Andrzej Cichocki

Andrzej Cichocki

In this paper we review basic and emerging models and associated algorithms for large-scale tensor networks, especially Tensor Train (TT) decompositions using novel mathematical and graphical representations. We discus the concept of tensorization (i.e., creating very high-order tensors from lower-order original data) and super compression of data achieved via quantized tensor train (QTT) networks. The purpose of a tensorization and quantization is to achieve, via low-rank tensor approximations "super" compression, and meaningful, compact representation of structured data. The main objective of this paper is to show how tensor networks can be used to solve a wide class of big data optimization problems (that are far from tractable by classical numerical methods) by applying tensorization and performing all operations using relatively small size matrices and tensors and applying iteratively optimized and approximative tensor contractions. Keywords: Tensor networks, tensor train (TT) decompositions, matrix product states (MPS), matrix product operators (MPO), basic tensor operations, tensorization, distributed representation od data optimization problems for very large-scale problems: generalized eigenvalue decomposition (GEVD), PCA/SVD, canonical correlation analysis (CCA).

Konstantin Sozykin, Andrei Chertkov, Roman Schutski et al.

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.

Namgil Lee, Andrzej Cichocki

We discuss extended definitions of linear and multilinear operations such as Kronecker, Hadamard, and contracted products, and establish links between them for tensor calculus. Then we introduce effective low-rank tensor approximation techniques including Candecomp/Parafac (CP), Tucker, and tensor train (TT) decompositions with a number of mathematical and graphical representations. We also provide a brief review of mathematical properties of the TT decomposition as a low-rank approximation technique. With the aim of breaking the curse-of-dimensionality in large-scale numerical analysis, we describe basic operations on large-scale vectors, matrices, and high-order tensors represented by TT decomposition. The proposed representations can be used for describing numerical methods based on TT decomposition for solving large-scale optimization problems such as systems of linear equations and symmetric eigenvalue problems.

Anh Huy Phan, Petr Tichavský, Andrzej Cichocki

Product between mode-$n$ unfolding $\bY_{(n)}$ of an $N$-D tensor $\tY$ and Khatri-Rao products of $(N-1)$ factor matrices $\bA^{(m)}$, $m = 1,..., n-1, n+1, ..., N$ exists in algorithms for CANDECOMP/PARAFAC (CP). If $\tY$ is an error tensor of a tensor approximation, this product is the gradient of a cost function with respect to factors, and has the largest workload in most CP algorithms. In this paper, a fast method to compute this product is proposed. Experimental verification shows that the fast CP gradient can accelerate the CP_ALS algorithm 2 times and 8 times faster for factorizations of 3-D and 4-D tensors, and the speed-up ratios can be 20-30 times for higher dimensional tensors.

Yixuan Zhou, Xing Xu, Zhe Sun et al.

Normalizing flows, a category of probabilistic models famed for their capabilities in modeling complex data distributions, have exhibited remarkable efficacy in unsupervised anomaly detection. This paper explores the potential of normalizing flows in multi-class anomaly detection, wherein the normal data is compounded with multiple classes without providing class labels. Through the integration of vector quantization (VQ), we empower the flow models to distinguish different concepts of multi-class normal data in an unsupervised manner, resulting in a novel flow-based unified method, named VQ-Flow. Specifically, our VQ-Flow leverages hierarchical vector quantization to estimate two relative codebooks: a Conceptual Prototype Codebook (CPC) for concept distinction and its concomitant Concept-Specific Pattern Codebook (CSPC) to capture concept-specific normal patterns. The flow models in VQ-Flow are conditioned on the concept-specific patterns captured in CSPC, capable of modeling specific normal patterns associated with different concepts. Moreover, CPC further enables our VQ-Flow for concept-aware distribution modeling, faithfully mimicking the intricate multi-class normal distribution through a mixed Gaussian distribution reparametrized on the conceptual prototypes. Through the introduction of vector quantization, the proposed VQ-Flow advances the state-of-the-art in multi-class anomaly detection within a unified training scheme, yielding the Det./Loc. AUROC of 99.5%/98.3% on MVTec AD. The codebase is publicly available at https://github.com/cool-xuan/vqflow.

Anh-Huy Phan, Andrzej Cichocki, Andre Uschmajew et al.

Decompositions of tensors into factor matrices, which interact through a core tensor, have found numerous applications in signal processing and machine learning. A more general tensor model which represents data as an ordered network of sub-tensors of order-2 or order-3 has, so far, not been widely considered in these fields, although this so-called tensor network decomposition has been long studied in quantum physics and scientific computing. In this study, we present novel algorithms and applications of tensor network decompositions, with a particular focus on the tensor train decomposition and its variants. The novel algorithms developed for the tensor train decomposition update, in an alternating way, one or several core tensors at each iteration, and exhibit enhanced mathematical tractability and scalability to exceedingly large-scale data tensors. The proposed algorithms are tested in classic paradigms of blind source separation from a single mixture, denoising, and feature extraction, and achieve superior performance over the widely used truncated algorithms for tensor train decomposition.

Daria Cherniuk, Stanislav Abukhovich, Anh-Huy Phan et al.

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.

Farnaz Sedighin, Andrzej Cichocki, Hossein Rabbani

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.

Ashish Jha, Dimitrii Ermilov, Konstantin Sobolev et al.

Pedestrian Attribute Recognition (PAR) focuses on identifying various attributes in pedestrian images, with key applications in person retrieval, suspect re-identification, and soft biometrics. However, Deep Neural Networks (DNNs) for PAR often suffer from over-parameterization and high computational complexity, making them unsuitable for resource-constrained devices. Traditional tensor-based compression methods typically factorize layers without adequately preserving the gradient direction during compression, leading to inefficient compression and a significant accuracy loss. In this work, we propose a novel approach for determining the optimal ranks of low-rank layers, ensuring that the gradient direction of the compressed model closely aligns with that of the original model. This means that the compressed model effectively preserves the update direction of the full model, enabling more efficient compression for PAR tasks. The proposed procedure optimizes the compression ranks for each layer within the ALM model, followed by compression using CPD-EPC or truncated SVD. This results in a reduction in model complexity while maintaining high performance.

Anh-Huy Phan, Konstantin Sobolev, Dmitry Ermilov et al.

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

Anh-Huy Phan, Petr Tichavský, Andrzej Cichocki

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.

Andrzej Cichocki, Piergiulio Tempesta

We introduce a comprehensive theoretical and algorithmic framework that bridges formal group theory and group entropies with modern machine learning, paving the way for an infinite, flexible family of Mirror Descent (MD) optimization algorithms. Our approach exploits the rich structure of group entropies, which are generalized entropic functionals governed by group composition laws, encompassing and significantly extending all trace-form entropies such as the Shannon, Tsallis, and Kaniadakis families. By leveraging group-theoretical mirror maps (or link functions) in MD, expressed via multi-parametric generalized logarithms and their inverses (group exponentials), we achieve highly flexible and adaptable MD updates that can be tailored to diverse data geometries and statistical distributions. To this end, we introduce the notion of \textit{mirror duality}, which allows us to seamlessly switch or interchange group-theoretical link functions with their inverses, subject to specific learning rate constraints. By tuning or learning the hyperparameters of the group logarithms enables us to adapt the model to the statistical properties of the training distribution, while simultaneously ensuring desirable convergence characteristics via fine-tuning. This generality not only provides greater flexibility and improved convergence properties, but also opens new perspectives for applications in machine learning and deep learning by expanding the design of regularizers and natural gradient algorithms. We extensively evaluate the validity, robustness, and performance of the proposed updates on large-scale, simplex-constrained quadratic programming problems.

Andrzej Cichocki, Michal Wietczak

We study Automatically Differentiable Nonlinear Tensor Networks (ADNTNs), a family of structured weight generators whose compact core tensors are trained end-to-end by reverse-mode automatic differentiation (AD). The approach can be viewed as a natural extension of low-rank adaptation and tensor factorisation: instead of using one low-rank matrix update, an ADNTN builds a large weight tensor through a hierarchy of small cores, nonlinear activations, and optional lateral mixing tensors. The paper focuses on three architectures: Tree Tensor Networks (TTNs), augmented TTNs (aTTNs) with boundary disentanglers, and Multi-scale Entanglement Renormalisation Ansatze (MERA). The formulation supports nonlinear activations, task-aware objectives, batching, and hardware-aware execution schedules. At the same time, the paper keeps a clear distinction between \emph{differentiating} a contraction program and making contraction free: AD does not remove the cost of large intermediates, poor contraction orders, or exact contraction of general loopy tensor networks. Extensive simulations on AlexNet and VGG-16 layers show per-layer compression ratios from roughly $2000\times$ to $77000\times$ in the studied settings, with accuracy often matching the dense baseline and, in several VGG-16 cases, improving it. These results are encouraging rather than final: they suggest that ADNTNs are a promising, mathematically structured, and hardware-aware route toward much smaller neural networks, provided that optimisation, contraction schedules, and deployment kernels are designed together.

Anh-Huy Phan, Petr Tichavský, Andrzej Cichocki

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.

Salman Ahmadi-Asl, Valentin Leplat, Anh-Huy Phan et al.

We introduce the tubal tensor train (TTT) decomposition, a tensor-network model that combines the t-product algebra of the tensor singular value decomposition (T-SVD) with the low-order core structure of the tensor train (TT) format. For an order-$(N+1)$ tensor with a distinguished tube mode, the proposed representation consists of two third-order boundary cores and $N-2$ fourth-order interior cores linked through the t-product. As a result, for bounded tubal ranks, the storage scales linearly with the number of modes, in contrast to direct high-order extensions of T-SVD. We present two computational strategies: a sequential fixed-rank construction, called TTT-SVD, and a Fourier-slice alternating scheme based on the alternating two-cores update (ATCU). We also state a TT-SVD-type error bound for TTT-SVD and illustrate the practical performance of the proposed model on image compression, video compression, tensor completion, and hyperspectral imaging.

Julia Gusak, Alexandr Katrutsa, Talgat Daulbaev et al.

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.

Evgeny Ponomarev, Ivan Oseledets, Andrzej Cichocki

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.

Andrzej Cichocki

IIn this paper we propose and investigate a new class of Generalized Exponentiated Gradient (GEG) algorithms using Mirror Descent (MD) updates, and applying the Bregman divergence with a two--parameter deformation of the logarithm as a link function. This link function (referred here to as the Euler logarithm) is associated with a relatively wide class of trace--form entropies. In order to derive novel GEG/MD updates, we estimate a deformed exponential function, which closely approximates the inverse of the Euler two--parameter deformed logarithm. The characteristic shape and properties of the Euler logarithm and its inverse--deformed exponential functions, are tuned by two hyperparameters. By learning these hyperparameters, we can adapt to the distribution of training data and adjust them to achieve desired properties of gradient descent algorithms. In the literature, there exist nowadays more than fifty mathematically well-established entropic functionals and associated deformed logarithms, so it is impossible to investigate all of them in one research paper. Therefore, we focus here on a class of trace-form entropies and the associated deformed two--parameters logarithms.

Andrzej Cichocki

In this paper, we develop a wide class Mirror Descent (MD) algorithms, which play a key role in machine learning. For this purpose we formulated the constrained optimization problem, in which we exploits the Bregman divergence with the Tempesta multi-parametric deformation logarithm as a link function. This link function called also mirror function defines the mapping between the primal and dual spaces and is associated with a very-wide (in fact, theoretically infinite) class of generalized trace-form entropies. In order to derive novel MD updates, we estimate generalized exponential function, which closely approximates the inverse of the multi-parametric Tempesta generalized logarithm. The shape and properties of the Tempesta logarithm and its inverse-deformed exponential functions can be tuned by several hyperparameters. By learning these hyperparameters, we can adapt to distribution or geometry of training data, and we can adjust them to achieve desired properties of MD algorithms. The concept of applying multi-parametric logarithms allow us to generate a new wide and flexible family of MD and mirror-less MD updates.

Weichen Dai, Yuxuan Huang, Li Zhu et al.

Humans possess a remarkable capacity for spatial cognition, allowing for self-localization even in novel or unfamiliar environments. While hippocampal neurons encoding position and orientation are well documented, the large-scale neural dynamics supporting spatial representation, particularly during naturalistic, passive experience, remain poorly understood. Here, we demonstrate for the first time that non-invasive brain-computer interfaces (BCIs) based on electroencephalography (EEG) can decode spontaneous, fine-grained egocentric 6D pose, comprising three-dimensional position and orientation, during passive viewing of egocentric video. Despite EEG's limited spatial resolution and high signal noise, we find that spatially coherent visual input (i.e., continuous and structured motion) reliably evokes decodable spatial representations, aligning with participants' subjective sense of spatial engagement. Decoding performance further improves when visual input is presented at a frame rate of 100 ms per image, suggesting alignment with intrinsic neural temporal dynamics. Using gradient-based backpropagation through a neural decoding model, we identify distinct EEG channels contributing to position -- and orientation specific -- components, revealing a distributed yet complementary neural encoding scheme. These findings indicate that the brain's spatial systems operate spontaneously and continuously, even under passive conditions, challenging traditional distinctions between active and passive spatial cognition. Our results offer a non-invasive window into the automatic construction of egocentric spatial maps and advance our understanding of how the human mind transforms everyday sensory experience into structured internal representations.

Andrzej Cichocki, Toshihisa Tanaka, Frank Nielsen et al.

This paper introduces a broad class of Mirror Descent (MD) and Generalized Exponentiated Gradient (GEG) algorithms derived from trace-form entropies defined via deformed logarithms. Leveraging these generalized entropies yields MD \& GEG algorithms with improved convergence behavior, robustness to vanishing and exploding gradients, and inherent adaptability to non-Euclidean geometries through mirror maps. We establish deep connections between these methods and Amari's natural gradient, revealing a unified geometric foundation for additive, multiplicative, and natural gradient updates. Focusing on the Tsallis, Kaniadakis, Sharma--Taneja--Mittal, and Kaniadakis--Lissia--Scarfone entropy families, we show that each entropy induces a distinct Riemannian metric on the parameter space, leading to GEG algorithms that preserve the natural statistical geometry. The tunable parameters of deformed logarithms enable adaptive geometric selection, providing enhanced robustness and convergence over classical Euclidean optimization. Overall, our framework unifies key first-order MD optimization methods under a single information-geometric perspective based on generalized Bregman divergences, where the choice of entropy determines the underlying metric and dual geometric structure.

Andrzej Cichocki, Sergio Cruces, Auxiliadora Sarmiento et al.

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: $α$, $β$, and the learning rate $η$. 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.

Maame G. Asante-Mensah, Anh Huy Phan, Salman Ahmadi-Asl et al.

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.

Zihao Huang, Zhe Sun, Feng Duan et al.

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.

M. Nazareth da Costa, R. Attux, A. Cichocki et al.

In this work, we firstly apply the Train-Tensor (TT) networks to construct a compact representation of the classical Multilayer Perceptron, representing a reduction of up to 95% of the coefficients. A comparative analysis between tensor model and standard multilayer neural networks is also carried out in the context of prediction of the Mackey-Glass noisy chaotic time series and NASDAQ index. We show that the weights of a multidimensional regression model can be learned by means of TT network and the optimization of TT weights is a more robust to the impact of coefficient initialization and hyper-parameter setting. Furthermore, an efficient algorithm based on alternating least squares has been proposed for approximating the weights in TT-format with a reduction of computational calculus, providing a much faster convergence than the well-known adaptive learning-method algorithms, widely applied for optimizing neural networks.

Yu Zhang, Tao Zhou, Wei Wu et al.

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.

Anh-Huy Phan, Konstantin Sobolev, Konstantin Sozykin et al.

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.

Andrzej Cichocki, Alexander P. Kuleshov

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.

Julia Gusak, Larisa Markeeva, Talgat Daulbaev et al.

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.

Talgat Daulbaev, Alexandr Katrutsa, Larisa Markeeva et al.

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.

Qiquan Shi, Jiaming Yin, Jiajun Cai et al.

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.

Julia Gusak, Talgat Daulbaev, Evgeny Ponomarev et al.

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.

Tatsuya Yokota, Hidekata Hontani, Qibin Zhao et al.

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''.

Tian Wang, Anastasios Bezerianos, Andrzej Cichocki et al.

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.

Julia Gusak, Maksym Kholiavchenko, Evgeny Ponomarev et al.

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.

Jordi Sole-Casals, Cesar F. Caiafa, Qibin Zhao et al.

One of the current issues in Brain-Computer Interface is how to deal with noisy Electroencephalography measurements organized as multidimensional datasets. On the other hand, recently, significant advances have been made in multidimensional signal completion algorithms that exploit tensor decomposition models to capture the intricate relationship among entries in a multidimensional signal. We propose to use tensor completion applied to EEG data for improving the classification performance in a motor imagery BCI system with corrupted measurements. Noisy measurements are considered as unknowns that are inferred from a tensor decomposition model. We evaluate the performance of four recently proposed tensor completion algorithms plus a simple interpolation strategy, first with random missing entries and then with missing samples constrained to have a specific structure (random missing channels), which is a more realistic assumption in BCI Applications. We measured the ability of these algorithms to reconstruct the tensor from observed data. Then, we tested the classification accuracy of imagined movement in a BCI experiment with missing samples. We show that for random missing entries, all tensor completion algorithms can recover missing samples increasing the classification performance compared to a simple interpolation approach. For the random missing channels case, we show that tensor completion algorithms help to reconstruct missing channels, significantly improving the accuracy in the classification of motor imagery, however, not at the same level as clean data. Tensor completion algorithms are useful in real BCI applications. The proposed strategy could allow using motor imagery BCI systems even when EEG data is highly affected by missing channels and/or samples, avoiding the need of new acquisitions in the calibration stage.

Anh-Huy Phan, Andrzej Cichocki, Ivan Oseledets et al.

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.

Jinshi Yu, Guoxu Zhou, Andrzej Cichocki et al.

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.

Qibin Zhao, Masashi Sugiyama, Andrzej Cichocki

Tensor train (TT) decomposition is a powerful representation for high-order tensors, which has been successfully applied to various machine learning tasks in recent years. However, since the tensor product is not commutative, permutation of data dimensions makes solutions and TT-ranks of TT decomposition inconsistent. To alleviate this problem, we propose a permutation symmetric network structure by employing circular multilinear products over a sequence of low-order core tensors. This network structure can be graphically interpreted as a cyclic interconnection of tensors, and thus we call it tensor ring (TR) representation. We develop several efficient algorithms to learn TR representation with adaptive TR-ranks by employing low-rank approximations. Furthermore, mathematical properties are investigated, which enables us to perform basic operations in a computationally efficiently way by using TR representations. Experimental results on synthetic signals and real-world datasets demonstrate that the proposed TR network is more expressive and consistently informative than existing TT networks.

Zhaoyang Qiu, Brendan Z. Allison, Jing Jin et al.

Motor imagery (MI) is a mental representation of motor behavior that has been widely used as a control method for a brain-computer interface (BCI), allowing communication for the physically impaired. The performance of MI based BCI mainly depends on the subject's ability to self-modulate EEG signals. Proper training can help naive subjects learn to modulate brain activity proficiently. However, training subjects typically involves abstract motor tasks and is time-consuming. To improve the performance of naive subjects during motor imagery, a novel paradigm was presented that would guide naive subjects to modulate brain activity effectively. In this new paradigm, pictures of the left or right hand were used as cues for subjects to finish the motor imagery task. Fourteen healthy subjects (11 male, aged 22-25 years, mean 23.6+/-1.16) participated in this study. The task was to imagine writing a Chinese character. Specifically, subjects could imagine hand movements following the sequence of writing strokes in the Chinese character. This paradigm was meant to find an effective and familiar action for most Chinese people, to provide them with a specific, extensively practiced task and help them modulate brain activity. Results showed that the writing task paradigm yielded significantly better performance than the traditional arrow paradigm (p<0.001). Questionnaire replies indicated that most subjects thought the new paradigm was easier and more comfortable. The proposed new motor imagery paradigm could guide subjects to help them modulate brain activity effectively. Results showed that there were significant improvements using new paradigm, both in classification accuracy and usability.

Qibin Zhao, Guoxu Zhou, Shengli Xie et al.

Tensor networks have in recent years emerged as the powerful tools for solving the large-scale optimization problems. One of the most popular tensor network is tensor train (TT) decomposition that acts as the building blocks for the complicated tensor networks. However, the TT decomposition highly depends on permutations of tensor dimensions, due to its strictly sequential multilinear products over latent cores, which leads to difficulties in finding the optimal TT representation. In this paper, we introduce a fundamental tensor decomposition model to represent a large dimensional tensor by a circular multilinear products over a sequence of low dimensional cores, which can be graphically interpreted as a cyclic interconnection of 3rd-order tensors, and thus termed as tensor ring (TR) decomposition. The key advantage of TR model is the circular dimensional permutation invariance which is gained by employing the trace operation and treating the latent cores equivalently. TR model can be viewed as a linear combination of TT decompositions, thus obtaining the powerful and generalized representation abilities. For optimization of latent cores, we present four different algorithms based on the sequential SVDs, ALS scheme, and block-wise ALS techniques. Furthermore, the mathematical properties of TR model are investigated, which shows that the basic multilinear algebra can be performed efficiently by using TR representaions and the classical tensor decompositions can be conveniently transformed into the TR representation. Finally, the experiments on both synthetic signals and real-world datasets were conducted to evaluate the performance of different algorithms.

Tatsuya Yokota, Qibin Zhao, Andrzej Cichocki

In recent years, low-rank based tensor completion, which is a higher-order extension of matrix completion, has received considerable attention. However, the low-rank assumption is not sufficient for the recovery of visual data, such as color and 3D images, where the ratio of missing data is extremely high. In this paper, we consider "smoothness" constraints as well as low-rank approximations, and propose an efficient algorithm for performing tensor completion that is particularly powerful regarding visual data. The proposed method admits significant advantages, owing to the integration of smooth PARAFAC decomposition for incomplete tensors and the efficient selection of models in order to minimize the tensor rank. Thus, our proposed method is termed as "smooth PARAFAC tensor completion (SPC)." In order to impose the smoothness constraints, we employ two strategies, total variation (SPC-TV) and quadratic variation (SPC-QV), and invoke the corresponding algorithms for model learning. Extensive experimental evaluations on both synthetic and real-world visual data illustrate the significant improvements of our method, in terms of both prediction performance and efficiency, compared with many state-of-the-art tensor completion methods.

Qibin Zhao, Liqing Zhang, Andrzej Cichocki

Tucker decomposition is the cornerstone of modern machine learning on tensorial data analysis, which have attracted considerable attention for multiway feature extraction, compressive sensing, and tensor completion. The most challenging problem is related to determination of model complexity (i.e., multilinear rank), especially when noise and missing data are present. In addition, existing methods cannot take into account uncertainty information of latent factors, resulting in low generalization performance. To address these issues, we present a class of probabilistic generative Tucker models for tensor decomposition and completion with structural sparsity over multilinear latent space. To exploit structural sparse modeling, we introduce two group sparsity inducing priors by hierarchial representation of Laplace and Student-t distributions, which facilitates fully posterior inference. For model learning, we derived variational Bayesian inferences over all model (hyper)parameters, and developed efficient and scalable algorithms based on multilinear operations. Our methods can automatically adapt model complexity and infer an optimal multilinear rank by the principle of maximum lower bound of model evidence. Experimental results and comparisons on synthetic, chemometrics and neuroimaging data demonstrate remarkable performance of our models for recovering ground-truth of multilinear rank and missing entries.

Anh-Huy Phan, Petr Tichavsky, Andrzej Cichocki

CANDECOMP/PARAFAC (CPD) approximates multiway data by sum of rank-1 tensors. Our recent study has presented a method to rank-1 tensor deflation, i.e. sequential extraction of the rank-1 components. In this paper, we extend the method to block deflation problem. When at least two factor matrices have full column rank, one can extract two rank-1 tensors simultaneously, and rank of the data tensor is reduced by 2. For decomposition of order-3 tensors of size R x R x R and rank-R, the block deflation has a complexity of O(R^3) per iteration which is lower than the cost O(R^4) of the ALS algorithm for the overall CPD.

Qibin Zhao, Guoxu Zhou, Liqing Zhang et al.

We propose a generative model for robust tensor factorization in the presence of both missing data and outliers. The objective is to explicitly infer the underlying low-CP-rank tensor capturing the global information and a sparse tensor capturing the local information (also considered as outliers), thus providing the robust predictive distribution over missing entries. The low-CP-rank tensor is modeled by multilinear interactions between multiple latent factors on which the column sparsity is enforced by a hierarchical prior, while the sparse tensor is modeled by a hierarchical view of Student-$t$ distribution that associates an individual hyperparameter with each element independently. For model learning, we develop an efficient closed-form variational inference under a fully Bayesian treatment, which can effectively prevent the overfitting problem and scales linearly with data size. In contrast to existing related works, our method can perform model selection automatically and implicitly without need of tuning parameters. More specifically, it can discover the groundtruth of CP rank and automatically adapt the sparsity inducing priors to various types of outliers. In addition, the tradeoff between the low-rank approximation and the sparse representation can be optimized in the sense of maximum model evidence. The extensive experiments and comparisons with many state-of-the-art algorithms on both synthetic and real-world datasets demonstrate the superiorities of our method from several perspectives.

Junhua Li, Zbigniew Struzik, Liqing Zhang et al.

An alternative pathway for the human brain to communicate with the outside world is by means of a brain computer interface (BCI). A BCI can decode electroencephalogram (EEG) signals of brain activities, and then send a command or an intent to an external interactive device, such as a wheelchair. The effectiveness of the BCI depends on the performance in decoding the EEG. Usually, the EEG is contaminated by different kinds of artefacts (e.g., electromyogram (EMG), background activity), which leads to a low decoding performance. A number of filtering methods can be utilized to remove or weaken the effects of artefacts, but they generally fail when the EEG contains extreme artefacts. In such cases, the most common approach is to discard the whole data segment containing extreme artefacts. This causes the fatal drawback that the BCI cannot output decoding results during that time. In order to solve this problem, we employ the Lomb-Scargle periodogram to estimate the spectral power from incomplete EEG (after removing only parts contaminated by artefacts), and Denoising Autoencoder (DAE) for learning. The proposed method is evaluated with motor imagery EEG data. The results show that our method can successfully decode incomplete EEG to good effect.

Guoxu Zhou, Andrzej Cichocki, Shengli Xie

Tensor decompositions are promising tools for big data analytics as they bring multiple modes and aspects of data to a unified framework, which allows us to discover complex internal structures and correlations of data. Unfortunately most existing approaches are not designed to meet the major challenges posed by big data analytics. This paper attempts to improve the scalability of tensor decompositions and provides two contributions: A flexible and fast algorithm for the CP decomposition (FFCP) of tensors based on their Tucker compression; A distributed randomized Tucker decomposition approach for arbitrarily big tensors but with relatively low multilinear rank. These two algorithms can deal with huge tensors, even if they are dense. Extensive simulations provide empirical evidence of the validity and efficiency of the proposed algorithms.

Chao Li, Lili Guo, Andrzej Cichocki

Many tensor-based data completion methods aim to solve image and video in-painting problems. But, all methods were only developed for a single dataset. In most of real applications, we can usually obtain more than one dataset to reflect one phenomenon, and all the datasets are mutually related in some sense. Thus one question raised whether such the relationship can improve the performance of data completion or not? In the paper, we proposed a novel and efficient method by exploiting the relationship among datasets for multi-video data completion. Numerical results show that the proposed method significantly improve the performance of video in-painting, particularly in the case of very high missing percentage.

Guoxu Zhou, Andrzej Cichocki, Qibin Zhao et al.

Nonnegative Tucker decomposition (NTD) is a powerful tool for the extraction of nonnegative parts-based and physically meaningful latent components from high-dimensional tensor data while preserving the natural multilinear structure of data. However, as the data tensor often has multiple modes and is large-scale, existing NTD algorithms suffer from a very high computational complexity in terms of both storage and computation time, which has been one major obstacle for practical applications of NTD. To overcome these disadvantages, we show how low (multilinear) rank approximation (LRA) of tensors is able to significantly simplify the computation of the gradients of the cost function, upon which a family of efficient first-order NTD algorithms are developed. Besides dramatically reducing the storage complexity and running time, the new algorithms are quite flexible and robust to noise because any well-established LRA approaches can be applied. We also show how nonnegativity incorporating sparsity substantially improves the uniqueness property and partially alleviates the curse of dimensionality of the Tucker decompositions. Simulation results on synthetic and real-world data justify the validity and high efficiency of the proposed NTD algorithms.

Qibin Zhao, Liqing Zhang, Andrzej Cichocki

CANDECOMP/PARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank. In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. To address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian treatment by incorporating a sparsity-inducing prior over multiple latent factors and the appropriate hyperpriors over all hyperparameters, resulting in automatic rank determination. To learn the model, we develop an efficient deterministic Bayesian inference algorithm, which scales linearly with data size. Our method is characterized as a tuning parameter-free approach, which can effectively infer underlying multilinear factors with a low-rank constraint, while also providing predictive distributions over missing entries. Extensive simulations on synthetic data illustrate the intrinsic capability of our method to recover the ground-truth of CP rank and prevent the overfitting problem, even when a large amount of entries are missing. Moreover, the results from real-world applications, including image inpainting and facial image synthesis, demonstrate that our method outperforms state-of-the-art approaches for both tensor factorization and tensor completion in terms of predictive performance.