2015].

2020-08-12

ArXiv (prépublication)

openreview.net

Tensor Networks for Language Modeling

John Anthony Terilla

The tensor network formalism has enjoyed over two decades of success in modeling the behavior of complex quantum-mechanical systems, but has… (voir plus) only recently and sporadically been leveraged in machine learning. Here we introduce a uniform matrix product state (u-MPS) model for probabilistic modeling of sequence data. We identify several distinctive features of this recurrent generative model, notably the ability to condition or marginalize sampling on characters at arbitrary locations within a sequence, with no need for approximate sampling methods. Despite the sequential architecture of u-MPS, we show that a recursive evaluation algorithm can be used to parallelize its inference and training, with a string of length n only requiring parallel time

2020-03-02

ArXiv (prépublication)

Tensor Networks for Probabilistic Sequence Modeling

John Anthony Terilla

Tensor networks are a powerful modeling framework developed for computational many-body physics, which have only recently been applied withi… (voir plus)n machine learning. In this work we utilize a uniform matrix product state (u-MPS) model for probabilistic modeling of sequence data. We first show that u-MPS enable sequence-level parallelism, with length-n sequences able to be evaluated in depth O(log n). We then introduce a novel generative algorithm giving trained u-MPS the ability to efficiently sample from a wide variety of conditional distributions, each one defined by a regular expression. Special cases of this algorithm correspond to autoregressive and fill-in-the-blank sampling, but more complex regular expressions permit the generation of richly structured text in a manner that has no direct analogue in current generative models. Experiments on synthetic text data find u-MPS outperforming LSTM baselines in several sampling tasks, and demonstrate strong generalization in the presence of limited data.

2020-03-02

International Conference on Artificial Intelligence and Statistics (publié)

proceedings.mlr.press

Tensor Networks for Probabilistic Sequence Modeling