Tensor networks (TN) have found a wide use in machine learning, and in particular, TN and deep learning bear striking similarities. In this work, we propose the quantum-classical hybrid tensor networks (HTN) which combine tensor networks with classical neural networks in a uniform deep learning framework to overcome the limitations of regular tensor networks in machine learning. We first analyze the limitations of regular tensor networks in the applications of machine learning involving the representation power and architecture scalability. We conclude that in fact the regular tensor networks are not competent to be the basic building blocks of deep learning. Then, we discuss the performance of HTN which overcome all the deficiency of regular tensor networks for machine learning. In this sense, we are able to train HTN in the deep learning way which is the standard combination of algorithms such as Back Propagation and Stochastic Gradient Descent. We finally provide two applicable cases to show the potential applications of HTN, including quantum states classification and quantum-classical autoencoder. These cases also demonstrate the great potentiality to design various HTN in deep learning way.
Given recent deep learning results that demonstrate the ability to effectively optimize high-dimensional non-convex functions with gradient descent optimization on GPUs, we ask in this paper whether symbolic gradient optimization tools such as Tensorflow can be effective for planning in hybrid (mixed discrete and continuous) nonlinear domains with high dimensional state and action spaces? To this end, we demonstrate that hybrid planning with Tensorflow and RMSProp gradient descent is competitive with mixed integer linear program (MILP) based optimization on piecewise linear planning domains (w…
Recent years have demonstrated that using random feature maps can significantly decrease the training and testing times of kernel-based algorithms without significantly lowering their accuracy. Regrettably, because random features are target-agnostic, typically thousands of such features are necessary to achieve acceptable accuracies. In this work, we consider the problem of learning a small number of explicit polynomial features. Our approach, named Tensor Machines, finds a parsimonious set of features by optimizing over the hypothesis class introduced by Kar and Karnick for random feature ma…
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 …
Multimodal data arise in various applications where information about the same phenomenon is acquired from multiple sensors and across different imaging modalities. Learning from multimodal data is of great interest in machine learning and statistics research as this offers the possibility of capturing complementary information among modalities. Multimodal modeling helps to explain the interdependence between heterogeneous data sources, discovers new insights that may not be available from a single modality, and improves decision-making. Recently, coupled matrix-tensor factorization has been i…
This work proposes a novel general-purpose estimator for supervised machine learning (ML) based on tensor trains (TT). The estimator uses TTs to parametrize discretized functions, which are then optimized using Riemannian gradient descent under the form of a tensor completion problem. Since this optimization is sensitive to initialization, it turns out that the use of other ML estimators for initialization is crucial. This results in a competitive, fast ML estimator with lower memory usage than many other ML estimators, like the ones used for the initialization.
Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；University of Bath（巴斯大学）；Technische Universität Chemnitz（开姆尼茨工业大学）
An increasing amount of collected data are high-dimensional multi-way arrays (tensors), and it is crucial for efficient learning algorithms to exploit this tensorial structure as much as possible. The ever-present curse of dimensionality for high dimensional data and the loss of structure when vectorizing the data motivates the use of tailored low-rank tensor classification methods. In the presence of small amounts of training data, kernel methods offer an attractive choice as they provide the possibility for a nonlinear decision boundary. We develop the Tensor Train Multi-way Multi-level Kern…
Hybrid quantum tensor networks for aeroelastic applications
Institute for AI Safety and Security（人工智能安全与保安研究所）；German Aerospace Center (DLR)（德国航空航天中心）；Institute of Aeroelasticity（气动弹性研究所）；Institute of Neural Information Processing（神经信息处理研究所）；University of Ulm（乌尔姆大学）
Guangdong University of Technology（广东工业大学）；RIKEN（理化学研究所）
Tensor robust principal component analysis (TRPCA) is a fundamental model in machine learning and computer vision. Recently, tensor train (TT) decomposition has been verified effective to capture the global low-rank correlation for tensor recovery tasks. However, due to the large-scale tensor data in real-world applications, previous TRPCA models often suffer from high computational complexity. In this letter, we propose an efficient TRPCA under hybrid model of Tucker and TT. Specifically, in theory we reveal that TT nuclear norm (TTNN) of the original big tensor can be equivalently converted …
Mila（米拉研究所）；Université de Montréal（蒙特利尔大学）；Princeton University（普林斯顿大学）；X - The Moonshot Factory（X——登月工厂）
We investigate a correspondence between two formalisms for discrete probabilistic modeling: probabilistic graphical models (PGMs) and tensor networks (TNs), a powerful modeling framework for simulating complex quantum systems. The graphical calculus of PGMs and TNs exhibits many similarities, with discrete undirected graphical models (UGMs) being a special case of TNs. However, more general probabilistic TN models such as Born machines (BMs) employ complex-valued hidden states to produce novel forms of correlation among the probabilities. While representing a new modeling resource for capturin…
Tensors are multidimensional arrays of numerical values and therefore generalize matrices to multiple dimensions. While tensors first emerged in the psychometrics community in the $20^{\text{th}}$ century, they have since then spread to numerous other disciplines, including machine learning. Tensors and their decompositions are especially beneficial in unsupervised learning settings, but are gaining popularity in other sub-disciplines like temporal and multi-relational data analysis, too. The scope of this paper is to give a broad overview of tensors, their decompositions, and how they are use…
Brookhaven National Laboratory（布鲁克海文国家实验室）；National Taiwan University（台湾大学）
One key step in performing quantum machine learning (QML) on noisy intermediate-scale quantum (NISQ) devices is the dimension reduction of the input data prior to their encoding. Traditional principle component analysis (PCA) and neural networks have been used to perform this task; however, the classical and quantum layers are usually trained separately. A framework that allows for a better integration of the two key components is thus highly desirable. Here we introduce a hybrid model combining the quantum-inspired tensor networks (TN) and the variational quantum circuits (VQC) to perform sup…
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Hangzhou Institute for Advanced Study, UCAS（国科大杭州高等研究院）；International Centre for Theoretical Physics Asia-Pacific（国际理论物理中心亚太地区分中心）
Restricted Boltzmann machines (RBM) and deep Boltzmann machines (DBM) are important models in machine learning, and recently found numerous applications in quantum many-body physics. We show that there are fundamental connections between them and tensor networks. In particular, we demonstrate that any RBM and DBM can be exactly represented as a two-dimensional tensor network. This representation gives an understanding of the expressive power of RBM and DBM using entanglement structures of the tensor networks, also provides an efficient tensor network contraction algorithm for the computing par…
Quantum machine learning (QML) is the use of quantum computing for the computation of machine learning algorithms. With the prevalence and importance of classical data, a hybrid quantum-classical approach to QML is called for. Parameterized Quantum Circuits (PQCs), and particularly Quantum Kernel PQCs, are generally used in the hybrid approach to QML. In this paper we discuss some important aspects of PQCs with quantum kernels including PQCs, quantum kernels, quantum kernels with quantum advantage, and the trainability of quantum kernels. We conclude that quantum kernels with hybrid kernel met…
Beijing Normal University（北京师范大学）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Swarma Research（集智研究院）；Hangzhou Institute for Advanced Study, UCAS（中国科学院大学杭州高等研究院）；International Centre for Theoretical Physics Asia-Pacific（亚太国际理论物理中心）
Modeling the joint distribution of high-dimensional data is a central task in unsupervised machine learning. In recent years, many interests have been attracted to developing learning models based on tensor networks, which have the advantages of a principle understanding of the expressive power using entanglement properties, and as a bridge connecting classical computation and quantum computation. Despite the great potential, however, existing tensor network models for unsupervised machine learning only work as a proof of principle, as their performance is much worse than the standard models s…
We introduce a hybrid model combining a quantum-inspired tensor network and a variational quantum circuit to perform supervised learning tasks. This architecture allows for the classical and quantum parts of the model to be trained simultaneously, providing an end-to-end training framework. We show that compared to the principal component analysis, a tensor network based on the matrix product state with low bond dimensions performs better as a feature extractor for the input data of the variational quantum circuit in the binary and ternary classification of MNIST and Fashion-MNIST datasets. Th…
Tensor2Tensor is a library for deep learning models that is well-suited for neural machine translation and includes the reference implementation of the state-of-the-art Transformer model.
Deep learning systems have become vital tools across many fields, but the increasing model sizes mean that training must be accelerated to maintain such systems' utility. Current systems like Tensorflow and MXNet focus on one specific parallelization strategy, data parallelism, which requires large training batch sizes in order to scale. We cast the problem of finding the best parallelization strategy as the problem of finding the best tiling to partition tensors with the least overall communication. We propose an algorithm that can find the optimal tiling. Our resulting parallelization soluti…
The University of Hong Kong（香港大学）；Delft University of Technology（代尔夫特理工大学）；Tsinghua University（清华大学）
Tensor, a multi-dimensional data structure, has been exploited recently in the machine learning community. Traditional machine learning approaches are vector- or matrix-based, and cannot handle tensorial data directly. In this paper, we propose a tensor train (TT)-based kernel technique for the first time, and apply it to the conventional support vector machine (SVM) for image classification. Specifically, we propose a kernelized support tensor train machine that accepts tensorial input and preserves the intrinsic kernel property. The main contributions are threefold. First, we propose a TT-ba…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）
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 disti…
tn4ml: Tensor Network Training and Customization for Machine Learning
Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Barcelona Supercomputing Center（巴塞罗那超级计算中心）；Qilimanjaro Quantum Tech（奇利曼哈罗量子科技公司）；European Organization for Nuclear Research (CERN)（欧洲核子研究组织（CERN））
Alphabet (Google) X（Alphabet（谷歌）X）；Perimeter Institute for Theoretical Physics（圆周理论物理研究所）；Google, Inc.（谷歌公司）
TensorNetwork is an open source library for implementing tensor network algorithms. Tensor networks are sparse data structures originally designed for simulating quantum many-body physics, but are currently also applied in a number of other research areas, including machine learning. We demonstrate the use of the API with applications both physics and machine learning, with details appearing in companion papers.
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院）
Tensor network algorithms seek to minimize correlations to compress the classical data representing quantum states. Tensor network algorithms and similar tools---called tensor network methods---form the backbone of modern numerical methods used to simulate many-body physics and have a further range of applications in machine learning. Finding and contracting tensor network states is a computational task which quantum computers might be used to accelerate. We present a quantum algorithm which returns a classical description of a rank-$r$ tensor network state satisfying an area law and approxima…
College of Information Science & Technology（信息科学与技术学院）；The Pennsylvania State University（宾夕法尼亚州立大学）
Two novel deep hybrid architectures, the Deep Hybrid Boltzmann Machine and the Deep Hybrid Denoising Auto-encoder, are proposed for handling semi-supervised learning problems. The models combine experts that model relevant distributions at different levels of abstraction to improve overall predictive performance on discriminative tasks. Theoretical motivations and algorithms for joint learning for each are presented. We apply the new models to the domain of data-streams in work towards life-long learning. The proposed architectures show improved performance compared to a pseudo-labeled, drop-o…
University of Maryland（马里兰大学）；Rice University（莱斯大学）；Haverford College（哈弗福德学院）
A model-based approach to forecasting chaotic dynamical systems utilizes knowledge of the physical processes governing the dynamics to build an approximate mathematical model of the system. In contrast, machine learning techniques have demonstrated promising results for forecasting chaotic systems purely from past time series measurements of system state variables (training data), without prior knowledge of the system dynamics. The motivation for this paper is the potential of machine learning for filling in the gaps in our underlying mechanistic knowledge that cause widely-used knowledge-base…
Università di Genova（热那亚大学）；CNR - SPIN Genova（意大利国家研究委员会热那亚自旋研究所）
We introduce a hybrid approach to solar flare prediction, whereby a supervised regularization method is used to realize feature importance and an unsupervised clustering method is used to realize the binary flare/no-flare decision. The approach is validated against NOAA SWPC data.
We implement a Tensor Train layer in the TensorFlow Neural Machine Translation (NMT) model using the t3f library. We perform training runs on the IWSLT English-Vietnamese '15 and WMT German-English '16 datasets with learning rates $\in \{0.0004,0.0008,0.0012\}$, maximum ranks $\in \{2,4,8,16\}$ and a range of core dimensions. We compare against a target BLEU test score of 24.0, obtained by our benchmark run. For the IWSLT English-Vietnamese training, we obtain BLEU test/dev scores of 24.0/21.9 and 24.2/21.9 using core dimensions $(2, 2, 256) \times (2, 2, 512)$ with learning rate 0.0012 and ra…
A Hybrid Tucker-LSTM Tensor Network Model for SOC Prediction in Electric Vehicles
College of Computer and Information Science（计算机与信息科学学院）；School of Culture Tourism（文化旅游学院）；Digital Intelligence Center（数字智能中心）；China Automotive Engineering Research Institute Co., Ltd.（中国汽车工程研究院股份有限公司）
Quantum-inspired tensor networks in machine learning models
Vicomtech Foundation, Basque Research and Technology Alliance (BRTA)（维科姆科技基金会，巴斯克研究与技术联盟（BRTA））；University of Deusto（德乌斯大学）；Escuela Técnica Superior de Ingeniería de Sistemas Informáticos, Universidad Politécnica de Madrid（马德里理工大学信息系统工程学院）；Department of Applied Physics, University of Geneva（日内瓦大学应用物理系）
Georgia Institute of Technology（佐治亚理工学院）；University of Enna（恩纳大学）；Tencent Corporation（腾讯公司）
This paper investigates different trade-offs between the number of model parameters and enhanced speech qualities by employing several deep tensor-to-vector regression models for speech enhancement. We find that a hybrid architecture, namely CNN-TT, is capable of maintaining a good quality performance with a reduced model parameter size. CNN-TT is composed of several convolutional layers at the bottom for feature extraction to improve speech quality and a tensor-train (TT) output layer on the top to reduce model parameters. We first derive a new upper bound on the generalization power of the c…
BrandMaker；FOM University of Applied Sciences for Economics and Management（FOM经济管理应用科学大学）；Ulm University（乌尔姆大学）
This book chapter describes a novel approach to training machine learning systems by means of a hybrid computer setup i.e. a digital computer tightly coupled with an analog computer. As an example a reinforcement learning system is trained to balance an inverted pendulum which is simulated on an analog computer, thus demonstrating a solution to the major challenge of adequately simulating the environment for reinforcement learning.
Rice University（莱斯大学）；Arizona State University（亚利桑那州立大学）
We consider the question: what is the abstraction that should be implemented by the computational engine of a machine learning system? Current machine learning systems typically push whole tensors through a series of compute kernels such as matrix multiplications or activation functions, where each kernel runs on an AI accelerator (ASIC) such as a GPU. This implementation abstraction provides little built-in support for ML systems to scale past a single machine, or for handling large models with matrices or tensors that do not easily fit into the RAM of an ASIC. In this paper, we present an al…
In this article, we perform a review of the state-of-the-art of hybrid machine learning in medical imaging. We start with a short summary of the general developments of the past in machine learning and how general and specialized approaches have been in competition in the past decades. A particular focus will be the theoretical and experimental evidence pro and contra hybrid modelling. Next, we inspect several new developments regarding hybrid machine learning with a particular focus on so-called known operator learning and how hybrid approaches gain more and more momentum across essentially a…
Georgia Institute of Technology（佐治亚理工学院）；Universidad San Pablo-CEU, CEU Universities（CEU大学圣巴布罗-CEU大学）
This work aims to design a low complexity spoken command recognition (SCR) system by considering different trade-offs between the number of model parameters and classification accuracy. More specifically, we exploit a deep hybrid architecture of a tensor-train (TT) network to build an end-to-end SRC pipeline. Our command recognition system, namely CNN+(TT-DNN), is composed of convolutional layers at the bottom for spectral feature extraction and TT layers at the top for command classification. Compared with a traditional end-to-end CNN baseline for SCR, our proposed CNN+(TT-DNN) model replaces…
Alphabet (Google) X（Alphabet（谷歌）X）；Max-Planck-Institut für Quantenoptik（马克斯·普朗克量子光学研究所）
We demonstrate the use of tensor networks for image classification with the TensorNetwork open source library. We explain in detail the encoding of image data into a matrix product state form, and describe how to contract the network in a way that is parallelizable and well-suited to automatic gradients for optimization. Applying the technique to the MNIST and Fashion-MNIST datasets we find out-of-the-box performance of 98% and 88% accuracy, respectively, using the same tensor network architecture. The TensorNetwork library allows us to seamlessly move from CPU to GPU hardware, and we see a fa…
The University of Adelaide（阿德莱德大学）；The Australian National University（澳大利亚国立大学）；NICTA（澳大利亚国家信息与通信技术研究院）
We propose a novel hybrid loss for multiclass and structured prediction problems that is a convex combination of log loss for Conditional Random Fields (CRFs) and a multiclass hinge loss for Support Vector Machines (SVMs). We provide a sufficient condition for when the hybrid loss is Fisher consistent for classification. This condition depends on a measure of dominance between labels - specifically, the gap in per observation probabilities between the most likely labels. We also prove Fisher consistency is necessary for parametric consistency when learning models such as CRFs. We demonstrate e…
We demonstrate Tensor Query Processor (TQP): a query processor that automatically compiles relational operators into tensor programs. By leveraging tensor runtimes such as PyTorch, TQP is able to: (1) integrate with ML tools (e.g., Pandas for data ingestion, Tensorboard for visualization); (2) target different hardware (e.g., CPU, GPU) and software (e.g., browser) backends; and (3) end-to-end accelerate queries containing both relational and ML operators. TQP is generic enough to support the TPC-H benchmark, and it provides performance that is comparable to, and often better than, that of spec…
Anisotropic Permeability Tensor Prediction from Porous Media Microstructure via Physics-Informed Progressive Transfer Learning with Hybrid CNN-Transformer
Faculty of Mathematics and Natural Sciences, University of Oslo（奥斯陆大学数学与自然科学学院）
Federal University of Juiz de Fora（茹伊斯迪福拉联邦大学）；Federal University of São João del-Rei（圣若昂德尔雷伊联邦大学）
This work presents a novel technique that integrates the methodologies of machine learning and system identification to solve multiclass problems. Such an approach allows to extract and select sets of representative features with reduced dimensionality, as well as predicts categorical outputs. The efficiency of the method was tested by running case studies investigated in machine learning, obtaining better absolute results when compared with classical classification algorithms.
Learning Mixtures of Linear Dynamical Systems via Hybrid Tensor-EM Method
Department of Biomedical Engineering（生物医学工程系）；Wu Tsai Institute, Center for Neurocomputation and Machine Intelligence（吴泰教授研究所、神经计算与人工智能中心）；Yale University（耶鲁大学）
International Institute of Information technology（国际信息技术学院）
Recent advances in cellular communication systems resulted in a huge increase in spectrum demand. To meet the requirements of the ever-growing need for spectrum, efficient utilization of the existing resources is of utmost importance. Channel Allocation, has thus become an inevitable research topic in wireless communications. In this paper, we propose an optimal channel allocation scheme, Optimal Hybrid Channel Allocation (OHCA) for an effective allocation of channels. We improvise upon the existing Fixed Channel Allocation (FCA) technique by imparting intelligence to the existing system by em…
As the volatility of electricity demand increases owing to climate change and electrification, the importance of accurate peak load forecasting is increasing. Traditional peak load forecasting has been conducted through time series-based models; however, recently, new models based on machine or deep learning are being introduced. This study performs a comparative analysis to determine the most accurate peak load-forecasting model for Korea, by comparing the performance of time series, machine learning, and hybrid models. Seasonal autoregressive integrated moving average with exogenous variable…
Neural machine translation - using neural networks to translate human language - is an area of active research exploring new neuron types and network topologies with the goal of dramatically improving machine translation performance. Current state-of-the-art approaches, such as the multi-head attention-based transformer, require very large translation corpuses and many epochs to produce models of reasonable quality. Recent attempts to parallelize the official TensorFlow "Transformer" model across multiple nodes have hit roadblocks due to excessive memory use and resulting out of memory errors …
Sorbonne Universitas（索邦大学）；University of Pittsburgh/Medical Center（匹兹堡大学医学中心）；Carnegie Mellon University（卡内基梅隆大学）
We introduce in this paper the principle of Deep Temporal Networks that allow to add time to convolutional networks by allowing deep integration principles not only using spatial information but also increasingly large temporal window. The concept can be used for conventional image inputs but also event based data. Although inspired by the architecture of brain that inegrates information over increasingly larger spatial but also temporal scales it can operate on conventional hardware using existing architectures. We introduce preliminary results to show the efficiency of the method. More in-de…
National Institute of Information and Communications Technology（信息通信研究机构）
Reduction of training time is an important issue in many tasks like patent translation involving neural networks. Data parallelism and model parallelism are two common approaches for reducing training time using multiple graphics processing units (GPUs) on one machine. In this paper, we propose a hybrid data-model parallel approach for sequence-to-sequence (Seq2Seq) recurrent neural network (RNN) machine translation. We apply a model parallel approach to the RNN encoder-decoder part of the Seq2Seq model and a data parallel approach to the attention-softmax part of the model. We achieved a spee…
