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

Discovering robust dependencies from data

Science revolves around forming hypotheses, designing experiments, collecting data, and tests. It was not until recently, with the advent of modern hardware and data analytics, that science shifted towards a big-data-driven paradigm that led to an unprecedented success across various fields. What is perhaps the most astounding feature of this new era, is that interesting hypotheses can now be automatically discovered from observational data. This dissertation investigates knowledge discovery procedures that do exactly this. In particular, we seek algorithms that discover the most informative models able to compactly “describe” aspects of the phenomena under investigation, in both supervised and unsupervised settings. We consider interpretable models in the form of subsets of the original variable set. We want the models to capture all possible interactions, e.g., linear, non-linear, between all types of variables, e.g., discrete, continuous, and lastly, we want their quality to be meaningfully assessed. For this, we employ information-theoretic measures, and particularly, the fraction of information for the supervised setting, and the normalized total correlation for the unsupervised. The former measures the uncertainty reduction of the target variable conditioned on a model, and the latter measures the information overlap of the variables included in a model. Without access to the true underlying data generating process, we estimate the aforementioned measures from observational data. This process is prone to statistical errors, and in our case, the errors manifest as biases towards larger models. This can lead to situations where the results are utterly random, hindering therefore further analysis. We correct this behavior with notions from statistical learning theory. In particular, we propose regularized estimators that are unbiased under the hypothesis of independence, leading to robust estimation from limited data samples and arbitrary dimensionalities. Moreover, we do this for models consisting of both discrete and continuous variables. Lastly, to discover the top scoring models, we derive effective optimization algorithms for exact, approximate, and heuristic search. These algorithms are powered by admissible, tight, and efficient-to-compute bounding functions for our proposed estimators that can be used to greatly prune the search space. Overall, the products of this dissertation can successfully assist data analysts with data exploration, discovering powerful description models, or concluding that no satisfactory models exist, implying therefore new experiments and data are required for the phenomena under investigation. This statement is supported by Materials Science researchers who corroborated our discoveries.In der Wissenschaft geht es um Hypothesenbildung, Entwerfen von Experimenten, Sammeln von Daten und Tests. Jüngst hat sich die Wissenschaft, durch das Aufkommen moderner Hardware und Datenanalyse, zu einem Big-Data-basierten Paradigma hin entwickelt, das zu einem beispiellosen Erfolg in verschiedenen Bereichen geführt hat. Ein erstaunliches Merkmal dieser neuen ra ist, dass interessante Hypothesen jetzt automatisch aus Beobachtungsdaten entdeckt werden k nnen. In dieser Dissertation werden Verfahren zur Wissensentdeckung untersucht, die genau dies tun. Insbesondere suchen wir nach Algorithmen, die Modelle identifizieren, die in der Lage sind, Aspekte der untersuchten Ph nomene sowohl in beaufsichtigten als auch in unbeaufsichtigten Szenarien kompakt zu “beschreiben”. Hierzu betrachten wir interpretierbare Modelle in Form von Untermengen der ursprünglichen Variablenmenge. Ziel ist es, dass diese Modelle alle m glichen Interaktionen erfassen (z.B. linear, nicht-lineare), zwischen allen Arten von Variablen unterscheiden (z.B. diskrete, kontinuierliche) und dass schlussendlich ihre Qualit t sinnvoll bewertet wird. Dazu setzen wir informationstheoretische Ma e ein, insbesondere den Informationsanteil für das überwachte und die normalisierte Gesamtkorrelation für das unüberwachte Szenario. Ersteres misst die Unsicherheitsreduktion der Zielvariablen, die durch ein Modell bedingt ist, und letztere misst die Informationsüberlappung der enthaltenen Variablen. Ohne Kontrolle des Datengenerierungsprozesses werden die oben genannten Ma e aus Beobachtungsdaten gesch tzt. Dies ist anf llig für statistische Fehler, die zu Verzerrungen in gr  eren Modellen führen. So entstehen Situationen, wobei die Ergebnisse v llig zuf llig sind und somit weitere Analysen st ren. Wir korrigieren dieses Verhalten mit Methoden aus der statistischen Lerntheorie. Insbesondere schlagen wir regularisierte Sch tzer vor, die unter der Hypothese der Unabh ngigkeit nicht verzerrt sind und somit zu einer robusten Sch tzung aus begrenzten Datenstichproben und willkürlichen-Dimensionalit ten führen. Darüber hinaus wenden wir dies für Modelle an, die sowohl aus diskreten als auch aus kontinuierlichen Variablen bestehen. Um die besten Modelle zu entdecken, leiten wir effektive Optimierungsalgorithmen mit verschiedenen Garantien ab. Diese Algorithmen basieren auf speziellen Begrenzungsfunktionen der vorgeschlagenen Sch tzer und erlauben es den Suchraum stark einzuschr nken. Insgesamt sind die Produkte dieser Arbeit sehr effektiv für die Wissensentdeckung. Letztere Aussage wurde von Materialwissenschaftlern best tigt

Detecting Baryon Acoustic Oscillations with Third-generation Gravitational Wave Observatories

We explore the possibility of detecting baryon acoustic oscillations (BAO) solely from gravitational wave (GW) observations of binary neutron star mergers with third-generation (3G) GW detectors such as the Cosmic Explorer and the Einstein Telescope. These measurements would provide a new independent probe of cosmology. The detection of the BAO peak with current-generation GW detectors (solely from GW observations) is not possible because i) unlike galaxies, the GW mergers are poorly localized, and ii) there are not enough merger events to probe the BAO length scale. With the 3G GW detector network, it is possible to observe 1/4 (1000) binary neutron star mergers per year that are localized well within one square degree in the sky for redshift z ≤ 0.3. We show that 3G observatories will enable precision measurements of the BAO feature in the large-scale two-point correlation function; the effect of BAO can be independently detected at different redshifts, with a log-evidence ratio of 1/423, 17, or 3, favoring a model with a BAO peak at redshift of 0.2, 0.25, or 0.3, respectively, using a redshift bin corresponding to a shell of thickness 150h -1 Mpc. © 2022. The Author(s). Published by the American Astronomical Society

Quantum machine learning: a classical perspective

Recently, increased computational power and data availability, as well as algorithmic advances, have led machine learning techniques to impressive results in regression, classification, data-generation and reinforcement learning tasks. Despite these successes, the proximity to the physical limits of chip fabrication alongside the increasing size of datasets are motivating a growing number of researchers to explore the possibility of harnessing the power of quantum computation to speed-up classical machine learning algorithms. Here we review the literature in quantum machine learning and discuss perspectives for a mixed readership of classical machine learning and quantum computation experts. Particular emphasis will be placed on clarifying the limitations of quantum algorithms, how they compare with their best classical counterparts and why quantum resources are expected to provide advantages for learning problems. Learning in the presence of noise and certain computationally hard problems in machine learning are identified as promising directions for the field. Practical questions, like how to upload classical data into quantum form, will also be addressed.Comment: v3 33 pages; typos corrected and references adde