This paper proposes StrEBM, a structured latent energy-based model for source-wise structured representation learning. The framework is motivated by a broader goal of promoting identifiable and decoupled latent organization by assigning different latent dimensions their own learnable structural biases, rather than constraining the entire latent representation with a single shared energy. In this sense, blind source separation is adopted here as a concrete and verifiable testbed, through which the evolution of latent dimensions toward distinct underlying components can be directly examined. In the proposed framework, latent trajectories are optimized directly together with an observation-generation map and source-wise structural parameters. Each latent dimension is associated with its own energy-based formulation, allowing different latent components to gradually evolve toward distinct source-like roles during training. In the present study, this source-wise energy design is instantiated using Gaussian-process-inspired energies with learnable length-scales, but the framework itself is not restricted to Gaussian processes and is intended as a more general structured latent EBM formulation. Experiments on synthetic multichannel signals under linear and nonlinear mixing settings show that the proposed model can recover source components effectively, providing an initial empirical validation of the framework. At the same time, the study reveals important optimization characteristics, including slow late-stage convergence and reduced stability under nonlinear observation mappings. These findings not only clarify the practical behavior of the current GP-based instantiation, but also establish a basis for future investigation of richer source-wise energy families and more robust nonlinear optimization strategies.

This paper presents a Structured Source-Wise Adaptive Diffusion Framework for linear and nonlinear blind source separation. The framework interprets each latent dimension as a source component and assigns to it an individual adaptive diffusion mechanism, thereby establishing source-wise latent modeling rather than relying on a single shared latent prior. The resulting formulation learns source recovery and the mixing/reconstruction process jointly within a unified end-to-end objective, allowing model parameters and latent sources to adapt simultaneously during training. This yields a common framework for both linear and nonlinear blind source separation. In the present instantiation, each source is further equipped with its own adaptive Gaussian process (GP) prior to impose source-wise temporal structure on the latent trajectories, while the overall framework is not restricted to Gaussian process priors and can in principle accommodate other structured source priors. The proposed model thus provides a general structured diffusion-based route to unsupervised source recovery, with potential relevance beyond blind source separation to interpretable latent modeling, source-wise disentanglement, and potentially identifiable nonlinear latent-variable learning under appropriate structural conditions.

Blind source separation (BSS) seeks to recover latent source signals from observed mixtures. Variational autoencoders (VAEs) offer a natural perspective for this problem: the latent variables can be interpreted as source components, the encoder can be viewed as a demixing mapping from observations to sources, and the decoder can be regarded as a remixing process from inferred sources back to observations. In this work, we propose AR-Flow VAE, a novel VAE-based framework for BSS in which each latent source is endowed with a parameter-adaptive autoregressive flow prior. This prior significantly enhances the flexibility of latent source modeling, enabling the framework to capture complex non-Gaussian behaviors and structured dependencies, such as temporal correlations, that are difficult to represent with conventional priors. In addition, the structured prior design assigns distinct priors to different latent dimensions, thereby encouraging the latent components to separate into different source signals under heterogeneous prior constraints. Experimental results validate the effectiveness of the proposed architecture for blind source separation. More importantly, this work provides a foundation for future investigations into the identifiability and interpretability of AR-Flow VAE.

Conventional echo state networks (ESNs) require supervised learning to train the readout layer, using the desired outputs as training data. In this study, we focus on input reconstruction (IR), which refers to training the readout layer to reproduce the input time series in its output. We reformulate the learning algorithm of the ESN readout layer to perform IR using unsupervised learning (UL). By conducting theoretical analysis and numerical experiments, we demonstrate that IR in ESNs can be effectively implemented under realistic conditions without explicitly using the desired outputs as training data; in this way, UL is enabled. Furthermore, we demonstrate that applications relying on IR, such as dynamical system replication and noise filtering, can be reformulated within the UL framework. Our findings establish a theoretically sound and universally applicable IR formulation, along with its related tasks in ESNs. This work paves the way for novel predictions and highlights unresolved theoretical challenges in ESNs, particularly in the context of time-series processing methods and computational models of the brain.

With the emergence of wireless sensor networks (WSNs), many traditional signal processing tasks are required to be computed in a distributed fashion, without transmissions of the raw data to a centralized processing unit, due to the limited energy and bandwidth resources available to the sensors. In this paper, we propose a distributed independent component analysis (ICA) algorithm, which aims at identifying the original signal sources based on observations of their mixtures measured at various sensor nodes. One of the most commonly used ICA algorithms is known as FastICA, which requires a spatial pre-whitening operation in the first step of the algorithm. Such a pre-whitening across all nodes of a WSN is impossible in a bandwidth-constrained distributed setting as it requires to correlate each channel with each other channel in the WSN. We show that an explicit network-wide pre-whitening step can be circumvented by leveraging the properties of the so-called Distributed Adaptive Signal Fusion (DASF) framework. Despite the lack of such a network-wide pre-whitening, we can still obtain the $Q$ least Gaussian independent components of the centralized ICA solution, where $Q$ scales linearly with the required communication load.

A major outstanding problem when interfacing with spinal motor neurons is how to accurately compensate for non-stationary effects in the signal during source separation routines, particularly when they cannot be estimated in advance. This forces current systems to instead use undifferentiated bulk signal, which limits the potential degrees of freedom for control. In this study we propose a potential solution, using an unsupervised learning algorithm to blindly correct for the effects of latent processes which drive the signal non-stationarities. We implement this methodology within the theoretical framework of a quasilinear version of independent component analysis (ICA). The proposed design, HarmonICA, sidesteps the identifiability problems of nonlinear ICA, allowing for equivalent predictability to linear ICA whilst retaining the ability to learn complex nonlinear relationships between non-stationary latents and their effects on the signal. We test HarmonICA on both invasive and non-invasive recordings both simulated and real, demonstrating an ability to blindly compensate for the non-stationary effects specific to each, and thus to significantly enhance the quality of a source separation routine.

While neural network approaches have made significant strides in resolving classical signal processing problems, it is often the case that hybrid approaches that draw insight from both signal processing and neural networks produce more complete solutions. In this paper, we present a hybrid classical digital signal processing/deep neural network (DSP/DNN) approach to source separation (SS) highlighting the theoretical link between variational autoencoder and classical approaches to SS. We propose a system that transforms the single channel under-determined SS task to an equivalent multichannel over-determined SS problem in a properly designed latent space. The separation task in the latent space is treated as finding a variational block-wise disentangled representation of the mixture. We show empirically, that the design choices and the variational formulation of the task at hand motivated by the classical signal processing theoretical results lead to robustness to unseen out-of-distribution data and reduction of the overfitting risk. To address the resulting permutation issue we explicitly incorporate a novel differentiable permutation loss function and augment the model with a memory mechanism to keep track of the statistics of the individual sources.

This paper describes an efficient unsupervised learning method for a neural source separation model that utilizes a probabilistic generative model of observed multichannel mixtures proposed for blind source separation (BSS). For this purpose, amortized variational inference (AVI) has been used for directly solving the inverse problem of BSS with full-rank spatial covariance analysis (FCA). Although this unsupervised technique called neural FCA is in principle free from the domain mismatch problem, it is computationally demanding due to the full rankness of the spatial model in exchange for robustness against relatively short reverberations. To reduce the model complexity without sacrificing performance, we propose neural FastFCA based on the jointly-diagonalizable yet full-rank spatial model. Our neural separation model introduced for AVI alternately performs neural network blocks and single steps of an efficient iterative algorithm called iterative source steering. This alternating architecture enables the separation model to quickly separate the mixture spectrogram by leveraging both the deep neural network and the multichannel optimization algorithm. The training objective with AVI is derived to maximize the marginalized likelihood of the observed mixtures. The experiment using mixture signals of two to four sound sources shows that neural FastFCA outperforms conventional BSS methods and reduces the computational time to about 2% of that for the neural FCA.

In this paper, sparse representation (factorization) of a data matrix is first discussed. An overcomplete basis matrix is estimated by using the K(cid:0)means method. We have proved that for the estimated overcom- plete basis matrix, the sparse solution (coefficient matrix) with minimum l1(cid:0)norm is unique with probability of one, which can be obtained using a linear programming algorithm. The comparisons of the l1(cid:0)norm so- lution and the l0(cid:0)norm solution are also presented, which can be used in recoverability analysis of blind source separation (BSS). Generally, if the sources are not sufficiently sparse, we perform blind separation in the time-frequency domain after preprocessing the observed data using the wavelet packets transformation.