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This tutorial paper surveys provably optimal alternatives to end-to-end backpropagation (E2EBP) — the de facto standard for training deep architectures. Modular training refers to strictly local training without both the forward and the backward pass, i.e., dividing a deep architecture into several nonoverlapping modules and training them separately without any end-to-end operation. Between the fully global E2EBP and the strictly local modular training, there are weakly modular hybrids performing training without the backward pass only. These alternatives can match or surpass the performance of E2EBP on challenging datasets such as ImageNet, and are gaining increasing attention primarily because they offer practical advantages over E2EBP, which will be enumerated herein. In particular, they allow for greater modularity and transparency in deep learning workflows, aligning deep learning with the mainstream computer science engineering that heavily exploits modularization for scalability. Modular training has also revealed novel insights about learning and has further implications on other important research domains. Specifically, it induces natural and effective solutions to some important practical problems such as data efficiency and transferability estimation. [ABSTRACT FROM AUTHOR]

To date, most linear and nonlinear Kalman filters (KFs) have been developed under the Gaussian assumption and the well-known minimum mean square error (MMSE) criterion. In order to improve the robustness with respect to impulsive (or heavy-tailed) non-Gaussian noises, the maximum correntropy criterion (MCC) has recently been used to replace the MMSE criterion in developing several robust Kalman-type filters. To deal with more complicated non-Gaussian noises such as noises from multimodal distributions, in this article, we develop a new Kalman-type filter, called minimum error entropy KF (MEE-KF), by using the minimum error entropy (MEE) criterion instead of the MMSE or MCC. Similar to the MCC-based KFs, the proposed filter is also an online algorithm with the recursive process, in which the propagation equations are used to give prior estimates of the state and covariance matrix, and a fixed-point algorithm is used to update the posterior estimates. In addition, the MEE extended KF (MEE-EKF) is also developed for performance improvement in the nonlinear situations. The high accuracy and strong robustness of MEE-KF and MEE-EKF are confirmed by experimental results. [ABSTRACT FROM AUTHOR]

Recently, maximum correntropy criterion (MCC) has been widely and successfully used in robust signal processing and machine learning, in which the correntropy is maximized instead of minimizing the popular mean square error (MSE) to improve the robustness with respect to outliers or impulsive noises. A lot of efforts have been devoted to derive different adaptive algorithms under MCC, but to date, little insight has been gained as to how the MCC solution will be influenced by outliers. In this paper, we investigate this problem and our focus is mainly on the parameter estimation of a simple linear errors-in-variables (EIVs) model with scalar variables. Under some conditions, we derive an upper bound on the absolute value of the estimation error and show that the MCC solution can get very close to the true value of the unknown parameter even with arbitrarily large outliers in both the input and output variables. Illustrative examples are provided to verify and clarify the theory. [ABSTRACT FROM AUTHOR]

The matrix-based Rényi's α-order entropy functional was recently introduced using the normalized eigenspectrum of a Hermitian matrix of the projected data in a reproducing kernel Hilbert space (RKHS). However, the current theory in the matrix-based Rényi's α-order entropy functional only defines the entropy of a single variable or mutual information between two random variables. In information theory and machine learning communities, one is also frequently interested in multivariate information quantities, such as the multivariate joint entropy and different interactive quantities among multiple variables. In this paper, we first define the matrix-based Rényi's α-order joint entropy among multiple variables. We then show how this definition can ease the estimation of various information quantities that measure the interactions among multiple variables, such as interactive information and total correlation. We finally present an application to feature selection to show how our definition provides a simple yet powerful way to estimate a widely-acknowledged intractable quantity from data. A real example on hyperspectral image (HSI) band selection is also provided. [ABSTRACT FROM AUTHOR]

Goal: We demonstrate an algorithm to automatically learn the time-limited waveforms associated with phasic events that repeatedly appear throughout an electroencephalogram. Methods: To learn the phasic event waveforms we propose a multiscale modeling process that is based on existing shift-invariant dictionary learning algorithms. For each channel, waveforms at different temporal scales are learned based on the assumption that only a few waveforms occur in any window of the time-series, but the same waveforms reoccur throughout the signal. Once the waveforms are learned, the timing and amplitude of the phasic event occurrences are estimated using matching pursuit. To summarize the waveforms learned across multiple channels and subjects, we analyze their frequency content, their similarity to Gabor–Morlet wavelets, and perform shift-invariant k-means to cluster the waveforms. A prototype waveform from each cluster is then tested for differential spatial patterns between different motor imagery conditions. Results: On multiple human EEG datasets, the learned waveforms capture key characteristics of signals they were trained to represent, with a consistency in waveform morphology and frequency content across multiple training sections and initializations. On multichannel datasets, the spatial amplitude patterns of the waveforms are also consistent and can be used to distinguish different modalities of motor imagery. Conclusion: We explored a methodology that can be used for modeling the recurrent waveforms in EEG traces. Significance: The methodology automatically identifies the most frequent phasic event waveforms in EEG, which could then be used as features for automatic evaluation and comparison of EEG during sleep, pathology, or mentally engaging tasks. [ABSTRACT FROM PUBLISHER]

Information theory provides principled ways to analyze different inference and learning problems, such as hypothesis testing, clustering, dimensionality reduction, classification, and so forth. However, the use of information theoretic quantities as test statistics, that is, as quantities obtained from empirical data, poses a challenging estimation problem that often leads to strong simplifications, such as Gaussian models, or the use of plug in density estimators that are restricted to certain representation of the data. In this paper, a framework to nonparametrically obtain measures of entropy directly from data using operators in reproducing kernel Hilbert spaces defined by infinitely divisible kernels is presented. The entropy functionals, which bear resemblance with quantum entropies, are defined on positive definite matrices and satisfy similar axioms to those of Renyi’s definition of entropy. Convergence of the proposed estimators follows from concentration results on the difference between the ordered spectrum of the Gram matrices and the integral operators associated to the population quantities. In this way, capitalizing on both the axiomatic definition of entropy and on the representation power of positive definite kernels, the proposed measure of entropy avoids the estimation of the probability distribution underlying the data. Moreover, estimators of kernel-based conditional entropy and mutual information are also defined. Numerical experiments on independence tests compare favorably with state-of-the-art. [ABSTRACT FROM PUBLISHER]

The minimum error entropy (MEE) criterion has been successfully used in fields such as parameter estimation, system identification and the supervised machine learning. There is in general no explicit expression for the optimal MEE estimate unless some constraints on the conditional distribution are imposed. A recent paper has proved that if the conditional density is conditionally symmetric and unimodal (CSUM), then the optimal MEE estimate (with Shannon entropy) equals the conditional median. In this study, we extend this result to the generalized MEE estimation where the optimality criterion is the Renyi entropy or equivalently, the a-order information potential (IP). [ABSTRACT FROM AUTHOR]

The minimum error entropy (MEE) estimation is concerned with the estimation of a certain random variable (unknown variable) based on another random variable (observation), so that the entropy of the estimation error is minimized. This estimation method may outperform the well-known minimum mean square error (MMSE) estimation especially for non-Gaussian situations. There is an important performance bound on the MEE estimation, namely the W-S lower bound, which is computed as the conditional entropy of the unknown variable given observation. Though it has been known in the literature for a considerable time, up to now there is little study on this performance bound. In this paper, we reexamine the W-S lower bound. Some basic properties of the W-S lower bound are presented, and the characterization of Gaussian distribution using the W-S lower bound is investigated. [ABSTRACT FROM AUTHOR]

The minimum error entropy (MEE) criterion has been receiving increasing attention due to its promising perspectives for applications in signal processing and machine learning. In the context of Bayesian estimation, the MEE criterion is concerned with the estimation of a certain random variable based on another random variable, so that the error's entropy is minimized. Several theoretical results on this topic have been reported. In this work, we present some further results on the MEE estimation. The contributions are twofold: (1) we extend a recent result on the minimum entropy of a mixture of unimodal and symmetric distributions to a more general case, and prove that if the conditional distributions are generalized uniformly dominated (GUD), the dominant alignment will be the MEE estimator; (2) we show by examples that the MEE estimator (not limited to singular cases) may be non-unique even if the error distribution is restricted to zero-mean (unbiased). [ABSTRACT FROM AUTHOR]

A novel spatiotemporal filtering method for single trial estimation of event-related potential (ERP) subcomponents is proposed here. Unlike some previous works in ERP estimation [1], refid="ref2" , the proposed method is able to estimate temporally correlated ERP subcomponents such as P3a and P3b. A new cost function is, therefore, defined which can deflate one of the correlated subcomponents. The method is applied to both simulated and real data and has shown to perform very well even in low signal-to-noise ratio situations. In addition, the method is compared to spatial principal component analysis and its superiority has been confirmed by using simulated signals. The approach can be especially useful in mental fatigue analysis where the relative variability of P300 subcomponents is the key factor in detecting the level of fatigue. [ABSTRACT FROM AUTHOR]