10 research outputs found
Tight Continuous Relaxation of the Balanced -Cut Problem
Spectral Clustering as a relaxation of the normalized/ratio cut has become
one of the standard graph-based clustering methods. Existing methods for the
computation of multiple clusters, corresponding to a balanced -cut of the
graph, are either based on greedy techniques or heuristics which have weak
connection to the original motivation of minimizing the normalized cut. In this
paper we propose a new tight continuous relaxation for any balanced -cut
problem and show that a related recently proposed relaxation is in most cases
loose leading to poor performance in practice. For the optimization of our
tight continuous relaxation we propose a new algorithm for the difficult
sum-of-ratios minimization problem which achieves monotonic descent. Extensive
comparisons show that our method outperforms all existing approaches for ratio
cut and other balanced -cut criteria.Comment: Long version of paper accepted at NIPS 201
Neural forecasting: Introduction and literature overview
Neural network based forecasting methods have become ubiquitous in
large-scale industrial forecasting applications over the last years. As the
prevalence of neural network based solutions among the best entries in the
recent M4 competition shows, the recent popularity of neural forecasting
methods is not limited to industry and has also reached academia. This article
aims at providing an introduction and an overview of some of the advances that
have permitted the resurgence of neural networks in machine learning. Building
on these foundations, the article then gives an overview of the recent
literature on neural networks for forecasting and applications.Comment: 66 pages, 5 figure
Graph-basierte Methoden zur unüberwachten und teilüberwachten Datenanalyse
Clustering and community detection are two important problems in data analysis with applications in various disciplines. Often in practice, there exists prior knowledge that helps the process of data analysis. In this thesis we develop generic graph-based methods for these data analysis problems both in unsupervised and semi-supervised settings. The main advantage of our methods is that they provide a common framework for integrating soft as well as hard prior knowledge. In the latter case, ours is the first method to have provable guarantees on the satisfaction of the given prior knowledge. The foundation of our methods is the exact continuous relaxation result that we derive for a class of combinatorial optimization problems. More specifically, we show that the (constrained) minimization of a ratio of set functions can be equivalently rewritten as a continuous optimization problem. We also present efficient algorithms for solving the continuous relaxations. While the global optimality is not guaranteed, in practice our methods consistently outperform the corresponding convex or spectral relaxations by a large margin. Moreover, our method has an additional guarantee that the solution respects the prior knowledge.Clustering und Community Detection sind zwei bedeutende Probleme in der Datenanalyse, mit vielfältigen Anwendungen in unterschiedlichen Bereichen. In der Praxis existiert häufig Vorwissen das in den Prozess der Datenanalyse einfließen kann. In dieser Arbeit entwickeln wir generische Graph-basierte Methoden für diese Problemstellungen der Datenanalyse, sowohl für den unüberwachten als auch den teilüberwachten Fall. Der Hauptvorteil unserer Verfahren ist dass sie ein allgemeines Framework zur Integration von weichen und harten Nebenbedingungen bereitstellen. In letzterem Fall ist unsere Methode die erste die beweisbare Garantien zur Erfüllung des gegebenen Vorwissen liefern kann. Die Grundlage unserer Methoden ist ein Resultat über exakte kontinuierliche Relaxierungen das wir für eine Klasse von kombinatorischen Optimierungsproblemen herleiten. Konkret zeigen wir dass die (beschränkte) Minimierung eines Bruches von Mengenfunktionen in ein äquivalentes kontinuierliches Optimierungsproblem umgeformt werden kann. Des Weiteren präsentieren wir effiziente Algorithmen zur Lösung der kontinuierlichen Relaxierungen. Während die globale Optimalität nicht garantiert werden kann, werden die entsprechenden konvexen oder spektralen Relaxierungen in der Praxis mit einem deutlichen Vorsprung übertroffen. Darüber hinaus hat unsere Methode eine zusätzliche Garantie dass die berechnete Lösung das Vorwissen stets berücksichtigt
Constrained fractional set programs and their application in local clustering and community detection
local clustering and community detectio
The Total Variation on Hypergraphs - Learning on Hypergraphs Revisited
Hypergraphs allow one to encode higher-order relationships in data and are
thus a very flexible modeling tool. Current learning methods are either based
on approximations of the hypergraphs via graphs or on tensor methods which are
only applicable under special conditions. In this paper, we present a new
learning framework on hypergraphs which fully uses the hypergraph structure.
The key element is a family of regularization functionals based on the total
variation on hypergraphs.Comment: Long version of paper accepted at NIPS 201