Correlation Clustering with Low-Rank Matrices

Correlation clustering is a technique for aggregating data based on qualitative information about which pairs of objects are labeled 'similar' or 'dissimilar.' Because the optimization problem is NP-hard, much of the previous literature focuses on finding approximation algorithms. In this paper we explore how to solve the correlation clustering objective exactly when the data to be clustered can be represented by a low-rank matrix. We prove in particular that correlation clustering can be solved in polynomial time when the underlying matrix is positive semidefinite with small constant rank, but that the task remains NP-hard in the presence of even one negative eigenvalue. Based on our theoretical results, we develop an algorithm for efficiently "solving" low-rank positive semidefinite correlation clustering by employing a procedure for zonotope vertex enumeration. We demonstrate the effectiveness and speed of our algorithm by using it to solve several clustering problems on both synthetic and real-world data

Automatic Speech Recognition for Low-resource Languages and Accents Using Multilingual and Crosslingual Information

This thesis explores methods to rapidly bootstrap automatic speech recognition systems for languages, which lack resources for speech and language processing. We focus on finding approaches which allow using data from multiple languages to improve the performance for those languages on different levels, such as feature extraction, acoustic modeling and language modeling. Under application aspects, this thesis also includes research work on non-native and Code-Switching speech

Local Guarantees in Graph Cuts and Clustering

Correlation Clustering is an elegant model that captures fundamental graph cut problems such as Min $s-t$ Cut, Multiway Cut, and Multicut, extensively studied in combinatorial optimization. Here, we are given a graph with edges labeled $+$ or $-$ and the goal is to produce a clustering that agrees with the labels as much as possible: $+$ edges within clusters and $-$ edges across clusters. The classical approach towards Correlation Clustering (and other graph cut problems) is to optimize a global objective. We depart from this and study local objectives: minimizing the maximum number of disagreements for edges incident on a single node, and the analogous max min agreements objective. This naturally gives rise to a family of basic min-max graph cut problems. A prototypical representative is Min Max $s-t$ Cut: find an $s-t$ cut minimizing the largest number of cut edges incident on any node. We present the following results: $(1)$ an $O(\sqrt{n})$-approximation for the problem of minimizing the maximum total weight of disagreement edges incident on any node (thus providing the first known approximation for the above family of min-max graph cut problems), $(2)$ a remarkably simple $7$-approximation for minimizing local disagreements in complete graphs (improving upon the previous best known approximation of $48$), and $(3)$ a $1/(2+\varepsilon)$-approximation for maximizing the minimum total weight of agreement edges incident on any node, hence improving upon the $1/(4+\varepsilon)$-approximation that follows from the study of approximate pure Nash equilibria in cut and party affiliation games

Empirical studies on word representations

One of the most fundamental tasks in natural language processing is representing words with mathematical objects (such as vectors). The word representations, which are most often estimated from data, allow capturing the meaning of words. They enable comparing words according to their semantic similarity, and have been shown to work extremely well when included in complex real-world applications. A large part of our work deals with ways of estimating word representations directly from large quantities of text. Our methods exploit the idea that words which occur in similar contexts have a similar meaning. How we define the context is an important focus of our thesis. The context can consist of a number of words to the left and to the right of the word in question, but, as we show, obtaining context words via syntactic links (such as the link between the verb and its subject) often works better. We furthermore investigate word representations that accurately capture multiple meanings of a single word. We show that translation of a word in context contains information that can be used to disambiguate the meaning of that word