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Tensor wheel (TW) decomposition combines the popular tensor ring and fully connected tensor network decompositions and has achieved excellent performance in tensor completion problem. A standard method to compute this decomposition is the alternating least squares (ALS). However, it usually suffers from slow convergence and numerical instability. In this work, the fast and robust SVD-based algorithms are investigated. Based on a result on TW-ranks, we first propose a deterministic algorithm that can estimate the TW decomposition of the target tensor under a controllable accuracy. Then, the randomized versions of this algorithm are presented, which can be divided into two categories and allow various types of sketching. Numerical results on synthetic and real data show that our algorithms have much better performance than the ALS-based method and are also quite robust. In addition, with one SVD-based algorithm, we also numerically explore the variability of TW decomposition with respect to TW-ranks and the comparisons between TW decomposition and other famous formats in terms of the performance on approximation and compression. [ABSTRACT FROM AUTHOR]

The popular fully-connected tensor network (FCTN) decomposition has achieved successful applications in many fields. A standard method to this decomposition is the alternating least squares. However, it often converges slowly and suffers from issues of numerical stability. In this work, we investigate the SVD-based algorithms for FCTN decomposition to tackle the aforementioned deficiencies. On the basis of a result about FCTN-ranks, a deterministic algorithm, namely FCTN-SVD, is first proposed, which can approximate the FCTN decomposition under a fixed accuracy. Then, we present the randomized version of the algorithm. Both synthetic and real data are used to test our algorithms. Numerical results show that they perform much better than the existing methods, and the randomized algorithm can indeed yield acceleration on FCTN-SVD. Moreover, we also apply our algorithms to tensor-on-vector regression and achieve quite decent performance. [ABSTRACT FROM AUTHOR]

Fully-connected tensor network (FCTN) decomposition is a generalization of the popular tensor train and tensor ring decompositions and has been applied to various fields with great success. The standard method for computing this decomposition is the well-known alternating least squares (ALS). However, it is very expensive, especially for large-scale tensors. To reduce the cost, we propose an ALS-based randomized algorithm. Specifically, by defining a new tensor product called subnetwork product and adjusting the sizes of FCTN factors suitably, the structure of the coefficient matrices of the ALS subproblems from FCTN decomposition is first figured out. Then, with the structure and the properties of subnetwork product, we devise the randomized algorithm based on leverage sampling. This algorithm enables sampling on FCTN factors and hence avoids the formation of full coefficient matrices of ALS subproblems. The computational complexity and numerical performance of our algorithm are presented. Experimental results show that it requires much less computation time to achieve similar accuracy compared with the deterministic ALS method. Further, we apply our algorithm to four famous problems, i.e., tensor-on-vector regression, multi-view subspace clustering, nonnegative tensor approximation and tensor completion, and the performances are quite decent. [ABSTRACT FROM AUTHOR]

Tensor wheel (TW) decomposition is an elegant compromise of the popular tensor ring decomposition and fully connected tensor network decomposition, and it has many applications. In this work, we investigate the computation of this decomposition. Three randomized algorithms based on random sampling or random projection are proposed. Specifically, by defining a new tensor product called the subwheel product, the structures of the coefficient matrices of the alternating least squares subproblems from the minimization problem of TW decomposition are first figured out. Then, using the structures and the properties of the subwheel product, a random sampling algorithm based on leverage sampling and two random projection algorithms respectively based on Kronecker subsampled randomized Fourier transform and TensorSketch are derived. These algorithms can implement the sampling and projection on TW factors and hence can avoid forming the full coefficient matrices of subproblems. We present the complexity analysis and numerical performance on synthetic data, real data, and image reconstruction for our algorithms. Experimental results show that, compared with the deterministic algorithm in the literature, they need much less computing time while achieving similar accuracy and reconstruction effect. We also apply the proposed algorithms to tensor completion and find that the sampling-based algorithm always has excellent performance and the projection-based algorithms behave well when the sampling rate is higher than 50%. [ABSTRACT FROM AUTHOR]

Tensor ring (TR) decomposition has been widely applied as an effective approach in a variety of applications to discover the hidden low‐rank patterns in multidimensional and higher‐order data. A well‐known method for TR decomposition is the alternating least squares (ALS). However, solving the ALS subproblems often suffers from high cost issue, especially for large‐scale tensors. In this paper, we provide two strategies to tackle this issue and design three ALS‐based algorithms. Specifically, the first strategy is used to simplify the calculation of the coefficient matrices of the normal equations for the ALS subproblems, which takes full advantage of the structure of the coefficient matrices of the subproblems and hence makes the corresponding algorithm perform much better than the regular ALS method in terms of computing time. The second strategy is to stabilize the ALS subproblems by QR factorizations on TR‐cores, and hence the corresponding algorithms are more numerically stable compared with our first algorithm. Extensive numerical experiments on synthetic and real data are given to illustrate and confirm the above results. In addition, we also present the complexity analyses of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this article, we develop additive autoregressive models (Add‐ARM) for the time series data with matrix valued predictors. The proposed models assume separable row, column and lag effects of the matrix variables, attaining stronger interpretability when compared with existing bilinear matrix autoregressive models. We utilize the Gershgorin's circle theorem to impose some certain conditions on the parameter matrices, which make the underlying process strictly stationary. We also introduce the alternating least squares estimation method to solve the involved equality constrained optimization problems. Asymptotic distributions of the parameter estimators are derived. In addition, we employ hypothesis tests to run diagnostics on the parameter matrices. The performance of the proposed models and methods is further demonstrated through simulations and real data analysis. [ABSTRACT FROM AUTHOR]

This research explored movie recommendation systems based on predicting top-rated and suitable movies for users. This research proposed a hybrid movie recommendation system that integrates both text-to-number conversion and cosine similarity approaches to predict the most top-rated and desired movies for the targeted users. The proposed movie recommendation employed the Alternating Least Squares (ALS) algorithm to reinforce the accuracy of movie recommendations. The performance analysis and evaluation were undertaken by employing the widely used "TMDB 5000 Movie Dataset" from the Kaggle dataset. Two experiments were conducted, categorizing the dataset into distinct modules, and the outcomes were contrasted with stateof-the-art models. The first experiment attained a Root Mean Squared Error (RMSE) of 0.97613, while the second experiment expanded predictions to 4800 movies, culminating in a substantially minimized RMSE of 0.8951, portraying a 97% accuracy enhancement. The findings underscore the essence of parameter selection in text-to-number conversion and cosine and the gap for other systems to maintain user preferences for comprehensive and precise data gathering. Overall, the proposed hybrid movie recommendation system demonstrated promising results in predicting top-rated movies and offering personalized and accurate recommendations to users. [ABSTRACT FROM AUTHOR]

The canonical polyadic (CP) decomposition of tensors is one of the most important tensor decompositions. While the well‐known alternating least squares (ALS) algorithm is often considered the workhorse algorithm for computing the CP decomposition, it is known to suffer from slow convergence in many cases and various algorithms have been proposed to accelerate it. In this article, we propose a new accelerated ALS algorithm that accelerates ALS in a blockwise manner using a simple momentum‐based extrapolation technique and a random perturbation technique. Specifically, our algorithm updates one factor matrix (i.e., block) at a time, as in ALS, with each update consisting of a minimization step that directly reduces the reconstruction error, an extrapolation step that moves the factor matrix along the previous update direction, and a random perturbation step for breaking convergence bottlenecks. Our extrapolation strategy takes a simpler form than the state‐of‐the‐art extrapolation strategies and is easier to implement. Our algorithm has negligible computational overheads relative to ALS and is simple to apply. Empirically, our proposed algorithm shows strong performance as compared to the state‐of‐the‐art acceleration techniques on both simulated and real tensors. [ABSTRACT FROM AUTHOR]

This study examines the psychometric properties of a screening protocol for dyslexia and demonstrates a special form of matrix factorization called Nous based on the Alternating Least Squares algorithm. Dyslexia presents an intrinsically multidimensional complex of cognitive loads. By building and enforcing a common 6-dimensional space, Nous extracts a multidimensional signal for each person and item from test data that increases the Shannon entropy of the dataset while at the same time being constrained to meet the special objectivity requirements of the Rasch model. The resulting Dyslexia Risk Scale (DRS) yields linear equal-interval measures that are comparable regardless of the subset of items taken by the examinee. Each measure and cell estimate is accompanied by an efficiently calculated standard error. By incorporating examinee age into the calibration process, the DRS can be generalized to all age groups to allow the tracking of individual dyslexia risk over time. The methodology was implemented using a 2019 calibration sample of 828 persons aged 7 to 82 with varying degrees of dyslexia risk. The analysis yielded high reliability (0.95) and excellent receiver operating characteristics (AUC = 0.96). The analysis is accompanied by a discussion of the information-theoretic properties of matrix factorization. [ABSTRACT FROM AUTHOR]

Canonical polyadic decomposition (CPD) is prevalent in chemometrics, signal processing, data mining, and many more fields. While many algorithms have been proposed to compute the CPD, alternating least squares (ALS) remains one of the most widely used algorithms for computing the decomposition. Recent works have introduced the notion of eigenvalues and singular values of a tensor and explored applications of eigenvectors and singular vectors in signal processing, data analytics, and various other fields. We introduce a new formulation for deriving singular values and vectors of a tensor by considering the critical points of a function differently from previous works. Computing these critical points in an alternating manner motivates an alternating optimization algorithm which corresponds to the ALS algorithm in the matrix case. However, for tensors with order greater than or equal to 3, it minimizes an objective function which is different from the commonly used least squares loss. Alternating optimization of this new objective leads to simple updates to the factor matrices with the same asymptotic computational cost as ALS. We show that a subsweep of this algorithm can achieve a superlinear convergence rate for exact CPD when the known rank is not larger than the mode lengths of the input tensor. We verify our theoretical arguments experimentally. We then view the algorithm as optimizing a Mahalanobis distance with respect to each factor with the ground metric dependent on the other factors. This perspective allows us to generalize our approach to interpolate between updates corresponding to the ALS and the new algorithm to manage the tradeoff between stability and fitness of the decomposition. Our experimental results show that for approximating synthetic and real-world tensors, this algorithm and its variants converge to a better conditioned decomposition with comparable and sometimes better fitness as compared to the ALS algorithm. [ABSTRACT FROM AUTHOR]

In several previously published studies, Lavine and coworkers have demonstrated that infrared (IR) spectra from all layers of an intact multilayered automotive paint chip can be collected in a single analysis by scanning across each layer of a cross sectioned paint chip using a Fourier transform IR imaging microscope. Applying alternating least squares to the spectral data, the IR spectrum of each layer of an original equipment manufacturer paint chip can be extracted from a line map of the spectral image. To further develop this imaging technique for automotive paint analysis, the capability to cross section "small" paint chips (1 mm or less) using an ultramicrotome has been incorporated into our current imaging methodology. An ultramicrotome does not require epoxy or other embedding media for the paint chip and will simplify the analysis. However, extracting the IR spectra for each layer of an original equipment manufacturer paint chip by alternating least squares can be problematic for thin peels (less than one micron thickness), necessitating the use of target testing factor analysis to determine whether a specific layer is present in the line map and modified alternating least squares to recover the IR spectrum of the layer. Using a new sample preparation technique and the appropriate multivariate curve resolution methods, high quality IR spectra of the layers of a modern automotive paint system can be obtained from paint fragments that are smaller than what is practical to analyze by conventional Fourier transform IR spectroscopy. Applying target testing factor analysis, alternating least squares and modified alternating least squares to the line map of an infrared image of an automotive paint chip cross sectioned using ultramicrotomy, the infrared spectrum of each layer of an original equipment manufacturer paint chip was successfully extracted from the spectral data, even for very thin peels less than one micron thick. [ABSTRACT FROM AUTHOR]

In this study, the problem of blind signal separation for coprime planar arrays is investigated. For coprime planar arrays comprising two uniform rectangular subarrays, we link the signal separation to the tensor-based model called coupled canonical polyadic decomposition (CPD) and propose an improved coupled trilinear decomposition approach. The output data of coprime planar arrays are modeled as a coupled tensor set that can be further interpreted as a coupled CPD model, allowing a signal separation to be achieved using coupled trilinear alternating least squares (TALS). Furthermore, in the procedure of the coupled TALS, a Vandermonde structure enforcing approach is explicitly applied, which is shown to ensure fast convergence. The results of Monto Carlo simulations show that our proposed algorithm has the same separation accuracy as the basic coupled TALS but with a faster convergence speed.

Paint smears represent a type of automotive paint sample found at a crime scene that is problematic for forensic automotive paint examiners to analyze as there are no reference materials present in automotive paint databases to generate hit-lists of potential suspect vehicles. Realistic paint smears are difficult to create in a laboratory and have also proven challenging to analyze because of the mixing of the various automotive paint layers. A procedure based on an impact tester has been developed to create smears to simulate paint transfer between vehicles during a collision. Data collected from 24 original equipment manufacturer (OEM) paints in simulated collisions using an impact tester with a steel (inert) substrate to simulate vehicle to vehicle collisions shows that attenuated total reflection infrared microscopy can isolate individual paint layers. For each OEM paint sample, the corresponding smear obtained was dependent upon the conditions used. By varying these conditions, the number of distinct layers obtained could be tuned for each of the OEM paints investigated. Furthermore, the IR spectrum of each layer extracted from the paint smear using alternating least squares was found to compare favorably to an in-house OEM paint infrared spectral library for each layer as the correct match (make and model of the vehicle from which the smear originated) was always found as a top five hit in the hit-list. The results of this study indicate that paint smears developed using an impactor can serve as the basis of realistic proficiency tests for forensic laboratories. Graphical Abstract [ABSTRACT FROM AUTHOR]

Alternating least squares is a classic, easily implemented, yet widely used method for tensor canonical polyadic approximation. Its subsequential and global convergence is ensured if the partial Hessians of the blocks during the whole sequence are uniformly positive definite. This paper shows that this positive definiteness assumption can be weakened in two ways. Firstly, if the smallest positive eigenvalues of the partial Hessians are uniformly positive, and the solutions of the subproblems are properly chosen, then global convergence holds. This allows the partial Hessians to be only positive semidefinite. Next, if at a limit point, the partial Hessians are positive definite, then global convergence also holds. We also discuss the connection of such an assumption to the uniqueness of exact CP decomposition. [ABSTRACT FROM AUTHOR]

In this study, the problem of blind signal separation for coprime planar arrays is investigated. For coprime planar arrays comprising two uniform rectangular subarrays, we link the signal separation to the tensor‐based model called coupled canonical polyadic decomposition (CPD) and propose an improved coupled trilinear decomposition approach. The output data of coprime planar arrays are modeled as a coupled tensor set that can be further interpreted as a coupled CPD model, allowing a signal separation to be achieved using coupled trilinear alternating least squares (TALS). Furthermore, in the procedure of the coupled TALS, a Vandermonde structure enforcing approach is explicitly applied, which is shown to ensure fast convergence. The results of Monto Carlo simulations show that our proposed algorithm has the same separation accuracy as the basic coupled TALS but with a faster convergence speed. [ABSTRACT FROM AUTHOR]

Traditional preschool education methods adopt standardized teaching models to solve the problem of neglecting the personalized needs and learning differences of each child in traditional preschool education methods. This article proposed a new algorithm for personalized learning in preschool education that integrated ALS (alternating least squares) and MLP (multilayer perceptron) to improve preschool education and provide personalized education tailored to the needs of preschool children. Firstly, learning data related to preschool children was collected and preprocessed. Then, the children-textbook interaction matrix was constructed, and the alternating least squares method was used for collaborative filtering. Finally, child data, textbook data, and collaborative filtering results were input into a multilayer perceptron for personalized textbook recommendation. The training set data was used for sufficient model training. The experimental results showed that the average accuracy of the ALS-MLP model in textbook Top-4 recommendation reached 95.9%, and recommending personalized textbooks for children could improve their academic performance by an average of 15.6 points. The application of the ALS-MLP model can accurately recommend textbooks based on children’s learning characteristics, providing new methods for personalized learning in preschool education. [ABSTRACT FROM AUTHOR]

Extraction of components from individual refinery streams (e.g., reformates and alkylates) in finished gasoline was undertaken using Raman spectroscopy to characterize the chemical content of the finished product. Modified alternating least squares (MALS) was used for separating Raman spectroscopic data sets of the finished product into its pure individual components. The advantages of MALS over alternating least squares (ALS) for multicomponent resolution are highlighted in this study using three Raman spectroscopic data sets which provide a suitable benchmark for comparing the performance of these two methods. MALS is superior to ALS in terms of accuracy and can better resolve components than ALS, and it is also more robust toward collinear data. Finally, components near the noise level usually cannot be extracted by ALS because of instability when inverting the covariance structure which inflates the noise present in the data. However, these same components can be extracted by MALS due to the stabilization of the least squares regression with respect to the matrix inversion using modified techniques from ridge regression. [ABSTRACT FROM AUTHOR]

In recent years, Raman spectroscopy has been used to study biological tissues. However, the analysis of experimental Raman spectra is still challenging, since the Raman spectra of most biological tissue components overlap significantly and it is difficult to separate individual components. New methods of analysis are needed that would allow for the decomposition of Raman spectra into components and the evaluation of their contribution. The aim of our work is to study the possibilities of the multivariate curve resolution alternating least squares (MCR-ALS) method for the analysis of skin tissues in vivo. We investigated the Raman spectra of human skin recorded using a portable conventional Raman spectroscopy setup. The MCR-ALS analysis was performed for the Raman spectra of normal skin, keratosis, basal cell carcinoma, malignant melanoma, and pigmented nevus. We obtained spectral profiles corresponding to the contribution of the optical system and skin components: melanin, proteins, lipids, water, etc. The obtained results show that the multivariate curve resolution alternating least squares analysis can provide new information on the biochemical profiles of skin tissues. Such information may be used in medical diagnostics to analyze Raman spectra with a low signal-to-noise ratio, as well as in various fields of science and industry for preprocessing Raman spectra to remove parasitic components. [ABSTRACT FROM AUTHOR]

Tensor models have been used extensively in signal processing applications to design different types of communication systems. This paper proposes, for the first time, the use of tensor models for optical communications. The signals of an optical dual-core coupler network are modeled as a multiway array (tensor), which satisfies a generalized Tucker train decomposition. This tensor model is then used to develop an estimation algorithm that enables the network parameters to be estimated from the input and output signals. The performance of this algorithm was evaluated by means of computer simulations, in terms of NMSE of the estimated parameters and convergence speed. For the tested configurations, good levels of NMSE with fast convergence were obtained, demonstrating the effectiveness of the proposed method as a promising tool for studying and designing optical devices, with a wide range of applications in the context of lightwave systems.

The alternating least squares (ALS) algorithm for CP and Tucker decomposition is dominated in cost by the tensor contractions necessary to set up the quadratic optimization subproblems. We introduce a novel family of algorithms that uses perturbative corrections to the subproblems rather than recomputing the tensor contractions. This approximation is accurate when the factor matrices are changing little across iterations, which occurs when ALS approaches convergence. We provide a theoretical analysis to bound the approximation error. Our numerical experiments demonstrate that the proposed pairwise perturbation algorithms are easy to control and converge to minima that are as good as ALS. The experimental results show improvements of up to 3.1× with respect to state‐of‐the‐art ALS approaches for various model tensor problems and real datasets. [ABSTRACT FROM AUTHOR]

Indscal models consider symmetric matrices B k = X W k X ′ for k = 1 , ... , K , where X : n × R is a compromise matrix termed the group-average and W k is a diagonal matrix of weights given by the kth individual to the R, specified in advance, columns of X ; non-negative weights are preferred and usually R < n . We propose a new two-phase alternating least squares (ALS) algorithm, which emphasizes the two main components (group average and weighting parameters) of the Indscal model and specifically helps with the interpretation of the model. Furthermore, it has thrown new light on the properties of the converged solution, that would be satisfied by any algorithm that minimizes the basic Indscal criterion: m i n ∑ k = 1 K ‖ B k - X W k X ′ ‖ 2 where the minimization is over X and the W k . The new algorithm has also proved to be a useful tool in unravelling the algebraic understanding of the role played by parameter constraints and their interpretation in variants of the Indscal model. The proposed analysis focusses on Indscal but the approach may be of more widespread interest, especially in the field of multidimensional data analysis. A major issue is that simultaneous least-squares estimates of the parameters may be found without imposing constraints. However, group average and individual weighting parameters may not be estimated uniquely, without imposing some subjective constraint that could encourage misleading interpretations. We encourage the use of linear constraints ∑ k = 1 K 1 ′ W k = 1 ′ , as it enables a comparison of the weights obtained (i) within group k and (ii) between the same item drawn from two or more groups. However, it is easy to exchange one system of constraints to another in a post- or pre-analysis. The new two-phase ALS algorithm (i) computes for fixed X : n × R the weights W k subject to ∑ k = 1 K 1 ′ W k = 1 ′ , and then (ii) keeping W k fixed, it updates X . At convergence, the estimates of X : n × R and the W k will apply to all algorithms that minimize the Indscal criterion. Furthermore, we show that only at convergence an analysis-of-variance property holds on the demarcation region between over- and under-fitting. When the analysis-of-variance is valid, its validity extends over the whole matrix domain, over trace operations, and to individual matrix elements. The optimization process is unusual in that optima and local optima occur on the edges of what seem to be closely related to Heywood cases in Factor analysis. [ABSTRACT FROM AUTHOR]

Spectroscopic measurements can show distorted spectral shapes arising from a mixture of absorbing and scattering contributions. These distortions (or baselines) often manifest themselves as non-constant offsets or low-frequency oscillations. As a result, these baselines can adversely affect analytical and quantitative results. Baseline correction is an umbrella term where one applies pre-processing methods to obtain baseline spectra (the unwanted distortions) and then remove the distortions by differencing. However, current state-of-the art baseline correction methods do not utilize analyte concentrations even if they are available, or even if they contribute significantly to the observed spectral variability. We modify a class of state-of-the-art methods (penalized baseline correction) that easily admit the incorporation of a priori analyte concentrations such that predictions can be enhanced. This modified approach will be deemed supervised and penalized baseline correction (SPBC). Performance will be assessed on two near infrared data sets across both classical penalized baseline correction methods (without analyte information) and modified penalized baseline correction methods (leveraging analyte information). There are cases of SPBC that provide useful baseline-corrected signals such that they outperform state-of-the-art penalized baseline correction algorithms such as AIRPLS. In particular, we observe that performance is conditional on the correlation between separate analytes: the analyte used for baseline correlation and the analyte used for prediction—the greater the correlation between the analyte used for baseline correlation and the analyte used for prediction, the better the prediction performance. • Exploit references (i.e., analyte concentrations) in baseline correction are novel. • Penalized baseline correction methods can easily accommodate reference measurements. • Analyte concentrations used for baseline correction and analyte concentrations used for prediction can be different. • Increased correlation between various analytes improves performance. [ABSTRACT FROM AUTHOR]

Few-shot learning poses a critical challenge due to the deviation problem caused by the scarcity of available samples. In this work, we aim to address deviations in both feature representations and prototypes. To achieve this, we propose a cross-modal de-deviation framework that leverages class semantic information to provide robust prior knowledge for the samples. This framework begins with a visual-to-semantic autoencoder trained on the labeled samples to predict semantic features for the unlabeled samples. Then, we devise a binary linear programming model to match the initial prototypes with the cluster centers of the unlabeled samples. To circumvent potential mismatches between the cluster centers and the initial prototypes, we perform the label assignment process in the semantic space by transforming the cluster centers into semantic representations and utilizing the class ground truth semantic features as reference points. Moreover, we model a linear classifier with the concatenation of the refined prototypes and the class ground truth semantic features serving as the initial weights. Then we propose a novel optimization strategy based on the alternating least squares (ALS) model. From the ALS model, we can derive two closed-form solutions regarding to the features and weights, facilitating alternative optimization of them. Extensive experiments conducted on few-shot learning benchmarks demonstrate the competitive advantages of our CMDD method over the state-of-the-art approaches, confirming its effectiveness in reducing deviation. The code is available at: https://github.com/pmhDL/CMDD.git. • Our CMDD method reduces prototype deviation through cross-modal label assignment, mitigating the risk of collapsing multiple clusters into one class and minimizing the impact of limited labeled samples on the refined prototypes. • An alternative optimization strategy based on the alternating least squares model is explored to optimize the features and classifier's weights, effectively promoting mutual enhancement between them. • Our CMDD method competes well with state-of-the-art approaches, demonstrated through comprehensive experiments and ablation studies on four few-shot benchmarks. [ABSTRACT FROM AUTHOR]

With the emergence of e-commerce and social networking systems, the use of recommendation systems gained popularity to predict the user ratings of an item. Since the large volume of data is generated from various sources at high speed, predicting the ratings accurately in real-time adds enormous benefit to the users while choosing the correct item. So a recommendation system must be capable enough to predict the rating accurately when the data are large. Apache Spark is a distributed framework well suited for processing large datasets and real-time data streams. In this paper, we propose an efficient matrix factorisation algorithm based on Spark MLlib alternating least squares (ALS) for collaborative filtering. The optimisations used for the proposed algorithm using Tungsten improved the performance of the algorithm significantly while doing the predictions. The experimental results prove that the proposed work is significantly faster for top-N recommendations and rating predictions compared with the existing works. [ABSTRACT FROM AUTHOR]

This paper describes an effcient reverse-mode differentiation algorithm for contraction operations for arbitrary and unconventional tensor network topologies. The approach leverages the tensor contraction tree of Evenbly and Pfeifer (2014), which provides an instruction set for the contraction sequence of a network. We show that this tree can be effciently leveraged for differentiation of a full tensor network contraction using a recursive scheme that exploits (1) the bilinear property of contraction and (2) the property that trees have a single path from root to leaves. While differentiation of tensor-tensor contraction is already possible in most automatic differentiation packages, we show that exploiting these two additional properties in the specific context of contraction sequences can improve effciency. Following a description of the algorithm and computational complexity analysis, we investigate its utility for gradient-based supervised learning for low-rank function recovery and for fitting real-world unstructured datasets. We demonstrate improved performance over alternating least-squares optimization approaches and the capability to handle heterogeneous and arbitrary tensor network formats. When compared to alternating minimization algorithms, we find that the gradient-based approach requires a smaller oversampling ratio (number of samples compared to number model parameters) for recovery. This increased effciency extends to fitting unstructured data of varying dimensionality and when employing a variety of tensor network formats. Here, we show improved learning using the hierarchical Tucker method over the tensor-train in high-dimensional settings on a number of benchmark problems. [ABSTRACT FROM AUTHOR]

Alternate least squares (ALS) reconstructions of the infrared (IR) spectra of the individual layers from original automotive paint were analyzed using machine learning methods to improve both the accuracy and speed of a forensic automotive paint examination. Twenty-six original equipment manufacturer (OEM) paints from vehicles sold in North America between 2000 and 2006 served as a test bed to validate the ALS procedure developed in a previous study for the spectral reconstruction of each layer from IR line maps of cross-sectioned OEM paint samples. An examination of the IR spectra from an in-house library (collected with a high-pressure transmission diamond cell) and the ALS reconstructed IR spectra of the same paint samples (obtained at ambient pressure using an IR transmission microscope equipped with a BaF2 cell) showed large peak shifts (approximately 10 cm−1) with some vibrational modes in many samples comprising the cohort. These peak shifts are attributed to differences in the residual polarization of the IR beam of the transmission IR microscope and the IR spectrometer used to collect the in-house IR spectral library. To solve the problem of frequency shifts encountered with some vibrational modes, IR spectra from the in-house spectral library and the IR microscope were transformed using a correction algorithm previously developed by our laboratory to simulate ATR spectra collected on an iS-50 FT-IR spectrometer. Applying this correction algorithm to both the ALS reconstructed spectra and in-house IR library spectra, the large peak shifts previously encountered with some vibrational modes were successfully mitigated. Using machine learning methods to identify the manufacturer and the assembly plant of the vehicle from which the OEM paint sample originated, each of the twenty-six cross-sectioned automotive paint samples was correctly classified as to the "make" and model of the vehicle and was also matched to the correct paint sample in the in-house IR spectral library. Graphical Abstract [ABSTRACT FROM AUTHOR]

The objective of this paper is to investigate a new numerical method for the approximation of the self-diffusion matrix of a tagged particle process defined on a grid. While standard numerical methods make use of long-time averages of empirical means of deviations of some stochastic processes, and are thus subject to statistical noise, we propose here a tensor method in order to compute an approximation of the solution of a high-dimensional quadratic optimization problem, which enables to obtain a numerical approximation of the self-diffusion matrix. The tensor method we use here relies on an iterative scheme which builds low-rank approximations of the quantity of interest and on a carefully tuned variance reduction method so as to evaluate the various terms arising in the functional to minimize. In particular, we numerically observe here that it is much less subject to statistical noise than classical approaches. • Novel approach for computing self-diffusion matrices of tagged particle processes. • Solution of a high-dimensional minimization problem using low-rank techniques. • Less statistical noise in the approximation compared to classical approaches. [ABSTRACT FROM AUTHOR]

We propose a novel approach for computing committor functions, which describe transitions of a stochastic process between metastable states. The committor function satisfies a backward Kolmogorov equation, and in typical high-dimensional settings of interest, it is intractable to compute and store the solution with traditional numerical methods. By parametrizing the committor function in a matrix product state/tensor train format and using a similar representation for the equilibrium probability density, we solve the variational formulation of the backward Kolmogorov equation with linear time and memory complexity in the number of dimensions. This approach bypasses the need for sampling the equilibrium distribution, which can be difficult when the distribution has multiple modes. Numerical results demonstrate the effectiveness of the proposed method for high-dimensional problems. • Parametrizes high-dimensional committor functions using tensor trains. • Achieves linear time and memory complexity in the number of dimensions. • Takes care of the boundary conditions within a tensor formulation. • Bypasses the need for sampling from the equilibrium distribution. • Produces transition paths during the optimization of the committor function. [ABSTRACT FROM AUTHOR]