We study low rank approximation of tensors, focusing on the tensor train and Tucker decompositions, as well as approximations with tree tensor networks and more general tensor networks. For tensor train decomposition, we give a bicriteria $(1 + \eps)$-approximation algorithm with a small bicriteria rank and $O(q \cdot \nnz(A))$ running time, up to lower order terms, which improves over the additive error algorithm of \cite{huber2017randomized}. We also show how to convert the algorithm of \cite{huber2017randomized} into a relative error algorithm, but their algorithm necessarily has a running time of $O(qr^2 \cdot \nnz(A)) + n \cdot \poly(qk/\eps)$ when converted to a $(1 + \eps)$-approximation algorithm with bicriteria rank $r$. To the best of our knowledge, our work is the first to achieve polynomial time relative error approximation for tensor train decomposition. Our key technique is a method for obtaining subspace embeddings with a number of rows polynomial in $q$ for a matrix which is the flattening of a tensor train of $q$ tensors. We extend our algorithm to tree tensor networks. In addition, we extend our algorithm to tensor networks with arbitrary graphs (which we refer to as general tensor networks), by using a result of \cite{ms08_simulating_quantum_tensor_contraction} and showing that a general tensor network of rank $k$ can be contracted to a binary tree network of rank $k^{O(\deg(G)\tw(G))}$, allowing us to reduce to the case of tree tensor networks. Finally, we give new fixed-parameter tractable algorithms for the tensor train, Tucker, and CP decompositions, which are simpler than those of \cite{swz19_tensor_low_rank} since they do not make use of polynomial system solvers. Our technique of Gaussian subspace embeddings with exactly $k$ rows (and thus exponentially small success probability) may be of independent interest.
A tensor network is a type of decomposition used to express and approximate large arrays of data. A given data-set, quantum state or higher dimensional multi-linear map is factored and approximated by a composition of smaller multi-linear maps. This is reminiscent to how a Boolean function might be decomposed into a gate array: this represents a special case of tensor decomposition, in which the tensor entries are replaced by 0, 1 and the factorisation becomes exact. The collection of associated techniques are called, tensor network methods: the subject developed independently in several distinct fields of study, which have more recently become interrelated through the language of tensor networks. The tantamount questions in the field relate to expressability of tensor networks and the reduction of computational overheads. A merger of tensor networks with machine learning is natural. On the one hand, machine learning can aid in determining a factorization of a tensor network approximating a data set. On the other hand, a given tensor network structure can be viewed as a machine learning model. Herein the tensor network parameters are adjusted to learn or classify a data-set. In this survey we recover the basics of tensor networks and explain the ongoing effort to develop the theory of tensor networks in machine learning.
Semantic image interpretation can vastly benefit from approaches that combine sub-symbolic distributed representation learning with the capability to reason at a higher level of abstraction. Logic Tensor Networks (LTNs) are a class of neuro-symbolic systems based on a differentiable, first-order logic grounded into a deep neural network. LTNs replace the classical concept of training set with a knowledge base of fuzzy logical axioms. By defining a set of differentiable operators to approximate the role of connectives, predicates, functions and quantifiers, a loss function is automatically specified so that LTNs can learn to satisfy the knowledge base. We focus here on the subsumption or \texttt{isOfClass} predicate, which is fundamental to encode most semantic image interpretation tasks. Unlike conventional LTNs, which rely on a separate predicate for each class (e.g., dog, cat), each with its own set of learnable weights, we propose a common \texttt{isOfClass} predicate, whose level of truth is a function of the distance between an object embedding and the corresponding class prototype. The PROTOtypical Logic Tensor Networks (PROTO-LTN) extend the current formulation by grounding abstract concepts as parametrized class prototypes in a high-dimensional embedding space, while reducing the number of parameters required to ground the knowledge base. We show how this architecture can be effectively trained in the few and zero-shot learning scenarios. Experiments on Generalized Zero Shot Learning benchmarks validate the proposed implementation as a competitive alternative to traditional embedding-based approaches. The proposed formulation opens up new opportunities in zero shot learning settings, as the LTN formalism allows to integrate background knowledge in the form of logical axioms to compensate for the lack of labelled examples.
Invariance has recently proven to be a powerful inductive bias in machine learning models. One such class of predictive or generative models are tensor networks. We introduce a new numerical algorithm to construct a basis of tensors that are invariant under the action of normal matrix representations of an arbitrary discrete group. This method can be up to several orders of magnitude faster than previous approaches. The group-invariant tensors are then combined into a group-invariant tensor train network, which can be used as a supervised machine learning model. We applied this model to a protein binding classification problem, taking into account problem-specific invariances, and obtained prediction accuracy in line with state-of-the-art deep learning approaches.
For the first time, this position paper introduces a fundamental link between tensor networks (TNs) and Green AI, highlighting their synergistic potential to enhance both the inclusivity and sustainability of AI research. We argue that TNs are valuable for Green AI due to their strong mathematical backbone and inherent logarithmic compression potential. We undertake a comprehensive review of the ongoing discussions on Green AI, emphasizing the importance of sustainability and inclusivity in AI research to demonstrate the significance of establishing the link between Green AI and TNs. To support our position, we first provide a comprehensive overview of efficiency metrics proposed in Green AI literature and then evaluate examples of TNs in the fields of kernel machines and deep learning using the proposed efficiency metrics. This position paper aims to incentivize meaningful, constructive discussions by bridging fundamental principles of Green AI and TNs. We advocate for researchers to seriously evaluate the integration of TNs into their research projects, and in alignment with the link established in this paper, we support prior calls encouraging researchers to treat Green AI principles as a research priority.
Random quantum circuits have been utilized in the contexts of quantum supremacy demonstrations, variational quantum algorithms for chemistry and machine learning, and blackhole information. The ability of random circuits to approximate any random unitaries has consequences on their complexity, expressibility, and trainability. To study this property of random circuits, we develop numerical protocols for estimating the frame potential, the distance between a given ensemble and the exact randomness. Our tensor-network-based algorithm has polynomial complexity for shallow circuits and is high-performing using CPU and GPU parallelism. We study 1. local and parallel random circuits to verify the linear growth in complexity as stated by the Brown-Susskind conjecture, and; 2. hardware-efficient ans\"atze to shed light on its expressibility and the barren plateau problem in the context of variational algorithms. Our work shows that large-scale tensor network simulations could provide important hints toward open problems in quantum information science.
Collaborative filtering models generally perform better than content-based filtering models and do not require careful feature engineering. However, in the cold-start scenario collaborative information may be scarce or even unavailable, whereas the content information may be abundant, but also noisy and expensive to acquire. Thus, selection of particular features that improve cold-start recommendations becomes an important and non-trivial task. In the recent approach by Nembrini et al., the feature selection is driven by the correlational compatibility between collaborative and content-based models. The problem is formulated as a Quadratic Unconstrained Binary Optimization (QUBO) which, due to its NP-hard complexity, is solved using Quantum Annealing on a quantum computer provided by D-Wave. Inspired by the reported results, we contend the idea that current quantum annealers are superior for this problem and instead focus on classical algorithms. In particular, we tackle QUBO via TTOpt, a recently proposed black-box optimizer based on tensor networks and multilinear algebra. We show the computational feasibility of this method for large problems with thousands of features, and empirically demonstrate that the solutions found are comparable to the ones obtained with D-Wave across all examined datasets.
We analyze the problem of high-order polynomial approximation from a many-body physics perspective, and demonstrate the descriptive power of entanglement entropy in capturing model capacity and task complexity. Instantiated with a high-order nonlinear dynamics modeling problem, tensor-network models are investigated and exhibit promising modeling advantages. This novel perspective establish a connection between quantum information and functional approximation, which worth further exploration in future research.
The tensor network, as a facterization of tensors, aims at performing the operations that are common for normal tensors, such as addition, contraction and stacking. However, due to its non-unique network structure, only the tensor network contraction is so far well defined. In this paper, we propose a mathematically rigorous definition for the tensor network stack approach, that compress a large amount of tensor networks into a single one without changing their structures and configurations. We illustrate the main ideas with the matrix product states based machine learning as an example. Our results are compared with the for loop and the efficient coding method on both CPU and GPU.
This work focuses on designing low complexity hybrid tensor networks by considering trade-offs between the model complexity and practical performance. Firstly, we exploit a low-rank tensor-train deep neural network (TT-DNN) to build an end-to-end deep learning pipeline, namely LR-TT-DNN. Secondly, a hybrid model combining LR-TT-DNN with a convolutional neural network (CNN), which is denoted as CNN+(LR-TT-DNN), is set up to boost the performance. Instead of randomly assigning large TT-ranks for TT-DNN, we leverage Riemannian gradient descent to determine a TT-DNN associated with small TT-ranks. Furthermore, CNN+(LR-TT-DNN) consists of convolutional layers at the bottom for feature extraction and several TT layers at the top to solve regression and classification problems. We separately assess the LR-TT-DNN and CNN+(LR-TT-DNN) models on speech enhancement and spoken command recognition tasks. Our empirical evidence demonstrates that the LR-TT-DNN and CNN+(LR-TT-DNN) models with fewer model parameters can outperform the TT-DNN and CNN+(TT-DNN) counterparts.
Tensor networks have demonstrated significant value for machine learning in a myriad of different applications. However, optimizing tensor networks using standard gradient descent has proven to be difficult in practice. Tensor networks suffer from initialization problems resulting in exploding or vanishing gradients and require extensive hyperparameter tuning. Efforts to overcome these problems usually depend on specific network architectures, or ad hoc prescriptions. In this paper we address the problems of initialization and hyperparameter tuning, making it possible to train tensor networks using established machine learning techniques. We introduce a `copy node' method that successfully initializes arbitrary tensor networks, in addition to a gradient based regularization technique for bond dimensions. We present numerical results that show that the combination of techniques presented here produces quantum inspired tensor network models with far fewer parameters, while improving generalization performance.
A rising problem in the compression of Deep Neural Networks is how to reduce the number of parameters in convolutional kernels and the complexity of these layers by low-rank tensor approximation. Canonical polyadic tensor decomposition (CPD) and Tucker tensor decomposition (TKD) are two solutions to this problem and provide promising results. However, CPD often fails due to degeneracy, making the networks unstable and hard to fine-tune. TKD does not provide much compression if the core tensor is big. This motivates using a hybrid model of CPD and TKD, a decomposition with multiple Tucker models with small core tensor, known as block term decomposition (BTD). This paper proposes a more compact model that further compresses the BTD by enforcing core tensors in BTD identical. We establish a link between the BTD with shared parameters and a looped chain tensor network (TC). Unfortunately, such strongly constrained tensor networks (with loop) encounter severe numerical instability, as proved by y (Landsberg, 2012) and (Handschuh, 2015a). We study perturbation of chain tensor networks, provide interpretation of instability in TC, demonstrate the problem. We propose novel methods to gain the stability of the decomposition results, keep the network robust and attain better approximation. Experimental results will confirm the superiority of the proposed methods in compression of well-known CNNs, and TC decomposition under challenging scenarios
Tensor networks, widely used for providing efficient representations of low-energy states of local quantum many-body systems, have been recently proposed as machine learning architectures which could present advantages with respect to traditional ones. In this work we show that tensor network architectures have especially prospective properties for privacy-preserving machine learning, which is important in tasks such as the processing of medical records. First, we describe a new privacy vulnerability that is present in feedforward neural networks, illustrating it in synthetic and real-world datasets. Then, we develop well-defined conditions to guarantee robustness to such vulnerability, which involve the characterization of models equivalent under gauge symmetry. We rigorously prove that such conditions are satisfied by tensor-network architectures. In doing so, we define a novel canonical form for matrix product states, which has a high degree of regularity and fixes the residual gauge that is left in the canonical forms based on singular value decompositions. We supplement the analytical findings with practical examples where matrix product states are trained on datasets of medical records, which show large reductions on the probability of an attacker extracting information about the training dataset from the model's parameters. Given the growing expertise in training tensor-network architectures, these results imply that one may not have to be forced to make a choice between accuracy in prediction and ensuring the privacy of the information processed.
Tensor Networks (TN) are approximations of high-dimensional tensors designed to represent locally entangled quantum many-body systems efficiently. This study provides a comprehensive comparison between classical TNs and TN-inspired quantum circuits in the context of Machine Learning on highly complex, simulated LHC data. We show that classical TNs require exponentially large bond dimensions and higher Hilbert-space mapping to perform comparably to their quantum counterparts. While such an expansion in the dimensionality allows better performance, we observe that, with increased dimensionality, classical TNs lead to a highly flat loss landscape, rendering the usage of gradient-based optimization methods highly challenging. Furthermore, by employing quantitative metrics, such as the Fisher information and effective dimensions, we show that classical TNs require a more extensive training sample to represent the data as efficiently as TN-inspired quantum circuits. We also engage with the idea of hybrid classical-quantum TNs and show possible architectures to employ a larger phase-space from the data. We offer our results using three main TN ansatz: Tree Tensor Networks, Matrix Product States, and Multi-scale Entanglement Renormalisation Ansatz.
We develop the first fast spectral algorithm to decompose a random third-order tensor over $\mathbb{R}^d$ of rank up to $O(d^{3/2}/\text{polylog}(d))$. Our algorithm only involves simple linear algebra operations and can recover all components in time $O(d^{6.05})$ under the current matrix multiplication time. Prior to this work, comparable guarantees could only be achieved via sum-of-squares [Ma, Shi, Steurer 2016]. In contrast, fast algorithms [Hopkins, Schramm, Shi, Steurer 2016] could only decompose tensors of rank at most $O(d^{4/3}/\text{polylog}(d))$. Our algorithmic result rests on two key ingredients. A clean lifting of the third-order tensor to a sixth-order tensor, which can be expressed in the language of tensor networks. A careful decomposition of the tensor network into a sequence of rectangular matrix multiplications, which allows us to have a fast implementation of the algorithm.
As the quantum computing community gravitates towards understanding the practical benefits of quantum computers, having a clear definition and evaluation scheme for assessing practical quantum advantage in the context of specific applications is paramount. Generative modeling, for example, is a widely accepted natural use case for quantum computers, and yet has lacked a concrete approach for quantifying success of quantum models over classical ones. In this work, we construct a simple and unambiguous approach to probe practical quantum advantage for generative modeling by measuring the algorithm's generalization performance. Using the sample-based approach proposed here, any generative model, from state-of-the-art classical generative models such as GANs to quantum models such as Quantum Circuit Born Machines, can be evaluated on the same ground on a concrete well-defined framework. In contrast to other sample-based metrics for probing practical generalization, we leverage constrained optimization problems (e.g., cardinality-constrained problems) and use these discrete datasets to define specific metrics capable of unambiguously measuring the quality of the samples and the model's generalization capabilities for generating data beyond the training set but still within the valid solution space. Additionally, our metrics can diagnose trainability issues such as mode collapse and overfitting, as we illustrate when comparing GANs to quantum-inspired models built out of tensor networks. Our simulation results show that our quantum-inspired models have up to a $68 \times$ enhancement in generating unseen unique and valid samples compared to GANs, and a ratio of 61:2 for generating samples with better quality than those observed in the training set. We foresee these metrics as valuable tools for rigorously defining practical quantum advantage in the domain of generative modeling.
The single-hidden-layer Randomly Weighted Feature Network (RWFN) introduced by Hong and Pavlic (2021) was developed as an alternative to neural tensor network approaches for relational learning tasks. Its relatively small footprint combined with the use of two randomized input projections -- an insect-brain-inspired input representation and random Fourier features -- allow it to achieve rich expressiveness for relational learning with relatively low training cost. In particular, when Hong and Pavlic compared RWFN to Logic Tensor Networks (LTNs) for Semantic Image Interpretation (SII) tasks to extract structured semantic descriptions from images, they showed that the RWFN integration of the two hidden, randomized representations better captures relationships among inputs with a faster training process even though it uses far fewer learnable parameters. In this paper, we use RWFNs to perform Visual Relationship Detection (VRD) tasks, which are more challenging SII tasks. A zero-shot learning approach is used with RWFN that can exploit similarities with other seen relationships and background knowledge -- expressed with logical constraints between subjects, relations, and objects -- to achieve the ability to predict triples that do not appear in the training set. The experiments on the Visual Relationship Dataset to compare the performance between RWFNs and LTNs, one of the leading Statistical Relational Learning frameworks, show that RWFNs outperform LTNs for the predicate-detection task while using fewer number of adaptable parameters (1:56 ratio). Furthermore, background knowledge represented by RWFNs can be used to alleviate the incompleteness of training sets even though the space complexity of RWFNs is much smaller than LTNs (1:27 ratio).
Random projection (RP) have recently emerged as popular techniques in the machine learning community for their ability in reducing the dimension of very high-dimensional tensors. Following the work in [30], we consider a tensorized random projection relying on Tensor Train (TT) decomposition where each element of the core tensors is drawn from a Rademacher distribution. Our theoretical results reveal that the Gaussian low-rank tensor represented in compressed form in TT format in [30] can be replaced by a TT tensor with core elements drawn from a Rademacher distribution with the same embedding size. Experiments on synthetic data demonstrate that tensorized Rademacher RP can outperform the tensorized Gaussian RP studied in [30]. In addition, we show both theoretically and experimentally, that the tensorized RP in the Matrix Product Operator (MPO) format is not a Johnson-Lindenstrauss transform (JLT) and therefore not a well-suited random projection map
Least squares support vector machines are a commonly used supervised learning method for nonlinear regression and classification. They can be implemented in either their primal or dual form. The latter requires solving a linear system, which can be advantageous as an explicit mapping of the data to a possibly infinite-dimensional feature space is avoided. However, for large-scale applications, current low-rank approximation methods can perform inadequately. For example, current methods are probabilistic due to their sampling procedures, and/or suffer from a poor trade-off between the ranks and approximation power. In this paper, a recursive Bayesian filtering framework based on tensor networks and the Kalman filter is presented to alleviate the demanding memory and computational complexities associated with solving large-scale dual problems. The proposed method is iterative, does not require explicit storage of the kernel matrix, and allows the formulation of early stopping conditions. Additionally, the framework yields confidence estimates of obtained models, unlike alternative methods. The performance is tested on two regression and three classification experiments, and compared to the Nystr\"om and fixed size LS-SVM methods. Results show that our method can achieve high performance and is particularly useful when alternative methods are computationally infeasible due to a slowly decaying kernel matrix spectrum.
Differentiable programming is a new programming paradigm which enables large scale optimization through automatic calculation of gradients also known as auto-differentiation. This concept emerges from deep learning, and has also been generalized to tensor network optimizations. Here, we extend the differentiable programming to tensor networks with isometric constraints with applications to multiscale entanglement renormalization ansatz (MERA) and tensor network renormalization (TNR). By introducing several gradient-based optimization methods for the isometric tensor network and comparing with Evenbly-Vidal method, we show that auto-differentiation has a better performance for both stability and accuracy. We numerically tested our methods on 1D critical quantum Ising spin chain and 2D classical Ising model. We calculate the ground state energy for the 1D quantum model and internal energy for the classical model, and scaling dimensions of scaling operators and find they all agree with the theory well.