Graph-based Estimation of Information Divergence Functions

abstract: Information divergence functions, such as the Kullback-Leibler divergence or the Hellinger distance, play a critical role in statistical signal processing and information theory; however estimating them can be challenge. Most often, parametric assumptions are made about the two distributions to estimate the divergence of interest. In cases where no parametric model fits the data, non-parametric density estimation is used. In statistical signal processing applications, Gaussianity is usually assumed since closed-form expressions for common divergence measures have been derived for this family of distributions. Parametric assumptions are preferred when it is known that the data follows the model, however this is rarely the case in real-word scenarios. Non-parametric density estimators are characterized by a very large number of parameters that have to be tuned with costly cross-validation. In this dissertation we focus on a specific family of non-parametric estimators, called direct estimators, that bypass density estimation completely and directly estimate the quantity of interest from the data. We introduce a new divergence measure, the $D_p$-divergence, that can be estimated directly from samples without parametric assumptions on the distribution. We show that the $D_p$-divergence bounds the binary, cross-domain, and multi-class Bayes error rates and, in certain cases, provides provably tighter bounds than the Hellinger divergence. In addition, we also propose a new methodology that allows the experimenter to construct direct estimators for existing divergence measures or to construct new divergence measures with custom properties that are tailored to the application. To examine the practical efficacy of these new methods, we evaluate them in a statistical learning framework on a series of real-world data science problems involving speech-based monitoring of neuro-motor disorders.Dissertation/ThesisDoctoral Dissertation Electrical Engineering 201

Practicality of generalization guarantees for unsupervised domain adaptation with neural networks

Understanding generalization is crucial to confidently engineer and deploy machine learningmodels, especially when deployment implies a shift in the data domain. For such domainadaptation problems, we seek generalization bounds which are tractably computable andtight. If these desiderata can be reached, the bounds can serve as guarantees for adequateperformance in deployment. However, in applications where deep neural networks are themodels of choice, deriving results which fulfill these remains an unresolved challenge; mostexisting bounds are either vacuous or has non-estimable terms, even in favorable conditions.In this work, we evaluate existing bounds from the literature with potential to satisfy ourdesiderata on domain adaptation image classification tasks, where deep neural networks arepreferred. We find that all bounds are vacuous and that sample generalization terms accountfor much of the observed looseness, especially when these terms interact with measures ofdomain shift. To overcome this and arrive at the tightest possible results, we combine eachbound with recent data-dependent PAC-Bayes analysis, greatly improving the guarantees.We find that, when domain overlap can be assumed, a simple importance weighting extensionof previous work provides the tightest estimable bound. Finally, we study which termsdominate the bounds and identify possible directions for further improvement

Equivalence between Time Series Predictability and Bayes Error Rate

Predictability is an emerging metric that quantifies the highest possible prediction accuracy for a given time series, being widely utilized in assessing known prediction algorithms and characterizing intrinsic regularities in human behaviors. Lately, increasing criticisms aim at the inaccuracy of the estimated predictability, caused by the original entropy-based method. In this brief report, we strictly prove that the time series predictability is equivalent to a seemingly unrelated metric called Bayes error rate that explores the lowest error rate unavoidable in classification. This proof bridges two independently developed fields, and thus each can immediately benefit from the other. For example, based on three theoretical models with known and controllable upper bounds of prediction accuracy, we show that the estimation based on Bayes error rate can largely solve the inaccuracy problem of predictability.Comment: 1 Figure, 1 Table, 5 Page

The intrinsic value of HFO features as a biomarker of epileptic activity

High frequency oscillations (HFOs) are a promising biomarker of epileptic brain tissue and activity. HFOs additionally serve as a prototypical example of challenges in the analysis of discrete events in high-temporal resolution, intracranial EEG data. Two primary challenges are 1) dimensionality reduction, and 2) assessing feasibility of classification. Dimensionality reduction assumes that the data lie on a manifold with dimension less than that of the feature space. However, previous HFO analyses have assumed a linear manifold, global across time, space (i.e. recording electrode/channel), and individual patients. Instead, we assess both a) whether linear methods are appropriate and b) the consistency of the manifold across time, space, and patients. We also estimate bounds on the Bayes classification error to quantify the distinction between two classes of HFOs (those occurring during seizures and those occurring due to other processes). This analysis provides the foundation for future clinical use of HFO features and buides the analysis for other discrete events, such as individual action potentials or multi-unit activity.Comment: 5 pages, 5 figure