On the Growth of Sobolev Norms for the Nonlinear Schrödinger Equation on Tori and Boundary Unique Continuation for Elliptic PDE

This dissertation is composed of two parts. The first part applies techniques from Harmonic and nonlinear Fourier Analysis to the nonlinear Schrödinger equation, and therefore tools from the study of Dispersive Partial Differential Equations (PDEs) will also be employed. The dissertation will apply the $\ell^2$ decoupling conjecture, proved recently by Bourgain and Demeter, to prove polynomial bounds on the growth of Sobolev norms of solutions to polynomial nonlinear Schrödinger equations. The first bound which is obtained applies to the cubic nonlinear Schrödinger equation and yields an improved bound for irrational tori in dimensions 2 and 3. For the 4 dimensional case an argument of Deng that relies on the upside-down I-method and long time Strichartz estimates on generic irrational tori is applied to get bounds in for the energy-critical nonlinear Schrödinger equation, assuming small energy. The second part of the thesis proves unique continuation at the boundary for solutions to a class of degenerate elliptic PDEs. Specifically, there is a weight at each point on the domain which is bounded by some fractional power of the distance to the boundary. Techniques from Calculus of Variations and Geometric PDEs are used to show that solutions to this class of PDEs are uniquely defined simply by specifying their values and some notion of their normal derivative on an open set with positive measure on the boundary

An Evaluative Measure of Clustering Methods Incorporating Hyperparameter Sensitivity

Clustering algorithms are often evaluated using metrics which compare with ground-truth cluster assignments, such as Rand index and NMI. Algorithm performance may vary widely for different hyperparameters, however, and thus model selection based on optimal performance for these metrics is discordant with how these algorithms are applied in practice, where labels are unavailable and tuning is often more art than science. It is therefore desirable to compare clustering algorithms not only on their optimally tuned performance, but also some notion of how realistic it would be to obtain this performance in practice. We propose an evaluation of clustering methods capturing this ease-of-tuning by modeling the expected best clustering score under a given computation budget. To encourage the adoption of the proposed metric alongside classic clustering evaluations, we provide an extensible benchmarking framework. We perform an extensive empirical evaluation of our proposed metric on popular clustering algorithms over a large collection of datasets from different domains, and observe that our new metric leads to several noteworthy observations

Word2Box: Capturing Set-Theoretic Semantics of Words using Box Embeddings

Learning representations of words in a continuous space is perhaps the most fundamental task in NLP, however words interact in ways much richer than vector dot product similarity can provide. Many relationships between words can be expressed set-theoretically, for example, adjective-noun compounds (eg. "red cars"$\subseteq$"cars") and homographs (eg. "tongue"$\cap$"body" should be similar to "mouth", while "tongue"$\cap$"language" should be similar to "dialect") have natural set-theoretic interpretations. Box embeddings are a novel region-based representation which provide the capability to perform these set-theoretic operations. In this work, we provide a fuzzy-set interpretation of box embeddings, and learn box representations of words using a set-theoretic training objective. We demonstrate improved performance on various word similarity tasks, particularly on less common words, and perform a quantitative and qualitative analysis exploring the additional unique expressivity provided by Word2Box