arXiv subjects

Tomasz Piotrowski

Publications and source records attributed to Tomasz Piotrowski.

13 recordsLinked to original sources

arXiv2025

Multi-Source Neural Activity Indices for EEG/MEG Localization: A Two-Stage Spatial Filtering Framework and Extension to MNE-Python

Accurate electroencephalography (EEG) and magnetoencephalography (MEG) source localization and reconstruction are essential for understanding brain function, yet remain challenging because the underlying EEG/MEG inverse problem is inherently ill-posed. Spatial filtering (beamforming) approaches, such as linearly constrained minimum variance (LCMV) spatial filters, are widely used and well supported by existing analysis software. In this work, we extend this framework by deriving a novel family of unbiased multi-source neural activity indices that form the localization stage of a two-stage spatial-filtering-based localization-reconstruction framework for the EEG/MEG inverse problem. In contrast to existing formulations, the proposed indices do not require knowledge of the target source covariance matrix, making them directly applicable in practical experimental settings. Their compact algebraic forms enable straightforward and numerically efficient implementation. The framework is validated on simulated EEG data and its applicability is illustrated through an example involving experimental EEG data from an oddball paradigm. To facilitate adoption, we provide a full open-source implementation extending MNE-Python, accompanied by a practical tutorial.

math.OC↗

arXiv2025

Cellular, Cell-less, and Everything in Between: A Unified Framework for Utility Region Analysis in Wireless Networks

We introduce a unified framework for analyzing utility regions of wireless networks, with a focus on signal-to-interference-plus-noise-ratio (SINR) and achievable rate regions. The framework provides valuable insights into interference patterns of modern network architectures, including extremely large MIMO and cell-less networks. A central contribution is a simple characterization of feasible utility regions using the concept of spectral radius of nonlinear mappings. This characterization provides a powerful mathematical tool for wireless system design and analysis. For example, it allows us to generalize existing characterizations of the weak Pareto boundary using compact notation. It also allows us to derive tractable sufficient conditions for the identification of convex utility regions. This property is particularly important because, on the weak Pareto boundary, it guarantees that time sharing (or user grouping) cannot simultaneously improve the utilities of all users. Beyond geometrical insights, these sufficient conditions have two key implications. First, they identify a family of (weighted) sum-rate maximization problems that are inherently convex, thus paving the way for the development of efficient, provably optimal solvers for this family. Second, they provide justification for formulating sum-rate maximization problems directly in terms of achievable rates, rather than SINR levels. Our theoretical insights also motivate an alternative to the concept of favorable propagation in the massive MIMO literature -- one that explicitly accounts for self-interference and the beamforming strategy.

eess.SP↗

arXiv2024

Reduced-Rank Estimation for Ill-Conditioned Stochastic Linear Model with High Signal-to-Noise Ratio

Reduced-rank approach has been used for decades in robust linear estimation of both deterministic and random vector of parameters in linear model y=Hx+\sqrt{epsilon}n. In practical settings, estimation is frequently performed under incomplete or inexact model knowledge, which in the stochastic case significantly increases mean-square-error (MSE) of an estimate obtained by the linear minimum mean-square-error (MMSE) estimator, which is MSE-optimal among linear estimators in the theoretical case of perfect model knowledge. However, the improved performance of reduced-rank estimators over MMSE estimator in estimation under incomplete or inexact model knowledge has been established to date only by means of numerical simulations and arguments indicating that the reduced-rank approach may provide improved performance over MMSE estimator in certain settings. In this paper we focus on the high signal-to-noise ratio (SNR) case, which has not been previously considered as a natural area of application of reduced-rank estimators. We first show explicit sufficient conditions under which familiar reduced-rank MMSE and truncated SVD estimators achieve lower MSE than MMSE estimator if singular values of array response matrix H are perturbed. We then extend these results to the case of a generic perturbation of array response matrix H, and demonstrate why MMSE estimator frequently attains higher MSE than reduced-rank MMSE and truncated SVD estimators if H is ill-conditioned. The main results of this paper are verified in numerical simulations.

math.OC↗

arXiv2024

Inverse Feasibility in Over-the-Air Federated Learning

We introduce the concept of inverse feasibility for linear forward models as a tool to enhance OTA FL algorithms. Inverse feasibility is defined as an upper bound on the condition number of the forward operator as a function of its parameters. We analyze an existing OTA FL model using this definition, identify areas for improvement, and propose a new OTA FL model. Numerical experiments illustrate the main implications of the theoretical results. The proposed framework, which is based on inverse problem theory, can potentially complement existing notions of security and privacy by providing additional desirable characteristics to networks.

stat.ML↗

arXiv2024

Positive concave deep equilibrium models

Deep equilibrium (DEQ) models are widely recognized as a memory efficient alternative to standard neural networks, achieving state-of-the-art performance in language modeling and computer vision tasks. These models solve a fixed point equation instead of explicitly computing the output, which sets them apart from standard neural networks. However, existing DEQ models often lack formal guarantees of the existence and uniqueness of the fixed point, and the convergence of the numerical scheme used for computing the fixed point is not formally established. As a result, DEQ models are potentially unstable in practice. To address these drawbacks, we introduce a novel class of DEQ models called positive concave deep equilibrium (pcDEQ) models. Our approach, which is based on nonlinear Perron-Frobenius theory, enforces nonnegative weights and activation functions that are concave on the positive orthant. By imposing these constraints, we can easily ensure the existence and uniqueness of the fixed point without relying on additional complex assumptions commonly found in the DEQ literature, such as those based on monotone operator theory in convex analysis. Furthermore, the fixed point can be computed with the standard fixed point algorithm, and we provide theoretical guarantees of its geometric convergence, which, in particular, simplifies the training process. Experiments demonstrate the competitiveness of our pcDEQ models against other implicit models.

cs.LG↗

arXiv2022

The fixed point iteration of positive concave mappings converges geometrically if a fixed point exists: implications to wireless systems

We prove that the fixed point iteration of arbitrary positive concave mappings with nonempty fixed point set converges geometrically for any starting point. We also show that positivity is crucial for this result to hold, and the concept of (nonlinear) spectral radius of asymptotic mappings provides us with information about the convergence factor. As a practical implication of the results shown here, we rigorously explain why some power control and load estimation algorithms in wireless networks, which are particular instances of the fixed point iteration, have shown geometric convergence in simulations. These algorithms have been typically derived by considering fixed point iterations of the general class of standard interference mappings, so the possibility of sublinear convergence rate could not be ruled out in previous studies, except in special cases that are often more restrictive than those considered here.

math.OC↗

arXiv2021

ToFFi -- Toolbox for Frequency-based Fingerprinting of Brain Signals

Spectral fingerprints (SFs) are unique power spectra signatures of human brain regions of interest (ROIs, Keitel & Gross, 2016). SFs allow for accurate ROI identification and can serve as biomarkers of differences exhibited by non-neurotypical groups. At present, there are no open-source, versatile tools to calculate spectral fingerprints. We have filled this gap by creating a modular, highly-configurable MATLAB Toolbox for Frequency-based Fingerprinting (ToFFi). It can transform MEG/EEG signals into unique spectral representations using ROIs provided by anatomical (AAL, Desikan-Killiany), functional (Schaefer), or other custom volumetric brain parcellations. Toolbox design supports reproducibility and parallel computations.

cs.LG↗

arXiv2021

Relaxed Zero-Forcing Beamformer under Temporally-Correlated Interference

The relaxed zero-forcing (RZF) beamformer is a quadratically-and-linearly constrained minimum variance beamformer. The central question addressed in this paper is whether RZF performs better than the widely-used minimum variance distortionless response and zero-forcing beamformers under temporally-correlated interference. First, RZF is rederived by imposing an ellipsoidal constraint that bounds the amount of interference leakage for mitigating the intrinsic gap between the output variance and the mean squared error (MSE) which stems from the temporal correlations. Second, an analysis of RZF is presented for the single-interference case, showing how the MSE is affected by the spatio-temporal correlations between the desired and interfering sources as well as by the signal and noise powers. Third, numerical studies are presented for the multiple-interference case, showing the remarkable advantages of RZF in its basic performance as well as in its application to brain activity reconstruction from EEG data. The analytical and experimental results clarify that the RZF beamformer gives near-optimal performance in some situations.

cs.IT↗

arXiv2019

Iterative Neural Networks with Bounded Weights

A recent analysis of a model of iterative neural network in Hilbert spaces established fundamental properties of such networks, such as existence of the fixed points sets, convergence analysis, and Lipschitz continuity. Building on these results, we show that under a single mild condition on the weights of the network, one is guaranteed to obtain a neural network converging to its unique fixed point. We provide a bound on the norm of this fixed point in terms of norms of weights and biases of the network. We also show why this model of a feed-forward neural network is not able to accomodate Hopfield networks under our assumption.

cs.LG↗

arXiv2019

supFunSim: : spatial filtering toolbox for EEG

Recognition and interpretation of brain activity patterns from EEG or MEG signals is one of the most important tasks in cognitive neuroscience, requiring sophisticated methods of signal processing. The supFunSim library is a new Matlab toolbox which generates accurate EEG forward models and implements a collection of spatial filters for EEG source reconstruction, including linearly constrained minimum-variance (LCMV), eigenspace LCMV, nulling (NL), and minimum-variance pseudo-unbiased reduced-rank (MV-PURE) filters in various versions. It also enables source-level directed connectivity analysis using partial directed coherence (PDC) and directed transfer function (DTF) measures. The supFunSim library is based on the well-known FieldTrip toolbox for EEG and MEG analysis and is written using object-oriented programming paradigm. The resulting modularity of the toolbox enables its simple extensibility. This paper gives a complete overview of the toolbox from both developer and end-user perspectives, including description of the installation process and some use cases.

eess.SP↗

arXiv2018

Localization of Brain Activity from EEG/MEG Using MV-PURE Framework

We consider the problem of localization of sources of brain electrical activity from electroencephalographic (EEG) and magnetoencephalographic (MEG) measurements using spatial filtering techniques. We propose novel reduced-rank activity indices based on the minimum-variance pseudo-unbiased reduced-rank estimation (MV-PURE) framework. The main results of this paper establish the key unbiasedness property of the proposed indices and their higher spatial resolution compared with full-rank indices in challenging task of localizing closely positioned and possibly highly correlated sources, especially in low signal-to-noise regime. Numerical examples are provided to illustrate the practical applicability of the proposed activity indices using both simulated and real data.

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arXiv2017

MV-PURE Spatial Filters with Application to EEG/MEG Source Reconstruction

In this paper we propose spatial filters for a linear regression model which are based on the minimum-variance pseudo-unbiased reduced-rank estimation (MV-PURE) framework. As a sample application, we consider the problem of reconstruction of brain activity from electroencephalographic (EEG) or magnetoencephalographic (MEG) measurements. The proposed filters come in two versions depending on whether or not the EEG/MEG forward model explicitly considers interfering activity in the way of brain activity originating in regions different to those of main interest, but measured as correlated with signals of interest by the EEG/MEG sensor array. In both cases, the proposed filters are equipped with a rank-selection criterion minimizing the mean-square error (MSE) of the filter output. Therefore, we consider them as novel nontrivial generalizations of well-known linearly constrained minimum variance (LCMV) and nulling filters. In order to facilitate reproducibility of our research, we provide (jointly with this paper) comprehensive simulation framework that allows for estimation of error of signal reconstruction for a number of spatial filters applied to MEG or EEG signals. Based on this framework, chief properties of proposed filters are verified in a set of detailed simulations.

eess.SP↗

arXiv2013

Performance of the stochastic MV-PURE estimator in highly noisy settings

The stochastic minimum-variance pseudo-unbiased reduced-rank estimator (stochastic MV-PURE estimator) has been developed to provide linear estimation with robustness against high noise levels, imperfections in model knowledge, and ill-conditioned systems. In this paper, we investigate the theoretical performance of the stochastic MV-PURE estimator under varying levels of additive noise. We prove that the mean-square-error (MSE) of this estimator in the low signal-to-noise (SNR) region is much smaller than that obtained with its full-rank version, the minimum-variance distortionless estimator, and the gap becomes larger as the noise level increases. These results shed light on the excellent performance of the stochastic MV-PURE estimator in highly noisy settings obtained in simulations so far. Furthermore, we extend previous numerical simulations to show how the insight gained from the results of this paper can be used in practice.

stat.AP↗