Neural networks have revolutionized many aspects of society but in the era of huge models with billions of parameters, optimizing and deploying them for commercial applications can require significant computational and financial resources. To address these challenges, we introduce the Tetra-AML toolbox, which automates neural architecture search and hyperparameter optimization via a custom-developed black-box Tensor train Optimization algorithm, TetraOpt. The toolbox also provides model compression through quantization and pruning, augmented by compression using tensor networks. Here, we analyze a unified benchmark for optimizing neural networks in computer vision tasks and show the superior performance of our approach compared to Bayesian optimization on the CIFAR-10 dataset. We also demonstrate the compression of ResNet-18 neural networks, where we use 14.5 times less memory while losing just 3.2% of accuracy. The presented framework is generic, not limited by computer vision problems, supports hardware acceleration (such as with GPUs and TPUs) and can be further extended to quantum hardware and to hybrid quantum machine learning models.
Despite the omnipresence of tensors and tensor operations in modern deep learning, the use of tensor mathematics to formally design and describe neural networks is still under-explored within the deep learning community. To this end, we introduce the Graph Tensor Network (GTN) framework, an intuitive yet rigorous graphical framework for systematically designing and implementing large-scale neural learning systems on both regular and irregular domains. The proposed framework is shown to be general enough to include many popular architectures as special cases, and flexible enough to handle data on any and many data domains. The power and flexibility of the proposed framework is demonstrated through real-data experiments, resulting in improved performance at a drastically lower complexity costs, by virtue of tensor algebra.
Once developed for quantum theory, tensor networks have been established as a successful machine learning paradigm. Now, they have been ported back to the quantum realm in the emerging field of quantum machine learning to assess problems that classical computers are unable to solve efficiently. Their nature at the interface between physics and machine learning makes tensor networks easily deployable on quantum computers. In this review article, we shed light on one of the major architectures considered to be predestined for variational quantum machine learning. In particular, we discuss how layouts like MPS, PEPS, TTNs and MERA can be mapped to a quantum computer, how they can be used for machine learning and data encoding and which implementation techniques improve their performance.
Many datasets in scientific and engineering applications are comprised of objects which have specific geometric structure. A common example is data which inhabits a representation of the group SO$(3)$ of 3D rotations: scalars, vectors, tensors, \textit{etc}. One way for a neural network to exploit prior knowledge of this structure is to enforce SO$(3)$-equivariance throughout its layers, and several such architectures have been proposed. While general methods for handling arbitrary SO$(3)$ representations exist, they computationally intensive and complicated to implement. We show that by judicious symmetry breaking, we can efficiently increase the expressiveness of a network operating only on vector and order-2 tensor representations of SO$(2)$. We demonstrate the method on an important problem from High Energy Physics known as \textit{b-tagging}, where particle jets originating from b-meson decays must be discriminated from an overwhelming QCD background. In this task, we find that augmenting a standard architecture with our method results in a \ensuremath{2.3\times} improvement in rejection score.
Living systems face both environmental complexity and limited access to free-energy resources. Survival under these conditions requires a control system that can activate, or deploy, available perception and action resources in a context specific way. We show here that when systems are described as executing active inference driven by the free-energy principle (and hence can be considered Bayesian prediction-error minimizers), their control flow systems can always be represented as tensor networks (TNs). We show how TNs as control systems can be implmented within the general framework of quantum topological neural networks, and discuss the implications of these results for modeling biological systems at multiple scales.
Tensor networks (TNs) and neural networks (NNs) are two fundamental data modeling approaches. TNs were introduced to solve the curse of dimensionality in large-scale tensors by converting an exponential number of dimensions to polynomial complexity. As a result, they have attracted significant attention in the fields of quantum physics and machine learning. Meanwhile, NNs have displayed exceptional performance in various applications, e.g., computer vision, natural language processing, and robotics research. Interestingly, although these two types of networks originate from different observations, they are inherently linked through the common multilinearity structure underlying both TNs and NNs, thereby motivating a significant number of intellectual developments regarding combinations of TNs and NNs. In this paper, we refer to these combinations as tensorial neural networks (TNNs), and present an introduction to TNNs in three primary aspects: network compression, information fusion, and quantum circuit simulation. Furthermore, this survey also explores methods for improving TNNs, examines flexible toolboxes for implementing TNNs, and documents TNN development while highlighting potential future directions. To the best of our knowledge, this is the first comprehensive survey that bridges the connections among NNs, TNs, and quantum circuits. We provide a curated list of TNNs at \url{https://github.com/tnbar/awesome-tensorial-neural-networks}.
Tensor networks, which have been traditionally used to simulate many-body physics, have recently gained significant attention in the field of machine learning due to their powerful representation capabilities. In this work, we propose a density-based clustering algorithm inspired by tensor networks. We encode classical data into tensor network states on an extended Hilbert space and train the tensor network states to capture the features of the clusters. Here, we define density and related concepts in terms of fidelity, rather than using a classical distance measure. We evaluate the performance of our algorithm on six synthetic data sets, four real world data sets, and three commonly used computer vision data sets. The results demonstrate that our method provides state-of-the-art performance on several synthetic data sets and real world data sets, even when the number of clusters is unknown. Additionally, our algorithm performs competitively with state-of-the-art algorithms on the MNIST, USPS, and Fashion-MNIST image data sets. These findings reveal the great potential of tensor networks for machine learning applications.
Machine learning (ML) has recently facilitated many advances in solving problems related to many-body physical systems. Given the intrinsic quantum nature of these problems, it is natural to speculate that quantum-enhanced machine learning will enable us to unveil even greater details than we currently have. With this motivation, this paper examines a quantum machine learning approach based on shallow variational ansatz inspired by tensor networks for supervised learning tasks. In particular, we first look at the standard image classification tasks using the Fashion-MNIST dataset and study the effect of repeating tensor network layers on ansatz's expressibility and performance. Finally, we use this strategy to tackle the problem of quantum phase recognition for the transverse-field Ising and Heisenberg spin models in one and two dimensions, where we were able to reach $\geq 98\%$ test-set accuracies with both multi-scale entanglement renormalization ansatz (MERA) and tree tensor network (TTN) inspired parametrized quantum circuits.
Machine Learning models capable of handling the large datasets collected in the financial world can often become black boxes expensive to run. The quantum computing paradigm suggests new optimization techniques, that combined with classical algorithms, may deliver competitive, faster and more interpretable models. In this work we propose a quantum-enhanced machine learning solution for the prediction of credit rating downgrades, also known as fallen-angels forecasting in the financial risk management field. We implement this solution on a neutral atom Quantum Processing Unit with up to 60 qubits on a real-life dataset. We report competitive performances against the state-of-the-art Random Forest benchmark whilst our model achieves better interpretability and comparable training times. We examine how to improve performance in the near-term validating our ideas with Tensor Networks-based numerical simulations.
Positive unlabeled learning is a binary classification problem with positive and unlabeled data. It is common in domains where negative labels are costly or impossible to obtain, e.g., medicine and personalized advertising. Most approaches to positive unlabeled learning apply to specific data types (e.g., images, categorical data) and can not generate new positive and negative samples. This work introduces a feature-space distance-based tensor network approach to the positive unlabeled learning problem. The presented method is not domain specific and significantly improves the state-of-the-art results on the MNIST image and 15 categorical/mixed datasets. The trained tensor network model is also a generative model and enables the generation of new positive and negative instances.
Many learning tasks, including learning potential energy surfaces from ab initio calculations, involve global spatial symmetries and permutational symmetry between atoms or general particles. Equivariant graph neural networks are a standard approach to such problems, with one of the most successful methods employing tensor products between various tensors that transform under the spatial group. However, as the number of different tensors and the complexity of relationships between them increase, maintaining parsimony and equivariance becomes increasingly challenging. In this paper, we propose using fusion diagrams, a technique widely employed in simulating SU($2$)-symmetric quantum many-body problems, to design new equivariant components for equivariant neural networks. This results in a diagrammatic approach to constructing novel neural network architectures. When applied to particles within a given local neighborhood, the resulting components, which we term "fusion blocks," serve as universal approximators of any continuous equivariant function defined in the neighborhood. We incorporate a fusion block into pre-existing equivariant architectures (Cormorant and MACE), leading to improved performance with fewer parameters on a range of challenging chemical problems. Furthermore, we apply group-equivariant neural networks to study non-adiabatic molecular dynamics of stilbene cis-trans isomerization. Our approach, which combines tensor networks with equivariant neural networks, suggests a potentially fruitful direction for designing more expressive equivariant neural networks.
We show how to develop sampling-based alternating least squares (ALS) algorithms for decomposition of tensors into any tensor network (TN) format. Provided the TN format satisfies certain mild assumptions, resulting algorithms will have input sublinear per-iteration cost. Unlike most previous works on sampling-based ALS methods for tensor decomposition, the sampling in our framework is done according to the exact leverage score distribution of the design matrices in the ALS subproblems. We implement and test two tensor decomposition algorithms that use our sampling framework in a feature extraction experiment where we compare them against a number of other decomposition algorithms.
We present an alternative approach to decompose non-negative tensors, called many-body approximation. Traditional decomposition methods assume low-rankness in the representation, resulting in difficulties in global optimization and target rank selection. We avoid these problems by energy-based modeling of tensors, where a tensor and its mode correspond to a probability distribution and a random variable, respectively. Our model can be globally optimized in terms of the KL divergence minimization by taking the interaction between variables (that is, modes), into account that can be tuned more intuitively than ranks. Furthermore, we visualize interactions between modes as tensor networks and reveal a nontrivial relationship between many-body approximation and low-rank approximation. We demonstrate the effectiveness of our approach in tensor completion and approximation.
The physics of a closed quantum mechanical system is governed by its Hamiltonian. However, in most practical situations, this Hamiltonian is not precisely known, and ultimately all there is are data obtained from measurements on the system. In this work, we introduce a highly scalable, data-driven approach to learning families of interacting many-body Hamiltonians from dynamical data, by bringing together techniques from gradient-based optimization from machine learning with efficient quantum state representations in terms of tensor networks. Our approach is highly practical, experimentally friendly, and intrinsically scalable to allow for system sizes of above 100 spins. In particular, we demonstrate on synthetic data that the algorithm works even if one is restricted to one simple initial state, a small number of single-qubit observables, and time evolution up to relatively short times. For the concrete example of the one-dimensional Heisenberg model our algorithm exhibits an error constant in the system size and scaling as the inverse square root of the size of the data set.
We present a framework to integrate tensor network (TN) methods with reinforcement learning (RL) for solving dynamical optimisation tasks. We consider the RL actor-critic method, a model-free approach for solving RL problems, and introduce TNs as the approximators for its policy and value functions. Our "actor-critic with tensor networks" (ACTeN) method is especially well suited to problems with large and factorisable state and action spaces. As an illustration of the applicability of ACTeN we solve the exponentially hard task of sampling rare trajectories in two paradigmatic stochastic models, the East model of glasses and the asymmetric simple exclusion process (ASEP), the latter being particularly challenging to other methods due to the absence of detailed balance. With substantial potential for further integration with the vast array of existing RL methods, the approach introduced here is promising both for applications in physics and to multi-agent RL problems more generally.
We introduce deep tensor networks, which are exponentially wide neural networks based on the tensor network representation of the weight matrices. We evaluate the proposed method on the image classification (MNIST, FashionMNIST) and sequence prediction (cellular automata) tasks. In the image classification case, deep tensor networks improve our matrix product state baselines and achieve 0.49% error rate on MNIST and 8.3% error rate on FashionMNIST. In the sequence prediction case, we demonstrate an exponential improvement in the number of parameters compared to the one-layer tensor network methods. In both cases, we discuss the non-uniform and the uniform tensor network models and show that the latter generalizes well to different input sizes.
Analyzing concentration of large random matrices is a common task in a wide variety of fields. Given independent random variables, many tools are available to analyze random matrices whose entries are linear in the variables, e.g. the matrix-Bernstein inequality. However, in many applications, we need to analyze random matrices whose entries are polynomials in the variables. These arise naturally in the analysis of spectral algorithms, e.g., Hopkins et al. [STOC 2016], Moitra-Wein [STOC 2019]; and in lower bounds for semidefinite programs based on the Sum of Squares hierarchy, e.g. Barak et al. [FOCS 2016], Jones et al. [FOCS 2021]. In this work, we present a general framework to obtain such bounds, based on the matrix Efron-Stein inequalities developed by Paulin-Mackey-Tropp [Annals of Probability 2016]. The Efron-Stein inequality bounds the norm of a random matrix by the norm of another simpler (but still random) matrix, which we view as arising by "differentiating" the starting matrix. By recursively differentiating, our framework reduces the main task to analyzing far simpler matrices. For Rademacher variables, these simpler matrices are in fact deterministic and hence, analyzing them is far easier. For general non-Rademacher variables, the task reduces to scalar concentration, which is much easier. Moreover, in the setting of polynomial matrices, our results generalize the work of Paulin-Mackey-Tropp. Using our basic framework, we recover known bounds in the literature for simple "tensor networks" and "dense graph matrices". Using our general framework, we derive bounds for "sparse graph matrices", which were obtained only recently by Jones et al. [FOCS 2021] using a nontrivial application of the trace power method, and was a core component in their work. We expect our framework to be helpful for other applications involving concentration phenomena for nonlinear random matrices.
We consider the problem of approximating a function in a general nonlinear subset of $L^2$, when only a weighted Monte Carlo estimate of the $L^2$-norm can be computed. Of particular interest in this setting is the concept of sample complexity, the number of sample points that are necessary to achieve a prescribed error with high probability. Reasonable worst-case bounds for this quantity exist only for particular model classes, like linear spaces or sets of sparse vectors. For more general sets, like tensor networks or neural networks, the currently existing bounds are very pessimistic. By restricting the model class to a neighbourhood of the best approximation, we can derive improved worst-case bounds for the sample complexity. When the considered neighbourhood is a manifold with positive local reach, its sample complexity can be estimated by means of the sample complexities of the tangent and normal spaces and the manifold's curvature.
Active inference provides a general framework for behavior and learning in autonomous agents. It states that an agent will attempt to minimize its variational free energy, defined in terms of beliefs over observations, internal states and policies. Traditionally, every aspect of a discrete active inference model must be specified by hand, i.e. by manually defining the hidden state space structure, as well as the required distributions such as likelihood and transition probabilities. Recently, efforts have been made to learn state space representations automatically from observations using deep neural networks. In this paper, we present a novel approach of learning state spaces using quantum physics-inspired tensor networks. The ability of tensor networks to represent the probabilistic nature of quantum states as well as to reduce large state spaces makes tensor networks a natural candidate for active inference. We show how tensor networks can be used as a generative model for sequential data. Furthermore, we show how one can obtain beliefs from such a generative model and how an active inference agent can use these to compute the expected free energy. Finally, we demonstrate our method on the classic T-maze environment.
Due to the beyond-classical capability of quantum computing, quantum machine learning is applied independently or embedded in classical models for decision making, especially in the field of finance. Fairness and other ethical issues are often one of the main concerns in decision making. In this work, we define a formal framework for the fairness verification and analysis of quantum machine learning decision models, where we adopt one of the most popular notions of fairness in the literature based on the intuition -- any two similar individuals must be treated similarly and are thus unbiased. We show that quantum noise can improve fairness and develop an algorithm to check whether a (noisy) quantum machine learning model is fair. In particular, this algorithm can find bias kernels of quantum data (encoding individuals) during checking. These bias kernels generate infinitely many bias pairs for investigating the unfairness of the model. Our algorithm is designed based on a highly efficient data structure -- Tensor Networks -- and implemented on Google's TensorFlow Quantum. The utility and effectiveness of our algorithm are confirmed by the experimental results, including income prediction and credit scoring on real-world data, for a class of random (noisy) quantum decision models with 27 qubits ($2^{27}$-dimensional state space) tripling ($2^{18}$ times more than) that of the state-of-the-art algorithms for verifying quantum machine learning models.