Approximation of Spatio-Temporal Random Processes Using Tensor Decomposition

In motor imagery (MI) tasks for brain computer interfaces (BCIs), the spatial covariance matrix (SCM) of electroencephalogram (EEG) signals plays a critical role in accurate classification. Given that SCMs are symmetric positive definite (SPD), Riemannian geometry is widely utilized to extract classification features. However, calculating distances between SCMs is computationally intensive due to operations like eigenvalue decomposition, and classical optimization techniques, such as gradient descent, cannot be directly applied to Riemannian manifolds, making the computation of the Riemannian mean more complex and reliant on iterative methods or approximations. In this paper, we propose a novel multiclass classification framework that integrates Riemannian geometry and neural networks to mitigate these challenges. The framework comprises two modules: a Riemannian module with multiple branches and a classification module. During training, a fusion loss function is introduced to update the branch corresponding to the true label, while other branches are updated using different loss functions along with the classification module. Comprehensive experiments on four sets of MI EEG data demonstrate the efficiency and effectiveness of the proposed model.

This paper proposes a practical way to solve the Uplink-Downlink Covariance Conversion (UDCC) problem in a frequency Division Duplex (FDD) communication system. The UDCC problem consists in the estimation of the Downlink (DL) spatial covariance matrix from the prior knowledge of the Uplink (UL) spatial covariance matrix without the need of a feedback transmission from the User Equipment (UE) to the Base Station (BS). Estimating the DL sample spatial covariance matrix is unfeasible in current massive Multiple-Input Multiple-Output (MIMO) deployments in frequency selective or fast fading channels due to the required large training overhead. Our method is based on the application of sparse filtering ideas to the estimation of a quantized version of the so-called Angular Power Spectrum (APS), being the common factor between the UL and DL spatial channel covariance matrices.

This paper presents a novel class of self-organizing sensing agents that adaptively learn an anisotropic, spatio-temporal Gaussian process using noisy measurements and move in order to improve the quality of the estimated covariance function. This approach is based on a class of anisotropic covariance functions of Gaussian processes introduced to model a broad range of spatio-temporal physical phenomena. The covariance function is assumed to be unknown a priori. Hence, it is estimated by the maximum a posteriori probability (MAP) estimator. The prediction of the field of interest is then obtained based on the MAP estimate of the covariance function. An optimal sampling strategy is proposed to minimize the information-theoretic cost function of the Fisher Information Matrix. Simulation results demonstrate the effectiveness and the adaptability of the proposed scheme.

The present study devotes to introducing and analyzing the use of the adaptive Fourier decomposition (AFD)‐type methods in the area of stochastic processes and random fields. We involve two types of algorithms, namely, stochastic AFD (SAFD) and stochastic pre‐orthogonal AFD (SPOAFD), for, respectively, the Hardy space format and non‐Hardy space ones, as may be regarded. We provide both their theoretical results and practical algorithms and compare them with the well adopted and, in fact, dominating Karhunen–Loève (KL)‐type expansions. The AFD methods involve a finite or infinite sequence of optimally chosen parameters; they, in contrast, do not rely on and hence not have to compute the eigenpairs of the second type Fredholm integral equation with the covariance function as the kernel. Apart from such computational conveniences, they with the same convergence rate have flexibility of choosing best suitable dictionaries for doing the specific task in the practice. We include a number of experiments showing that in terms of effectiveness, the AFD methods give better approximations before all the positive eigenvalues running out in the case the integral operator being of a finite rank or before the KL iteration step becomes excessively large: The AFD methods normally give better approximations from the very beginning of the iterations.

The spatial sign covariance matrix with unknown location

The well-known spatial sign covariance matrix (SSCM) carries out a radial transform which moves all data points to a sphere, followed by computing the classical covariance matrix of the transformed data. Its popularity stems from its robustness to outliers, fast computation, and applications to correlation and principal component analysis. In this paper we study more general radial functions. It is shown that the eigenvectors of the generalized SSCM are still consistent and the ranks of the eigenvalues are preserved. The influence function of the resulting scatter matrix is derived, and it is shown that its asymptotic breakdown value is as high as that of the original SSCM. A simulation study indicates that the best results are obtained when the inner half of the data points are not transformed and points lying far away are moved to the center.

This paper proposes a novel method for simultaneously solving the problems of underdetermined blind source separation (BSS), source activity detection, dereverberation and direction-of-arrival (DOA) estimation by introducing an extension of the "multichannel factorial hidden Markov model (MFH-MM)." The MFHMM is an extension of the multichannel non-negative matrix factorization (NMF) modeL in which the basis spectra are allowed to vary over time according to the transitions of the hidden states. This model has allowed us to perform source separation, source activity detection and dereverberation in a unified manner. In our previous model, the spatial covariance of each source has been treated as a model parameter. This has led the entire generative model to have an unnecessarily high degree of freedom, and thus the parameter inference has been prone to getting trapped into undesired local optima. To reasonably restrict the solution space of the spatial covariance matrix of each source, we propose to describe it as a weighted sum of the fixed spatial covariance matrix corresponding to the discrete set of DOAs. Through the parameter inference, the proposed model allows us to simultaneously solve the problems of underdetermined BSS, source activity detection, dereverberation and DOA estimation. Experimental results revealed that the proposed method was superior to a previous method in terms of the signal-to-distortion ratios of separated signals.

Spatial channel covariance information can replace full channel state information for designing analog precoders in millimeter wave (mmWave) hybrid MIMO systems. Hybrid MIMO architectures, however, make it challenging to estimate the spatial channel covariance matrix because the estimator in baseband can only see the low-dimensional projections of the original channel. In this paper, we propose two key ideas for developing the covariance estimation techniques based on compressive sensing techniques. One is to use the Hermitian property of the covariance matrix, and the other is to use a time-varying analog combining matrix to effectively extend the measurement size.

Studies on the riding modes of shared bicycles have aimed to heighten the understanding of cycling characteristics. This paper analyzes the spatiotemporal riding modes of shared bicycles based on tensor decomposition in Beijing, China. Two third-order tensors are constructed for the origin and destination points of shared bicycles in the day, hour, and space dimensions. Three factor matrices explicitly reveal two modes, three modes, and six modes in the day dimension, hour dimension, and space dimension, respectively. The relationships among the different modes in the three dimensions are demonstrated in an interaction table. Further, the density for different types of points of interest (POIs) are calculated to further analyze the potential riding purpose for different riding modes. Notably, the main POI types for the areas of O2 and D2 modes are consistent with the areas of D3 and O3 modes, which reflects the tidal characteristics of the commuting activities of shared bicycles. The main functional areas are inferred according to the riding modes and POIs, which enables verification of the correctness of the obtained riding modes to some extent. By method comparison, tensor decomposition shows the advantage of being able to reveal the spatiotemporal modes among multiple dimensions.

In order to solve users' rating sparsely problem existing in present recommender systems, this article proposes a personalized recommendation algorithm based on contextual awareness and tensor decomposition. Via this algorithm, it was first constructed two third-order tensors to represent six types of entities, including the user-user-item contexts and the item-item-user contexts. And then, this article uses a high order singular value decomposition method to mine the potential semantic association of the two third-order tensors above. Finally, the resulting tensors were combined to reach the recommendation list to respond the users' personalized query requests. Experimental results show that the proposed algorithm can effectively improve the effectiveness of the recommendation system. Especially in the case of sparse data, it can significantly improve the quality of the recommendation.

A note on the product correlation rule

The one-sample multivariate shape problem is considered. The shape matrix is a scatter matrix rescaled or normalized in a certain way. The procedures here are based on multivariate spatial signs and spatial ranks. Tests and estimates based on the spatial sign covariance matrix and the spatial rank covariance matrices of different types are considered. The asymptotic efficiencies show a good performance of these methods, particularly for heavy-tailed distributions.

In this article, we propose a new blind speech extraction (BSE) method that robustly extracts a directional speech from background diffuse noise by combining independent low-rank matrix analysis (ILRMA) and efficient rank-constrained spatial covariance matrix (SCM) estimation. To achieve more accurate BSE than ILRMA, which assumes each source to be a point source (rank-1 spatial model), the proposed method restores the lost spatial basis for the full-rank SCM of diffuse noise. We adopt the multivariate complex generalized Gaussian distribution (GGD) as the statistical generative model to express various types of observed signal. To estimate the model parameters for an arbitrary shape parameter of the multivariate GGD, we derive a new inequality for rank-constrained SCMs. Also, we propose new acceleration methods to accomplish much faster extraction than conventional blind source separation methods. In BSE experiments using simulated and real recorded data, we confirm that the proposed method achieves more accurate and faster speech extraction than conventional methods.

To mitigate the asynchronous ICI (Inter-Cell Interference), the CFR (Channel Frequency Response) and SCM (Spatial Covariance Matrix) should be accurately estimated. Therefore, the enhanced asynchronous ICI mitigation scheme which can effectively reduce the estimation error of both CFR and SCM, and improve performance for MIMO (Multiple-Input Multiple-Output)-OFDMA (Orthogonal Frequency Division Multiple Access) system is derived. In particular, the lower-bound performance is approached to within 1 dB, by using the proposed method for both MMSE (Minimum Mean Square Error) and ML (Maximum Likelihood) receiver in the presence of asynchronous ICI.

Tensor-based approach for underdetermined operational modal identification