I am a Ph.D. student at MPI Tuebingen and ETH Zurich, and supervised by Prof. Jeff Bilmes. For my Diplom (Master), I studied Bioinformatics in Tuebingen and Austin. I am particularly interested in

• submodular functions
• more generally, discrete and combinatorial problems in machine learning and computer vision
• (approximate) optimization (that is, algorithms that are useable and also come with some theoretical guarantees)
• online learning
• kernel methods, e.g. for clustering and ICA
• theoretical aspects of clustering
• applications in computer vision and biology

I have co-organized the NIPS workshop on Discrete Optimization in Machine Learning and given invited talks at the COSA workshop at the Technical University in Munich, Cornell University, TTI, UC Berkeley and TU Munich.

Below is a summary of projects and links to code.

Structured problems with submodular cost

Our generic setting is as follows: take a combinatorial optimization problem, and replace the usual sum-of-weights cost by a nonlinear, submodular function. This makes the problem much harder but has very useful applications and relations to questions in machine learning. In particular, we worked on this setting with graph cuts. The resulting cooperative cuts find applications wherever graph cuts can be used: for example in computer vision (e.g. for image segmentation), approximate inference, or approximate faster minimization of submodular functions. (Download data/code for image segmentation with cooperative cuts)
In addition, we derived online algorithms for structured sequential decision problems (e.g., online spanning tree, or s-t cut) with submodular cost (in contrast, almost all previous results are for linear costs). (All this was work with Jeff Bilmes.)
During our work, we came across several functions that are actually not submodular (but maybe some people hoped them to be), and thus I am collecting those functions in a bag of submodular non-examples.

Clustering & Graph Cuts

Nearest Neighbor Clustering: we developed a generic algorithm to minimize popular clustering objective functions, such as Normalized Cut or k-means. This method is consistent, thanks to pooling neighboring points. (with Ulrike von Luxburg, Sebastien Bubeck, Michael Kaufmann).
Download Demo Code

Generalized Clustering via Kernel Embeddings: The concept of clustering can be generalized to finding distributions that are maximally separated. The standard case is to measure separation by means of the distribution (k-means). MMD, the maximum mean discrepancy in a feature space, allows to also take higher-order moments into account. As a result, it is possible to e.g. separate two Gaussians with the same mean but different variance. Interestingly, maximizing MMD is closely related to kernel k-means and various other clustering criteria. (with Ulrike von Luxburg, Bernhard Schoelkopf, Arthur Gretton, Bharath Sriperumbudur)

Approximation Algorithms for Tensor Clustering: We prove an approximation factor for tensor clustering (i.e., "cutting a big cube into little cubes") for maybe the simplest possible algorithm. Various divergences are possible, such as Euclidean distance, Bregman divergences etc. (with Suvrit Sra, Arindam Banerjee)

Solution Stability in Linear Programming Relaxations: Graph Partitioning and Unsupervised Learning. Data is often noisy. If there are several good solutions to an optimization problem (here, graph partitioning), then the noise can be the determining part to make one solution optimal. However, how much do we want to trust this solution, if a small perturbation in the data suddenly makes another solution the best? this work proposes a method to test the stability of an optimal graph cut solution to perturbation of the edge weights (i.e., noise).We also show that several common clustering and graph partitioning objectives fall in our framework. (with Sebastian Nowozin)

Kernel Independent Component Analysis

Fast Kernel ICA: We developed a Newton-like method for kernel ICA that uses HSIC as the independence criterion. The algorithm is faster than previous gradient descent algorithms, and retains the advantages of Kernel ICA (with Hao Shen, Arthur Gretton).
Download Code for Fast kernel ICA

available upon request