Efficient modelling of feature interactions underpins supervised learning for non-sequential tasks, characterized by a lack of inherent ordering of features (variables). The brute force approach of learning a parameter for each interaction of every order comes at an exponential computational and memory cost (Curse of Dimensionality). To alleviate this issue, it has been proposed to implicitly represent the model parameters as a tensor, the order of which is equal to the number of features; for efficiency, it can be further factorized into a compact Tensor Train (TT) format. However, both TT and other Tensor Networks (TNs), such as Tensor Ring and Hierarchical Tucker, are sensitive to the ordering of their indices (and hence to the features). To establish the desired invariance to feature ordering, we propose to represent the weight tensor through the Canonical Polyadic (CP) Decomposition (CPD), and introduce the associated inference and learning algorithms, including suitable regularization and initialization schemes. It is demonstrated that the proposed CP-based predictor significantly outperforms other TN-based predictors on sparse data while exhibiting comparable performance on dense non-sequential tasks. Furthermore, for enhanced expressiveness, we generalize the framework to allow feature mapping to arbitrarily high-dimensional feature vectors. In conjunction with feature vector normalization, this is shown to yield dramatic improvements in performance for dense non-sequential tasks, matching models such as fully-connected neural networks.
This work proposes to learn fair low-rank tensor decompositions by regularizing the Canonical Polyadic Decomposition factorization with the kernel Hilbert-Schmidt independence criterion (KHSIC). It is shown, theoretically and empirically, that a small KHSIC between a latent factor and the sensitive features guarantees approximate statistical parity. The proposed algorithm surpasses the state-of-the-art algorithm, FATR (Zhu et al., 2018), in controlling the trade-off between fairness and residual fit on synthetic and real data sets.
Recently, there has been a trend to combine independent component analysis and canonical polyadic decomposition (ICA-CPD) for an enhanced robustness for the computation of CPD, and ICA-CPD could be further converted into CPD of a 5th-order partially symmetric tensor, by calculating the eigenmatrices of the 4th-order cumulant slices of a trilinear mixture. In this study, we propose a new 5th-order CPD algorithm constrained with partial symmetry based on joint diagonalization. As the main steps involved in the proposed algorithm undergo no updating iterations for the loading matrices, it is much…
We consider the problem of structured canonical polyadic decomposition. If the size of the problem is very big, then stochastic gradient approaches are viable alternatives to classical methods, such as Alternating Optimization and All-At-Once optimization. We extend a recent stochastic gradient approach by employing an acceleration step (Nesterov momentum) in each iteration. We compare our approach with state-of-the-art alternatives, using both synthetic and real-world data, and find it to be very competitive.
Osaka University（大阪大学）；The University of Tokyo（东京大学）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；Nara Institute of Science and Technology（奈良先端科学技术大学院大学）
Methods based on vector embeddings of knowledge graphs have been actively pursued as a promising approach to knowledge graph completion.However, embedding models generate storage-inefficient representations, particularly when the number of entities and relations, and the dimensionality of the real-valued embedding vectors are large. We present a binarized CANDECOMP/PARAFAC(CP) decomposition algorithm, which we refer to as B-CP, where real-valued parameters are replaced by binary values to reduce model size. Moreover, we show that a fast score computation technique can be developed with bitwise…
Dalian University of Technology（大连理工大学）；Xiaomi Inc.（小米公司）
We propose a novel algorithm for the computation of canonical polyadic decomposition (CPD) of large-scale tensors. The proposed algorithm generalizes the random projection (RAP) technique, which is often used to compute large-scale decompositions, from one single projection to multiple but coupled random projections (CoRAP). The proposed CoRAP technique yields a set of tensors that together admits a coupled CPD (C-CPD) and a C-CPD algorithm is then used to jointly decompose these tensors. The results of C-CPD are finally fused to obtain factor matrices of the original large-scale data tensor. …
Computer Science Research Institute, Sandia National Laboratories（桑迪亚国家实验室计算机科学研究所）；Department of Statistics & Data Science, University of California, Los Angeles（加州大学洛杉矶分校统计与数据科学系）
ETH Zurich, Swiss Federal Institute of Technology（苏黎世联邦理工学院）
Acoustic monitoring for machine fault detection is a recent and expanding research path that has already provided promising results for industries. However, it is impossible to collect enough data to learn all types of faults from a machine. Thus, new algorithms, trained using data from healthy conditions only, were developed to perform unsupervised anomaly detection. A key issue in the development of these algorithms is the noise in the signals, as it impacts the anomaly detection performance. In this work, we propose a powerful data-driven and quasi non-parametric denoising strategy for spec…
In this paper, we develop a method which we call OnlineGCP for computing the Generalized Canonical Polyadic (GCP) tensor decomposition of streaming data. GCP differs from traditional canonical polyadic (CP) tensor decompositions as it allows for arbitrary objective functions which the CP model attempts to minimize. This approach can provide better fits and more interpretable models when the observed tensor data is strongly non-Gaussian. In the streaming case, tensor data is gradually observed over time and the algorithm must incrementally update a GCP factorization with limited access to prior…
As electric grids experience high penetration levels of renewable generation, fundamental changes are required to address real-time situational awareness. This paper uses unique traits of tensors to devise a model-free situational awareness and energy forecasting framework for distribution networks. This work formulates the state of the network at multiple time instants as a three-way tensor; hence, recovering full state information of the network is tantamount to estimating all the values of the tensor. Given measurements received from $μ$phasor measurement units and/or smart meters, the reco…
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；IQVIA（艾昆纬）
Low-rank tensor factorization or completion is well-studied and applied in various online settings, such as online tensor factorization (where the temporal mode grows) and online tensor completion (where incomplete slices arrive gradually). However, in many real-world settings, tensors may have more complex evolving patterns: (i) one or more modes can grow; (ii) missing entries may be filled; (iii) existing tensor elements can change. Existing methods cannot support such complex scenarios. To fill the gap, this paper proposes a Generalized Online Canonical Polyadic (CP) Tensor factorization an…
RWTH Aachen University（亚琛工业大学）；Umeå Universitet（于默奥大学）；University of Copenhagen（哥本哈根大学）
Tensor decompositions, such as CANDECOMP/PARAFAC (CP), are widely used in a variety of applications, such as chemometrics, signal processing, and machine learning. A broadly used method for computing such decompositions relies on the Alternating Least Squares (ALS) algorithm. When the number of components is small, regardless of its implementation, ALS exhibits low arithmetic intensity, which severely hinders its performance and makes GPU offloading ineffective. We observe that, in practice, experts often have to compute multiple decompositions of the same tensor, each with a small number of c…
Université de Mons（蒙斯大学）；Univ. Côte d’Azur（蔚蓝海岸大学）；CNRS（法国国家科学研究中心）；I3S（索菲亚安提波利斯信息学、信号与系统研究所）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Grenoble INP（格勒诺布尔国立理工学院）；GIPSA-lab（格勒诺布尔图像、语音信号与自动化实验室）
We propose an extension of the canonical polyadic (CP) tensor model where one of the latent factors is allowed to vary through data slices in a constrained way. The components of the latent factors, which we want to retrieve from data, can vary from one slice to another up to a diffeomorphism. We suppose that the diffeomorphisms are also unknown, thus merging curve registration and tensor decomposition in one model, which we call registered CP. We present an algorithm to retrieve both the latent factors and the diffeomorphism, which is assumed to be in a parametrized form. At the end of the pa…
Tensor decompositions are powerful tools for large data analytics as they jointly model multiple aspects of data into one framework and enable the discovery of the latent structures and higher-order correlations within the data. One of the most widely studied and used decompositions, especially in data mining and machine learning, is the Canonical Polyadic or CP decomposition. However, today's datasets are not static and these datasets often dynamically growing and changing with time. To operate on such large data, we present OCTen the first ever compression-based online parallel implementatio…
University of Southern California（南加州大学）；Princeton University（普林斯顿大学）；University of Minnesota（明尼苏达大学）
We consider the problem of factorizing a structured 3-way tensor into its constituent Canonical Polyadic (CP) factors. This decomposition, which can be viewed as a generalization of singular value decomposition (SVD) for tensors, reveals how the tensor dimensions (features) interact with each other. However, since the factors are a priori unknown, the corresponding optimization problems are inherently non-convex. The existing guaranteed algorithms which handle this non-convexity incur an irreducible error (bias), and only apply to cases where all factors have the same structure. To this end, w…
Sandia National Laboratories（桑迪亚国家实验室）；University of Michigan（密歇根大学）
Tensor decomposition is a well-known tool for multiway data analysis. This work proposes using stochastic gradients for efficient generalized canonical polyadic (GCP) tensor decomposition of large-scale tensors. GCP tensor decomposition is a recently proposed version of tensor decomposition that allows for a variety of loss functions such as Bernoulli loss for binary data or Huber loss for robust estimation. The stochastic gradient is formed from randomly sampled elements of the tensor and is efficient because it can be computed using the sparse matricized-tensor-times-Khatri-Rao product (MTTK…
Department of Statistics, University of Wisconsin-Madison（威斯康星大学麦迪逊分校统计学系）；Departments of Biostatistics & Bioinformatics and Computer Science, Duke University（杜克大学生物统计学与生物信息学系、计算机科学系）
University of Malta（马耳他大学）；Technical University of Crete（克里特理工大学）；DeepLab；University of West Attica（西阿提卡大学）；National Technical University of Athens（雅典国立技术大学）
An increasing number of emerging applications in data science and engineering are based on multidimensional and structurally rich data. The irregularities, however, of high-dimensional data often compromise the effectiveness of standard machine learning algorithms. We hereby propose the Rank-R Feedforward Neural Network (FNN), a tensor-based nonlinear learning model that imposes Canonical/Polyadic decomposition on its parameters, thereby offering two core advantages compared to typical machine learning methods. First, it handles inputs as multilinear arrays, bypassing the need for vectorizatio…
Learning the multivariate distribution of data is a core challenge in statistics and machine learning. Traditional methods aim for the probability density function (PDF) and are limited by the curse of dimensionality. Modern neural methods are mostly based on black-box models, lacking identifiability guarantees. In this work, we aim to learn multivariate cumulative distribution functions (CDFs), as they can handle mixed random variables, allow efficient box probability evaluation, and have the potential to overcome local sample scarcity owing to their cumulative nature. We show that any grid s…
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…
Shenzhen Research Institute of Big Data（深圳大数据研究院）；Oregon State University（俄勒冈州立大学）；University of Minnesota（明尼苏达大学）
This work considers low-rank canonical polyadic decomposition (CPD) under a class of non-Euclidean loss functions that frequently arise in statistical machine learning and signal processing. These loss functions are often used for certain types of tensor data, e.g., count and binary tensors, where the least squares loss is considered unnatural.Compared to the least squares loss, the non-Euclidean losses are generally more challenging to handle. Non-Euclidean CPD has attracted considerable interests and a number of prior works exist. However, pressing computational and theoretical challenges, s…
A Unified MDL-based Binning and Tensor Factorization Framework for PDF Estimation
Communications Research Laboratory（通信研究中心）；Ilmenau University of Technology（伊门瑙技术大学）；School of Electrical Engineering（电气工程学院）；Tel Aviv University（特拉维夫大学）
This paper provides a novel framework that learns canonical embeddings for non-rigid shape matching. In contrast to prior work in this direction, our framework is trained end-to-end and thus avoids instabilities and constraints associated with the commonly-used Laplace-Beltrami basis or sequential optimization schemes. On multiple datasets, we demonstrate that learning self symmetry maps with a deep functional map projects 3D shapes into a low dimensional canonical embedding that facilitates non-rigid shape correspondence via a simple nearest neighbor search. Our framework outperforms multiple…
Southern University of Science and Technology（南方科技大学）；Sun Yat-sen University（中山大学）
Tensor decomposition is an effective tool for learning multi-way structures and heterogeneous features from high-dimensional data, such as the multi-view images and multichannel electroencephalography (EEG) signals, are often represented by tensors. However, most of tensor decomposition methods are the linear feature extraction techniques, which are unable to reveal the nonlinear structure within high-dimensional data. To address such problem, a lot of algorithms have been proposed for simultaneously performs linear and non-linear feature extraction. A representative algorithm is the Graph Reg…
Anisotropic Tensor Deconvolution of Hyperspectral Images
Department of Information and Communications Engineering, Aalto University, Finland（信息与通信工程系，艾尔沃大学，芬兰）；Université de Lorraine, CNRS, CRAN, France（洛林大学，法国国家科学研究中心，CRAN，法国）
