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arXiv:2208.11382

Quantum algorithm for Markov Random Fields structure learning by information theoretic properties

Liming Zhao, Lin‐Chun Wan, Luo, Ming-Xing

Published Aug 24, 2022• 0 citations
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Abstract

Probabilistic graphical models play a crucial role in machine learning and have wide applications in various fields. One pivotal subset is undirected graphical models, also known as Markov random fields. In this work, we investigate the structure learning methods of Markov random fields on quantum computers. We propose a quantum algorithm for structure learning of an $r$-wise Markov Random Field with a bounded degree underlying graph, based on a nearly optimal classical greedy algorithm. The quantum algorithm provides a polynomial speed-up over the classical counterpart in terms of the number of variables. Our work demonstrates the potential merits of quantum computation over classical computation in solving some problems in machine learning.

DOI
10.48550/arxiv.2208.11382

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Probabilistic graphical models play a crucial role in machine learning and have wide applications in various fields. One pivotal subset is undirected graphical models, also known as Markov random fields. In this work, we investigate the structure learning methods of Markov random fields on quantum computers. We propose a quantum algorithm for structure learning of an rrr-wise Markov Random Field with a bounded degree underlying graph, based on a nearly optimal classical greedy algorithm. The quantum algorithm provides a polynomial speed-up over the classical counterpart in terms of the number of variables. Our work demonstrates the potential merits of quantum computation over classical computation in solving some problems in machine learning.
Abstract Probabilistic graphical models play a crucial role in machine learning and have wide applications across various fields. A pivotal subset within this realm is undirected graphical models, also known as Markov random fields. In this work, we explore the application of quantum computing to the field of machine learning, specifically focusing on the task of structure learning in Markov random fields. We propose a quantum algorithm for structure learning of an r-wise Markov Random Field with a bounded degree underlying graph, based on a nearly optimal classical greedy algorithm. Our quantum algorithm provides a polynomial speed-up over the classical counterpart, with a runtime of O ˜ 2 L M n r + 1 log ( 1 / w ) compared to the classical O ˜ M n r , where n is the number of variables. It is nearly a quadratic speedup in terms of n. Through this research, we aim to showcase the potential advantages of quantum computation in tackling complex machine-learning problems. Furthermore, the sample complexity O ˜ 2 2 L τ 2 δ 2 L ( log ( 1 / w ) + log n ) remains consistent between classical and quantum approaches, ensuring the efficiency and robustness of our method. By demonstrating the efficiency and scalability of our quantum algorithm for structure learning, we contribute to the growing body of work that highlights the role of quantum computing in advancing the field of machine learning and probabilistic graphical models.
Markov random fields (MRFs) appear in many problems in machine learning and statistics. From a computational learning theory point of view, a natural problem of learning MRFs arises: given samples from an MRF from a restricted class, learn the structure of the MRF, that is the neighbors of each node of the underlying graph. In this work, we start at a known near-optimal classical algorithm for this learning problem and develop a modified classical algorithm. This classical algorithm retains the run time and guarantee of the previous algorithm and enables the use of quantum subroutines. Adapting a previous quantum algorithm, the Quantum Sparsitron, we provide a polynomial quantum speedup in terms of the number of variables for learning the structure of an MRF, if the MRF has bounded degree.
Restricted Boltzmann Machines (RBMs) are widely used probabilistic undirected graphical models with visible and latent nodes, playing an important role in statistics and machine learning. The task of structure learning for RBMs involves inferring the underlying graph by using samples from the visible nodes. Specifically, learning the two-hop neighbors of each visible node allows for the inference of the graph structure. Prior research has addressed the structure learning problem for specific classes of RBMs, namely ferromagnetic and locally consistent RBMs. In this paper, we extend the scope to the quantum computing domain and propose corresponding quantum algorithms for this problem. Our study demonstrates that the proposed quantum algorithms yield a polynomial speedup compared to the classical algorithms for learning the structure of these two classes of RBMs.
Abstract Leveraging the intrinsic probabilistic nature of quantum systems, generative quantum machine learning (QML) offers the potential to outperform classical learning models. Current generative QML algorithms mostly rely on general-purpose models that, while being very expressive, face several training challenges. One potential way to address these setbacks is by constructing problem-informed models that are capable of more efficient training on structured problems. In particular, probabilistic graphical models provide a flexible framework for representing structure in generative learning problems and can thus be exploited to incorporate inductive bias into QML algorithms. In this work, we propose a problem-informed quantum circuit Born machine Ansatz for learning the joint probability distribution of random variables, with independence relations efficiently represented by a Markov network (MN). We further demonstrate the applicability of the MN framework in constructing generative learning benchmarks and compare our model’s performance to previous designs, showing that it outperforms problem-agnostic circuits. Based on a preliminary analysis of trainability, we narrow down the class of MNs to those exhibiting favourable trainability properties. Finally, we discuss the potential of our model to offer quantum advantage in the context of generative learning, and present a practical scenario demonstrating the scalability of the approach.
The theories of optimization and machine learning answer foundational questions in computer science and lead to new algorithms for practical applications. While these topics have been extensively studied in the context of classical computing, their quantum counterparts are far from well-understood. In this thesis, we explore algorithms that bridge the gap between the fields of quantum computing and machine learning. First, we consider general optimization problems with only function evaluations. For two core problems, namely general convex optimization and volume estimation of convex bodies, we give quantum algorithms as well as quantum lower bounds that constitute the quantum speedups of both problems to be polynomial compared to their classical counterparts. We then consider machine learning and optimization problems with input data stored explicitly as matrices. We first look at semidefinite programs and provide quantum algorithms with polynomial speedup compared to the classical state-of-the-art. We then move to machine learning and give the optimal quantum algorithms for linear and kernel-based classifications. To complement with our quantum algorithms, we also introduce a framework for quantum-inspired classical algorithms, showing that for low-rank matrix arithmetics there can only be polynomial quantum speedup. Finally, we study statistical problems on quantum computers, with the focus on testing properties of probability distributions. We show that for testing various properties including L1-distance, L2-distance, Shannon and Renyi entropies, etc., there are polynomial quantum speedups compared to their classical counterparts. We also extend these results to testing properties of quantum states.
Improvising the performance of machine learning for applications in the field of computer science leads to create new algorithms. As these are being optimized, using the algorithms of the classical machine learning in the area of quantum computing are being widely researched. First, we can consider general optimization problems with only functional Evaluations. Quantum Speedups of the problems can be improved compared to the classical algorithms. A frame work for quantum inspired classical algorithms can be introduced. To study the various problems on quantum computers on testing properties and compare the quantum speed ups with the classical counter parts.