Presence Of Heavy-tailed Noise Research Articles

VB-T PHD-SLAM: efficient SLAM under heavy-tailed noise

To address the challenge of simultaneous localization and mapping (SLAM) in the presence of heavy-tailed noise, this paper introduces a robust probability hypothesis density (PHD) SLAM algorithm. This algorithm models measurement noise using the Student's t-distribution, which better captures the heavy-tailed nature of the noise. Since the prior density is assumed to be Gaussian mixture form, the posterior density is no longer Gaussian mixture form after the likelihood update of the t-distribution. A variational Bayesian approach is employed to ensure computable multi-target densities during filtering, minimizing the Kullback-Leibler divergence to obtain an approximate solution for the new marginal likelihood function. Then a new closed-form recursion of PHD-SLAM is derived by using t-distribution. Simulation results and real-world validations demonstrate that the proposed algorithm outperforms PHD-SLAM 1.0 and PHD-SLAM 2.0 in terms of both localization and mapping accuracy while maintaining computational efficiency in SLAM scenarios affected by heavy-tailed noise.

Robust Bilinear form identification: A subgradient method with geometrically decaying stepsize in the presence of heavy-tailed noise

Robust Bilinear form identification: A subgradient method with geometrically decaying stepsize in the presence of heavy-tailed noise

Alpha-Stable Autoregressive Modeling of Chua's Circuit in the Presence of Heavy-Tailed Noise

This study presents alpha-stable autoregressive (AR) modeling of the dynamics of Chua's circuit in the presence of heavy-tailed noise. The parameters of the AR time series are estimated using the covariation-based Yule-Walker method, and the parameters of alpha-stable distributed residuals are calculated using the regression type method. Visual depictions of the calculated parameters of the AR model and alpha-stable distributions of residuals are presented. The medians of the estimated parameters of the AR model and alpha-stable distributions parameters of residuals are presented for heavy-tailed noise with various stability index parameters. Thus, the impulsive behavior of Chua's circuit can be modeled as alpha-stable AR time series, and the model can provide an alternative approach to describe the chaotic systems driven by heavy-tailed noise.

On Estimating Rank-One Spiked Tensors in the Presence of Heavy Tailed Errors

In this paper, we study the estimation of a rank-one spiked tensor in the presence of heavy tailed noise. Our results highlight some of the fundamental similarities and differences in the tradeoff between statistical and computational efficiencies under heavy tailed and Gaussian noise. In particular, we show that, for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> th order tensors, the tradeoff manifests in an identical fashion as the Gaussian case when the noise has finite <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4(p-1)$ </tex-math></inline-formula> th moment. The difference in signal strength requirements, with or without computational constraints, for us to estimate the singular vectors at the optimal rate, interestingly, narrows for noise with heavier tails and vanishes when the noise only has finite fourth moment. Moreover, if the noise has less than fourth moment, tensor SVD, perhaps the most natural approach, is suboptimal even though it is computationally intractable. Our analysis exploits a close connection between estimating the rank-one spikes and the spectral norm of a random tensor with iid entries. In particular, we show that the order of the spectral norm of a random tensor can be precisely characterized by the moment of its entries, generalizing classical results for random matrices. In addition to the theoretical guarantees, we propose estimation procedures for the heavy tailed regime, which are easy to implement and efficient to run. Numerical experiments are presented to demonstrate their practical merits.

Fast Rates of Gaussian Empirical Gain Maximization With Heavy-Tailed Noise.

In a regression setup, we study in this brief the performance of Gaussian empirical gain maximization (EGM), which includes a broad variety of well-established robust estimation approaches. In particular, we conduct a refined learning theory analysis for Gaussian EGM, investigate its regression calibration properties, and develop improved convergence rates in the presence of heavy-tailed noise. To achieve these purposes, we first introduce a new weak moment condition that could accommodate the cases where the noise distribution may be heavy-tailed. Based on the moment condition, we then develop a novel comparison theorem that can be used to characterize the regression calibration properties of Gaussian EGM. It also plays an essential role in deriving improved convergence rates. Therefore, the present study broadens our theoretical understanding of Gaussian EGM.

Robust learning of Huber loss under weak conditional moment

In this paper, we study the performance of robust learning with Huber loss. As an alternative to traditional empirical risk minimization schemes, Huber regression has been extensively used in machine learning. A new comparison theorem is established in the paper, which characterizes the gap between the excess generalization error and the prediction error. In addition, we refine the error bounds from the perspective of statistical learning theory and improve the convergence rates in the presence of heavy-tailed noise. It is worth mentioning that a new moment condition E[|Y|1+∊|X=x]∈LρX2 is employed in analysis of error bound and learning rates from a theoretical viewpoint.

Robust Spectrum Sensing Via Probability Measure Transform

In this paper, we develop a new robust spectrum sensing method for MIMO cognitive radios in the presence of heavy-tailed noise. The proposed sensing technique, called measure-transformed covariance test (MTCT), operates by applying a transform to the probability measure of the data. The considered probability measure transform is structured by a non-negative function, called MT-function, that weights the data points. We show that proper selection of the MT-function, under the class of zero-centered spherically contoured Gaussian functions, can lead to significant mitigation of heavy-tailed noise effects on the sensing performance. Simulation studies illustrate the advantages of the proposed MTCT comparing to other robust MIMO and SIMO spectrum sensing techniques.

Local Defect Detection in Bearings in the Presence of Heavy-Tailed Noise and Spectral Overlapping of Informative and Non-Informative Impulses.

The problem of the informative frequency band (IFB) selection for local fault detection is considered in the paper. There are various approaches that are very effective in this issue. Most of the techniques are vibration-based and they are related to the cyclic impulses detection (associated with the local fault) in the background noise. However, when the background noise in the vibration signal has non-Gaussian impulsive behavior, the classical methods seem to be insufficient. Recently, new techniques were proposed by several authors and interesting approaches were tested for different non-Gaussian signals. We demonstrate the comparative analysis related to the results for three selected techniques proposed in recent years, namely the Alpha selector, Conditional Variance-based selector, and Spearman selector. The techniques seem to be effective for the IFB selection for the non-Gaussian distributed vibration signals. The main purpose of this article is to investigate how spectral overlapping of informative and non-informative impulsive components will affect diagnostic procedures. According to our knowledge, this problem was not considered in the literature for the non-Gaussian signals. Nevertheless, as we demonstrated by the simulations, the level of overlapping and the location of a center frequency of the mentioned frequency bands have a significant influence on the behavior of the considered selectors. The discussion about the effectiveness of each analyzed method is conducted. The considered problem is supported by real-world examples.

A novel optimization algorithm for MIMO Hammerstein model identification under heavy-tailed noise

In this paper, we study the system identification of multi-input multi-output (MIMO) Hammerstein processes under the typical heavy-tailed noise. To the best of our knowledge, there is no general analytical method to solve this identification problem. Motivated by this, we propose a general identification method to solve this problem based on a Gaussian-Mixture Distribution intelligent optimization algorithm (GMDA). The nonlinear part of Hammerstein process is modeled by a Radial Basis Function (RBF) neural network, and the identification problem is converted to an optimization problem. To overcome the drawbacks of analytical identification method in the presence of heavy-tailed noise, a meta-heuristic optimization algorithm, Cuckoo search (CS) algorithm is used. To improve its performance for this identification problem, the Gaussian-mixture Distribution (GMD) and the GMD sequences are introduced to improve the performance of the standard CS algorithm. Numerical simulations for different MIMO Hammerstein models are carried out, and the simulation results verify the effectiveness of the proposed GMDA.

Inexact Alternating Optimization for Phase Retrieval in the Presence of Outliers

Phase retrieval has been mainly considered in the presence of Gaussian noise. However, the performance of the algorithms proposed under the Gaussian noise model severely degrades when grossly corrupted data, i.e., outliers, exist. This paper investigates techniques for phase retrieval in the presence of heavy-tailed noise -- which is considered a better model for situations where outliers exist. An $\ell_p$-norm ($0<p<2$) based estimator is proposed for fending against such noise, and two-block inexact alternating optimization is proposed as the algorithmic framework to tackle the resulting optimization problem. Two specific algorithms are devised by exploring different local approximations within this framework. Interestingly, the core conditional minimization steps can be interpreted as iteratively reweighted least squares and gradient descent. Convergence properties of the algorithms are discussed, and the Cram\'er-Rao bound (CRB) is derived. Simulations demonstrate that the proposed algorithms approach the CRB and outperform state-of-the-art algorithms in heavy-tailed noise.

Robust sequential learning of feedforward neural networks in the presence of heavy-tailed noise

Feedforward neural networks (FFNN) are among the most used neural networks for modeling of various nonlinear problems in engineering. In sequential and especially real time processing all neural networks models fail when faced with outliers. Outliers are found across a wide range of engineering problems. Recent research results in the field have shown that to avoid overfitting or divergence of the model, new approach is needed especially if FFNN is to run sequentially or in real time. To accommodate limitations of FFNN when training data contains a certain number of outliers, this paper presents new learning algorithm based on improvement of conventional extended Kalman filter (EKF). Extended Kalman filter robust to outliers (EKF-OR) is probabilistic generative model in which measurement noise covariance is not constant; the sequence of noise measurement covariance is modeled as stochastic process over the set of symmetric positive-definite matrices in which prior is modeled as inverse Wishart distribution. In each iteration EKF-OR simultaneously estimates noise estimates and current best estimate of FFNN parameters. Bayesian framework enables one to mathematically derive expressions, while analytical intractability of the Bayes’ update step is solved by using structured variational approximation. All mathematical expressions in the paper are derived using the first principles. Extensive experimental study shows that FFNN trained with developed learning algorithm, achieves low prediction error and good generalization quality regardless of outliers’ presence in training data.

Robust linear least squares regression

We consider the problem of robustly predicting as well as the best linear combination of d given functions in least squares regression, and variants of this problem including constraints on the parameters of the linear combination. For the ridge estimator and the ordinary least squares estimator, and their variants, we provide new risk bounds of order d/n without logarithmic factor unlike some standard results, where n is the size of the training data. We also provide a new estimator with better deviations in the presence of heavy-tailed noise. It is based on truncating differences of losses in a min–max framework and satisfies a d/n risk bound both in expectation and in deviations. The key common surprising factor of these results is the absence of exponential moment condition on the output distribution while achieving exponential deviations. All risk bounds are obtained through a PAC-Bayesian analysis on truncated differences of losses. Experimental results strongly back up our truncated min–max estimator.

Polarity-Coincidence-Array Based Spectrum Sensing for Multiple Antenna Cognitive Radios in the Presence of Non-Gaussian Noise

One of the main requirements of cognitive radio (CR) systems is the ability to perform spectrum sensing in a reliable manner in challenging environments that arise due to propagation channels which undergo multipath fading and non-Gaussian noise. While most existing literature on spectrum sensing has focused on impairments introduced by additive white Gaussian noise (AWGN), this assumption fails to model the behavior of certain types of noise found in practice. In this paper, the use of a non-parametric and easily implementable detection device, namely the polarity-coincidence-array (PCA) detector, is proposed for the detection of weak primary signals with a cognitive radio equipped with multiple antennas. Its performance is evaluated in the presence of heavy-tailed noise. The detector performance in terms of the probabilities of detection and false alarm is derived when the communication channels between the primary user transmitter and the multiple antennas at the cognitive radio are AWGN as well as when they undergo Rayleigh fading. From the numerical results, it is observed that a significant performance enhancement is achieved by the PCA detector compared to that of the energy detector with AWGN as well as fading channels as the heaviness of the tail of the non-Gaussian noise increases.