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This paper introduces a novel family of generalized exponentiated gradient (EG) updates derived from an Alpha-Beta divergence regularization function. Collectively referred to as EGAB, the proposed updates belong to the category of multiplicative gradient algorithms for positive data and demonstrate considerable flexibility by controlling iteration behavior and performance through three hyperparameters: $\alpha$, $\beta$, and the learning rate $\eta$. To enforce a unit $l_1$ norm constraint for nonnegative weight vectors within generalized EGAB algorithms, we develop two slightly distinct approaches. One method exploits scale-invariant loss functions, while the other relies on gradient projections onto the feasible domain. As an illustration of their applicability, we evaluate the proposed updates in addressing the online portfolio selection problem (OLPS) using gradient-based methods. Here, they not only offer a unified perspective on the search directions of various OLPS algorithms (including the standard exponentiated gradient and diverse mean-reversion strategies), but also facilitate smooth interpolation and extension of these updates due to the flexibility in hyperparameter selection. Simulation results confirm that the adaptability of these generalized gradient updates can effectively enhance the performance for some portfolios, particularly in scenarios involving transaction costs.

Tensor decomposition of convolutional and fully-connected layers is an effective way to reduce parameters and FLOP in neural networks. Due to memory and power consumption limitations of mobile or embedded devices, the quantization step is usually necessary when pre-trained models are deployed. A conventional post-training quantization approach applied to networks with decomposed weights yields a drop in accuracy. This motivated us to develop an algorithm that finds tensor approximation directly with quantized factors and thus benefit from both compression techniques while keeping the prediction quality of the model. Namely, we propose to use Alternating Direction Method of Multipliers (ADMM) for Canonical Polyadic (CP) decomposition with factors whose elements lie on a specified quantization grid. We compress neural network weights with a devised algorithm and evaluate it's prediction quality and performance. We compare our approach to state-of-the-art post-training quantization methods and demonstrate competitive results and high flexibility in achiving a desirable quality-performance tradeoff.

In this paper, the problem of image super-resolution for Optical Coherence Tomography (OCT) has been addressed. Due to the motion artifacts, OCT imaging is usually done with a low sampling rate and the resulting images are often noisy and have low resolution. Therefore, reconstruction of high resolution OCT images from the low resolution versions is an essential step for better OCT based diagnosis. In this paper, we propose a novel OCT super-resolution technique using Tensor Ring decomposition in the embedded space. A new tensorization method based on a block Hankelization approach with overlapped patches, called overlapped patch Hankelization, has been proposed which allows us to employ Tensor Ring decomposition. The Hankelization method enables us to better exploit the inter connection of pixels and consequently achieve better super-resolution of images. The low resolution image was first patch Hankelized and then its Tensor Ring decomposition with rank incremental has been computed. Simulation results confirm that the proposed approach is effective in OCT super-resolution.

Pedestrian Attribute Recognition (PAR) deals with the problem of identifying features in a pedestrian image. It has found interesting applications in person retrieval, suspect re-identification and soft biometrics. In the past few years, several Deep Neural Networks (DNNs) have been designed to solve the task; however, the developed DNNs predominantly suffer from over-parameterization and high computational complexity. These problems hinder them from being exploited in resource-constrained embedded devices with limited memory and computational capacity. By reducing a network's layers using effective compression techniques, such as tensor decomposition, neural network compression is an effective method to tackle these problems. We propose novel Lightweight Attribute Localizing Models (LWALM) for Pedestrian Attribute Recognition (PAR). LWALM is a compressed neural network obtained after effective layer-wise compression of the Attribute Localization Model (ALM) using the Canonical Polyadic Decomposition with Error Preserving Correction (CPD-EPC) algorithm.

Semi-supervised anomaly detection (SSAD) methods have demonstrated their effectiveness in enhancing unsupervised anomaly detection (UAD) by leveraging few-shot but instructive abnormal instances. However, the dominance of homogeneous normal data over anomalies biases the SSAD models against effectively perceiving anomalies. To address this issue and achieve balanced supervision between heavily imbalanced normal and abnormal data, we develop a novel framework called AnoOnly (Anomaly Only). Unlike existing SSAD methods that resort to strict loss supervision, AnoOnly suspends it and introduces a form of weak supervision for normal data. This weak supervision is instantiated through the utilization of batch normalization, which implicitly performs cluster learning on normal data. When integrated into existing SSAD methods, the proposed AnoOnly demonstrates remarkable performance enhancements across various models and datasets, achieving new state-of-the-art performance. Additionally, our AnoOnly is natively robust to label noise when suffering from data contamination. Our code is publicly available at https://github.com/cool-xuan/AnoOnly.

This paper presents a pixel selection method for compact image representation based on superpixel segmentation and tensor completion. Our method divides the image into several regions that capture important textures or semantics and selects a representative pixel from each region to store. We experiment with different criteria for choosing the representative pixel and find that the centroid pixel performs the best. We also propose two smooth tensor completion algorithms that can effectively reconstruct different types of images from the selected pixels. Our experiments show that our superpixel-based method achieves better results than uniform sampling for various missing ratios.

In this paper, we propose a new adaptive cross algorithm for computing a low tubal rank approximation of third-order tensors, with less memory and lower computational complexity than the truncated tensor SVD (t-SVD). This makes it applicable for decomposing large-scale tensors. We conduct numerical experiments on synthetic and real-world datasets to confirm the efficiency and feasibility of the proposed algorithm. The simulation results show more than one order of magnitude acceleration in the computation of low tubal rank (t-SVD) for large-scale tensors. An application to pedestrian attribute recognition is also presented.

In this paper, we extend the Discrete Empirical Interpolation Method (DEIM) to the third-order tensor case based on the t-product and use it to select important/ significant lateral and horizontal slices/features. The proposed Tubal DEIM (TDEIM) is investigated both theoretically and numerically. The experimental results show that the TDEIM can provide more accurate approximations than the existing methods. An application of the proposed method to the supervised classification task is also presented.

Tensor networks (TNs) and neural networks (NNs) are two fundamental data modeling approaches. TNs were introduced to solve the curse of dimensionality in large-scale tensors by converting an exponential number of dimensions to polynomial complexity. As a result, they have attracted significant attention in the fields of quantum physics and machine learning. Meanwhile, NNs have displayed exceptional performance in various applications, e.g., computer vision, natural language processing, and robotics research. Interestingly, although these two types of networks originate from different observations, they are inherently linked through the common multilinearity structure underlying both TNs and NNs, thereby motivating a significant number of intellectual developments regarding combinations of TNs and NNs. In this paper, we refer to these combinations as tensorial neural networks (TNNs), and present an introduction to TNNs in three primary aspects: network compression, information fusion, and quantum circuit simulation. Furthermore, this survey also explores methods for improving TNNs, examines flexible toolboxes for implementing TNNs, and documents TNN development while highlighting potential future directions. To the best of our knowledge, this is the first comprehensive survey that bridges the connections among NNs, TNs, and quantum circuits. We provide a curated list of TNNs at \url{https://github.com/tnbar/awesome-tensorial-neural-networks}.

Efficient and fast computation of a tensor singular value decomposition (t-SVD) with a few passes over the underlying data tensor is crucial because of its many potential applications. The current/existing subspace randomized algorithms need (2q+2) passes over the data tensor to compute a t-SVD, where q is a non-negative integer number (power iteration parameter). In this paper, we propose an efficient and flexible randomized algorithm that can handle any number of passes q, which not necessary need be even. The flexibility of the proposed algorithm in using fewer passes naturally leads to lower computational and communication costs. This advantage makes it particularly appropriate when our task calls for several tensor decompositions or when the data tensors are huge. The proposed algorithm is a generalization of the methods developed for matrices to tensors. The expected/ average error bound of the proposed algorithm is derived. Extensive numerical experiments on random and real-world data sets are conducted, and the proposed algorithm is compared with some baseline algorithms. The extensive computer simulation experiments demonstrate that the proposed algorithm is practical, efficient, and in general outperforms the state of the arts algorithms. We also demonstrate how to use the proposed method to develop a fast algorithm for the tensor completion problem.

This paper proposes a general framework to use the cross tensor approximation or tensor ColUmn-Row (CUR) approximation for reconstructing incomplete images and videos. The key importance of the new algorithms is their simplicity and ease of implementation with low computational complexity. For the case of data tensors with 1) structural missing components or 2) a high missing rate, we propose an efficient smooth tensor CUR algorithms which first make the sampled fibers smooth and then apply the proposed CUR algorithms. The numerical experiments show the significant benefit of this smoothing procedure. The main contribution of this paper is to develop/investigate improved multistage CUR algorithms with filtering (smoothing ) preprocessing for tensor completion. The second contribution is a detailed comparison of the performance of image recovery for four different CUR strategies via extensive computer simulations. Our simulations clearly indicated that the proposed algorithms are much faster than most of the existing state-of-the-art algorithms developed for tensor completion, while performance is comparable and often even better. Furthermore, we will provide in GitHub the MATLAB codes which can be used for various applications. Moreover, to our best knowledge, the CUR (cross approximation) algorithms have not been investigated nor compared till now for image and video completion.

We present a novel procedure for optimization based on the combination of efficient quantized tensor train representation and a generalized maximum matrix volume principle. We demonstrate the applicability of the new Tensor Train Optimizer (TTOpt) method for various tasks, ranging from minimization of multidimensional functions to reinforcement learning. Our algorithm compares favorably to popular evolutionary-based methods and outperforms them by the number of function evaluations or execution time, often by a significant margin., Comment: 26 pages, 8 figures, accepted to Thirty-sixth Conference on Neural Information Processing Systems (NeurIPS 2022). Pre camera-ready version

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

A large number of autonomous driving tasks need high-definition stereo images, which requires a large amount of storage space. Efficiently executing lossless compression has become a practical problem. Commonly, it is hard to make accurate probability estimates for each pixel. To tackle this, we propose L3C-Stereo, a multi-scale lossless compression model consisting of two main modules: the warping module and the probability estimation module. The warping module takes advantage of two view feature maps from the same domain to generate a disparity map, which is used to reconstruct the right view so as to improve the confidence of the probability estimate of the right view. The probability estimation module provides pixel-wise logistic mixture distributions for adaptive arithmetic coding. In the experiments, our method outperforms the hand-crafted compression methods and the learning-based method on all three datasets used. Then, we show that a better maximum disparity can lead to a better compression effect. Furthermore, thanks to a compression property of our model, it naturally generates a disparity map of an acceptable quality for the subsequent stereo tasks.

A conventional approach to train neural ordinary differential equations (ODEs) is to fix an ODE solver and then learn the neural network's weights to optimize a target loss function. However, such an approach is tailored for a specific discretization method and its properties, which may not be optimal for the selected application and yield the overfitting to the given solver. In our paper, we investigate how the variability in solvers' space can improve neural ODEs performance. We consider a family of Runge-Kutta methods that are parameterized by no more than two scalar variables. Based on the solvers' properties, we propose an approach to decrease neural ODEs overfitting to the pre-defined solver, along with a criterion to evaluate such behaviour. Moreover, we show that the right choice of solver parameterization can significantly affect neural ODEs models in terms of robustness to adversarial attacks. Recently it was shown that neural ODEs demonstrate superiority over conventional CNNs in terms of robustness. Our work demonstrates that the model robustness can be further improved by optimizing solver choice for a given task. The source code to reproduce our experiments is available at https://github.com/juliagusak/neural-ode-metasolver.

Accurate electroencephalogram (EEG) pattern decoding for specific mental tasks is one of the key steps for the development of brain-computer interface (BCI), which is quite challenging due to the considerably low signal-to-noise ratio of EEG collected at the brain scalp. Machine learning provides a promising technique to optimize EEG patterns toward better decoding accuracy. However, existing algorithms do not effectively explore the underlying data structure capturing the true EEG sample distribution, and hence can only yield a suboptimal decoding accuracy. To uncover the intrinsic distribution structure of EEG data, we propose a clustering-based multi-task feature learning algorithm for improved EEG pattern decoding. Specifically, we perform affinity propagation-based clustering to explore the subclasses (i.e., clusters) in each of the original classes, and then assign each subclass a unique label based on a one-versus-all encoding strategy. With the encoded label matrix, we devise a novel multi-task learning algorithm by exploiting the subclass relationship to jointly optimize the EEG pattern features from the uncovered subclasses. We then train a linear support vector machine with the optimized features for EEG pattern decoding. Extensive experimental studies are conducted on three EEG datasets to validate the effectiveness of our algorithm in comparison with other state-of-the-art approaches. The improved experimental results demonstrate the outstanding superiority of our algorithm, suggesting its prominent performance for EEG pattern decoding in BCI applications.

Deep learning has achieved excellent performance in a wide range of domains, especially in speech recognition and computer vision. Relatively less work has been done for EEG, but there is still significant progress attained in the last decade. Due to the lack of a comprehensive and topic widely covered survey for deep learning in EEG, we attempt to summarize recent progress to provide an overview, as well as perspectives for future developments. We first briefly mention the artifacts removal for EEG signal and then introduce deep learning models that have been utilized in EEG processing and classification. Subsequently, the applications of deep learning in EEG are reviewed by categorizing them into groups such as brain-computer interface, disease detection, and emotion recognition. They are followed by the discussion, in which the pros and cons of deep learning are presented and future directions and challenges for deep learning in EEG are proposed. We hope that this paper could serve as a summary of past work for deep learning in EEG and the beginning of further developments and achievements of EEG studies based on deep learning., Comment: Accepted for publication in the IEEE Transactions on Cognitive and Developmental Systems

Most state of the art deep neural networks are overparameterized and exhibit a high computational cost. A straightforward approach to this problem is to replace convolutional kernels with its low-rank tensor approximations, whereas the Canonical Polyadic tensor Decomposition is one of the most suited models. However, fitting the convolutional tensors by numerical optimization algorithms often encounters diverging components, i.e., extremely large rank-one tensors but canceling each other. Such degeneracy often causes the non-interpretable result and numerical instability for the neural network fine-tuning. This paper is the first study on degeneracy in the tensor decomposition of convolutional kernels. We present a novel method, which can stabilize the low-rank approximation of convolutional kernels and ensure efficient compression while preserving the high-quality performance of the neural networks. We evaluate our approach on popular CNN architectures for image classification and show that our method results in much lower accuracy degradation and provides consistent performance., Comment: This paper is accepted to ECCV2020

This article discusses some trends and concepts in developing new generation of future Artificial General Intelligence (AGI) systems which relate to complex facets and different types of human intelligence, especially social, emotional, attentional and ethical intelligence. We describe various aspects of multiple human intelligences and learning styles, which may impact on a variety of AI problem domains. Using the concept of 'multiple intelligences' rather than a single type of intelligence, we categorize and provide working definitions of various AGI depending on their cognitive skills or capacities. Future AI systems will be able not only to communicate with human users and each other, but also to efficiently exchange knowledge and wisdom with abilities of cooperation, collaboration and even co-creating something new and valuable and have meta-learning capacities. Multi-agent systems such as these can be used to solve problems that would be difficult to solve by any individual intelligent agent. Key words: Artificial General Intelligence (AGI), multiple intelligences, learning styles, physical intelligence, emotional intelligence, social intelligence, attentional intelligence, moral-ethical intelligence, responsible decision making, creative-innovative intelligence, cognitive functions, meta-learning of AI systems., Comment: 19 Figures, 27 pages

Normalization is an important and vastly investigated technique in deep learning. However, its role for Ordinary Differential Equation based networks (neural ODEs) is still poorly understood. This paper investigates how different normalization techniques affect the performance of neural ODEs. Particularly, we show that it is possible to achieve 93% accuracy in the CIFAR-10 classification task, and to the best of our knowledge, this is the highest reported accuracy among neural ODEs tested on this problem.

We propose a simple interpolation-based method for the efficient approximation of gradients in neural ODE models. We compare it with the reverse dynamic method (known in the literature as "adjoint method") to train neural ODEs on classification, density estimation, and inference approximation tasks. We also propose a theoretical justification of our approach using logarithmic norm formalism. As a result, our method allows faster model training than the reverse dynamic method that was confirmed and validated by extensive numerical experiments for several standard benchmarks.

The development of reinforced learning methods has extended application to many areas including algorithmic trading. In this paper trading on the stock exchange is interpreted into a game with a Markov property consisting of states, actions, and rewards. A system for trading the fixed volume of a financial instrument is proposed and experimentally tested; this is based on the asynchronous advantage actor-critic method with the use of several neural network architectures. The application of recurrent layers in this approach is investigated. The experiments were performed on real anonymized data. The best architecture demonstrated a trading strategy for the RTS Index futures (MOEX:RTSI) with a profitability of 66% per annum accounting for commission. The project source code is available via the following link: http://github.com/evgps/a3c_trading.

This work proposes a novel approach for multiple time series forecasting. At first, multi-way delay embedding transform (MDT) is employed to represent time series as low-rank block Hankel tensors (BHT). Then, the higher-order tensors are projected to compressed core tensors by applying Tucker decomposition. At the same time, the generalized tensor Autoregressive Integrated Moving Average (ARIMA) is explicitly used on consecutive core tensors to predict future samples. In this manner, the proposed approach tactically incorporates the unique advantages of MDT tensorization (to exploit mutual correlations) and tensor ARIMA coupled with low-rank Tucker decomposition into a unified framework. This framework exploits the low-rank structure of block Hankel tensors in the embedded space and captures the intrinsic correlations among multiple TS, which thus can improve the forecasting results, especially for multiple short time series. Experiments conducted on three public datasets and two industrial datasets verify that the proposed BHT-ARIMA effectively improves forecasting accuracy and reduces computational cost compared with the state-of-the-art methods., Comment: Accepted by AAAI 2020

Big data analysis has become a crucial part of new emerging technologies such as the internet of things, cyber-physical analysis, deep learning, anomaly detection, etc. Among many other techniques, dimensionality reduction plays a key role in such analyses and facilitates feature selection and feature extraction. Randomized algorithms are efficient tools for handling big data tensors. They accelerate decomposing large-scale data tensors by reducing the computational complexity of deterministic algorithms and the communication among different levels of the memory hierarchy, which is the main bottleneck in modern computing environments and architectures. In this paper, we review recent advances in randomization for the computation of Tucker decomposition and Higher Order SVD (HOSVD). We discuss random projection and sampling approaches, single-pass, and multi-pass randomized algorithms, and how to utilize them in the computation of the Tucker decomposition and the HOSVD. Simulations on synthetic and real datasets are provided to compare the performance of some of the best and most promising algorithms.

In this article two new algorithms are presented that convert a given data tensor train into either a Tucker decomposition with orthogonal matrix factors or a multi-scale entanglement renormalization ansatz (MERA). The Tucker core tensor is never explicitly computed but stored as a tensor train instead, resulting in both computationally and storage efficient algorithms. Both the multilinear Tucker-ranks as well as the MERA-ranks are automatically determined by the algorithm for a given upper bound on the relative approximation error. In addition, an iterative algorithm with low computational complexity based on solving an orthogonal Procrustes problem is proposed for the first time to retrieve optimal rank-lowering disentangler tensors, which are a crucial component in the construction of a low-rank MERA. Numerical experiments demonstrate the effectiveness of the proposed algorithms together with the potential storage benefit of a low-rank MERA over a tensor train.

We introduce a new method for speeding up the inference of deep neural networks. It is somewhat inspired by the reduced-order modeling techniques for dynamical systems.The cornerstone of the proposed method is the maximum volume algorithm. We demonstrate efficiency on neural networks pre-trained on different datasets. We show that in many practical cases it is possible to replace convolutional layers with much smaller fully-connected layers with a relatively small drop in accuracy.

Deep image prior (DIP), which utilizes a deep convolutional network (ConvNet) structure itself as an image prior, has attracted attentions in computer vision and machine learning communities. It empirically shows the effectiveness of ConvNet structure for various image restoration applications. However, why the DIP works so well is still unknown, and why convolution operation is useful for image reconstruction or enhancement is not very clear. In this study, we tackle these questions. The proposed approach is dividing the convolution into ``delay-embedding'' and ``transformation (\ie encoder-decoder)'', and proposing a simple, but essential, image/tensor modeling method which is closely related to dynamical systems and self-similarity. The proposed method named as manifold modeling in embedded space (MMES) is implemented by using a novel denoising-auto-encoder in combination with multi-way delay-embedding transform. In spite of its simplicity, the image/tensor completion, super-resolution, deconvolution, and denoising results of MMES are quite similar even competitive to DIP in our extensive experiments, and these results would help us for reinterpreting/characterizing the DIP from a perspective of ``low-dimensional patch-manifold prior''.

Objective: Schizophrenia seriously affects the quality of life. To date, both simple (linear discriminant analysis) and complex (deep neural network) machine learning methods have been utilized to identify schizophrenia based on functional connectivity features. The existing simple methods need two separate steps (i.e., feature extraction and classification) to achieve the identification, which disables simultaneous tuning for the best feature extraction and classifier training. The complex methods integrate two steps and can be simultaneously tuned to achieve optimal performance, but these methods require a much larger amount of data for model training. Methods: To overcome the aforementioned drawbacks, we proposed a multi-kernel capsule network (MKCapsnet), which was developed by considering the brain anatomical structure. Kernels were set to match with partition sizes of brain anatomical structure in order to capture interregional connectivities at the varying scales. With the inspiration of widely-used dropout strategy in deep learning, we developed vector dropout in the capsule layer to prevent overfitting of the model. Results: The comparison results showed that the proposed method outperformed the state-of-the-art methods. Besides, we compared performances using different parameters and illustrated the routing process to reveal characteristics of the proposed method. Conclusion: MKCapsnet is promising for schizophrenia identification. Significance: Our study not only proposed a multi-kernel capsule network but also provided useful information in the parameter setting, which is informative for further studies using a capsule network for neurophysiological signal classification., Comment: IEEE Transactions on Cybernetics (2020)

The low-rank tensor approximation is very promising for the compression of deep neural networks. We propose a new simple and efficient iterative approach, which alternates low-rank factorization with a smart rank selection and fine-tuning. We demonstrate the efficiency of our method comparing to non-iterative ones. Our approach improves the compression rate while maintaining the accuracy for a variety of tasks.

The Canonical Polyadic decomposition (CPD) is a convenient and intuitive tool for tensor factorization; however, for higher-order tensors, it often exhibits high computational cost and permutation of tensor entries, these undesirable effects grow exponentially with the tensor order. Prior compression of tensor in-hand can reduce the computational cost of CPD, but this is only applicable when the rank $R$ of the decomposition does not exceed the tensor dimensions. To resolve these issues, we present a novel method for CPD of higher-order tensors, which rests upon a simple tensor network of representative inter-connected core tensors of orders not higher than 3. For rigour, we develop an exact conversion scheme from the core tensors to the factor matrices in CPD, and an iterative algorithm with low complexity to estimate these factor matrices for the inexact case. Comprehensive simulations over a variety of scenarios support the approach., Comment: 13 pages

Nonsmooth Nonnegative Matrix Factorization (nsNMF) is capable of producing more localized, less overlapped feature representations than other variants of NMF while keeping satisfactory fit to data. However, nsNMF as well as other existing NMF methods is incompetent to learn hierarchical features of complex data due to its shallow structure. To fill this gap, we propose a deep nsNMF method coined by the fact that it possesses a deeper architecture compared with standard nsNMF. The deep nsNMF not only gives parts-based features due to the nonnegativity constraints, but also creates higher-level, more abstract features by combing lower-level ones. The in-depth description of how deep architecture can help to efficiently discover abstract features in dnsNMF is presented. And we also show that the deep nsNMF has close relationship with the deep autoencoder, suggesting that the proposed model inherits the major advantages from both deep learning and NMF. Extensive experiments demonstrate the standout performance of the proposed method in clustering analysis.

A novel algorithm to solve the quadratic programming problem over ellipsoids is proposed. This is achieved by splitting the problem into two optimisation sub-problems, quadratic programming over a sphere and orthogonal projection. Next, an augmented-Lagrangian algorithm is developed for this multiple constraint optimisation. Benefit from the fact that the QP over a single sphere can be solved in a closed form by solving a secular equation, we derive a tighter bound of the minimiser of the secular equation. We also propose to generate a new psd matrix with a low condition number from the matrices in the quadratic constraints. This correction method improves convergence of the proposed augmented-Lagrangian algorithm. Finally, applications of the quadratically constrained QP to bounded linear regression and tensor decompositions are presented.

A novel algorithm is proposed for CANDECOMP/PARAFAC tensor decomposition to exploit best rank-1 tensor approximation. Different from the existing algorithms, our algorithm updates rank-1 tensors simultaneously in parallel. In order to achieve this, we develop new all-at-once algorithms for best rank-1 tensor approximation based on the Levenberg-Marquardt method and the rotational update. We show that the LM algorithm has the same complexity of first-order optimisation algorithms, while the rotational method leads to solving the best rank-1 approximation of tensors of size $2 \times 2 \times \cdots \times 2$. We derive a closed-form expression of the best rank-1 tensor of $2\times 2 \times 2$ tensors and present an ALS algorithm which updates 3 component at a time for higher order tensors. The proposed algorithm is illustrated in decomposition of difficult tensors which are associated with multiplication of two matrices., Comment: 33 pages

In CANDECOMP/PARAFAC tensor decomposition, degeneracy often occurs in some difficult scenarios, e.g., when the rank exceeds the tensor dimension, or when the loading components are highly collinear in several or all modes, or when CPD does not have an optimal solution. In such the cases, norms of some rank-1 terms become significantly large and cancel each other. This makes algorithms getting stuck in local minima while running a huge number of iterations does not improve the decomposition. In this paper, we propose an error preservation correction method to deal with such problem. Our aim is to seek a new tensor whose norms of rank-1 tensor components are minimised in an optimization problem, while it preserves the approximation error. An alternating correction algorithm and an all-atone algorithm have been developed for the problem. In addition, we propose a novel CPD with a bound constraint on a norm of the rank-one tensors. The method can be useful for decomposing tensors that cannot be analyzed by traditional algorithms, such as tensors corresponding to the matrix multiplication., Comment: 27 pages