--We introduce V ariational Latent Mode Decomposition (VLMD), a new algorithm for extracting oscillatory modes and associated connectivity structures from multivariate signals. VLMD addresses key limitations of existing Multivariate Mode Decomposition (MMD) techniques--including high computational cost, sensitivity to parameter choices, and weak modeling of interchannel dependencies. Its improved performance is driven by a novel underlying model, Latent Mode Decomposition (LMD), which blends sparse coding and mode decomposition to represent multichannel signals as sparse linear combinations of shared latent components composed of AM-FM oscillatory modes. This formulation enables VLMD to operate in a lower-dimensional latent space, enhancing robustness to noise, scalability, and interpretability. The algorithm solves a constrained variational optimization problem that jointly enforces reconstruction fidelity, sparsity, and frequency regularization. Experiments on synthetic and real-world datasets demonstrate that VLMD outperforms state-of-the-art MMD methods in accuracy, efficiency, and the interpretability of extracted structures. ONST A TIONARY signal decomposition techniques constitute an essential tool in Signal Processing for analyzing complex signals. Among them, Mode Decomposition (MD) has emerged as a fundamental framework, enabling the extraction of meaningful intrinsic oscillatory components [1]. Over the last couple of decades, a wide range of MD methods and algorithms have been developed and successfully applied across a wide range of interdisciplinary applications, such as biomedical signal analysis, structural health monitoring, and financial time-series analysis. This particular trend dates back to the late nineties with the introduction of Empirical Mode Decomposition (EMD) [1], which was followed by the development of other similar alternatives, such as Synchro-squeezed Transform (SST) [2], V ariational Mode Decomposition (VMD) [3] and Sliding-window Singular Spectrum Analysis (SSA) [4]. Originally designed for single-channel time series analysis, some of these methods were later extended to handle multivariate time series. Notable multivariate algorithms include Multivariate Empirical Mode Decomposition (MEMD) [5], multivariate nonlinear chirp mode decomposition [6], iterative filtering [7], as well as Multivariate V ariational Mode Decomposition (MVMD) [8].

The decomposition of a signal is a fundamental tool in many fields of research, including signal processing, geophysics, astrophysics, engineering, medicine, and many more. By breaking down complex signals into simpler oscillatory components we can enhance the understanding and processing of the data, unveiling hidden information contained in them. Traditional methods, such as Fourier analysis and wavelet transforms, which are effective in handling mono-dimensional stationary signals struggle with non-stationary data sets and they require, this is the case of the wavelet, the selection of predefined basis functions. In contrast, the Empirical Mode Decomposition (EMD) method and its variants, such as Iterative Filtering (IF), have emerged as effective nonlinear approaches, adapting to signals without any need for a priori assumptions. To accelerate these methods, the Fast Iterative Filtering (FIF) algorithm was developed, and further extensions, such as Multivariate FIF (MvFIF) and Multidimensional FIF (FIF2), have been proposed to handle higher-dimensional data. In this work, we introduce the Multidimensional and Multivariate Fast Iterative Filtering (MdMvFIF) technique, an innovative method that extends FIF to handle data that vary simultaneously in space and time. This new algorithm is capable of extracting Intrinsic Mode Functions (IMFs) from complex signals that vary in both space and time, overcoming limitations found in prior methods. The potentiality of the proposed method is demonstrated through applications to artificial and real-life signals, highlighting its versatility and effectiveness in decomposing multidimensional and multivariate nonstationary signals. The MdMvFIF method offers a powerful tool for advanced signal analysis across many scientific and engineering disciplines.

The interdependence and high dimensionality of multivariate signals present significant challenges for denoising, as conventional univariate methods often struggle to capture the complex interactions between variables. A successful approach must consider not only the multivariate dependencies of the desired signal but also the multivariate dependencies of the interfering noise. In our previous research, we introduced a method using machine learning to extract the maximum portion of ``predictable information" from univariate signal. We extend this approach to multivariate signals, with the key idea being to properly incorporate the interdependencies of the noise back into the interdependent reconstruction of the signal. The method works successfully for various multivariate signals, including chaotic signals and highly oscillating sinusoidal signals which are corrupted by spatially correlated intensive noise. It consistently outperforms other existing multivariate denoising methods across a wide range of scenarios.

Digitalizing real-world analog signals typically involves sampling in time and discretizing in amplitude. Subsequent signal reconstructions inevitably incur an error that depends on the amplitude resolution and the temporal density of the acquired samples. From an implementation viewpoint, consistent signal reconstruction methods have proven a profitable error-rate decay as the sampling rate increases. Despite that, these results are obtained under offline settings. Therefore, a research gap exists regarding methods for consistent signal reconstruction from data streams. This paper presents a method that consistently reconstructs streamed multivariate time series of quantization intervals under a zero-delay response requirement. On the other hand, previous work has shown that the temporal dependencies within univariate time series can be exploited to reduce the roughness of zero-delay signal reconstructions. This work shows that the spatiotemporal dependencies within multivariate time series can also be exploited to achieve improved results. Specifically, the spatiotemporal dependencies of the multivariate time series are learned, with the assistance of a recurrent neural network, to reduce the roughness of the signal reconstruction on average while ensuring consistency. Our experiments show that our proposed method achieves a favorable error-rate decay with the sampling rate compared to a similar but non-consistent reconstruction.

We propose a greedy variational method for decomposing a non-negative multivariate signal as a weighted sum of Gaussians which, borrowing the terminology from statistics, we refer to as a Gaussian mixture model (GMM). Mixture components are added one at the time in two steps. In the first step, a new Gaussian atom and an amplitude are chosen based on a heuristic that aims to minimize the 2-norm of the residual. In the second step the 2-norm of the residual is further decreased by simultaneously adjusting all current Gaussians. Notably, our method has the following features: (1) It accepts multivariate signals, i.e. sampled multivariate function, histograms, time series, images, etc. as input. (2) The method can handle general (i.e. ellipsoidal) Gaussians. (3) No prior assumption on the number of mixture components is needed. To the best of our knowledge, no previous method for GMM decomposition simultaneously enjoys all these features. Since finding the optimal atom is a non-convex problem, an important point is how to initialize each new atom. We initialize the mean at the maximum of the residual. As a motivation for this initialization procedure, we prove an upper bound, which cannot be improved by a global constant, for the distance from any mode of a GMM to the set of corresponding means. For mixtures of spherical Gaussians with common variance $\sigma^2$, the bound takes the simple form $\sqrt{n}\sigma$. We evaluate our method on one- and two-dimensional signals. We also discuss the relation between clustering and signal decomposition, and compare our method to the baseline expectation maximization algorithm.

Clustering of high-dimensional signals, sequences or functional data is a common task that arises in many domains [18, 19]. Such data come up in diverse fields, as in speech analysis, genomics, mass spectrometry, MRI or EEG measurements, and many more. Clustering seeks to partition data into groups with high overall similarity between members (instances) of the same group and dissimilarity to members of other groups. For time-series signals, this means partitioning the instances into groups of similarly behaving functions over time, where the measure of similarity is crucial and often application-specific. In many real-world scenarios, signals are high-dimensional (such as in genomics), noisy (as in low-quality speech recordings), and exhibit non-stationary behavior: for example peaks and other non-smooth local patterns, or changes in frequency over time.