Improved Cheeger's Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap

Let \phi(G) be the minimum conductance of an undirected graph G, and let 0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, \phi(G) = O(k) \lambda_2 / \sqrt{\lambda_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if \lambda_k$ is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut

An Incremental Reseeding Strategy for Clustering

In this work we propose a simple and easily parallelizable algorithm for multiway graph partitioning. The algorithm alternates between three basic components: diffusing seed vertices over the graph, thresholding the diffused seeds, and then randomly reseeding the thresholded clusters. We demonstrate experimentally that the proper combination of these ingredients leads to an algorithm that achieves state-of-the-art performance in terms of cluster purity on standard benchmarks datasets. Moreover, the algorithm runs an order of magnitude faster than the other algorithms that achieve comparable results in terms of accuracy. We also describe a coarsen, cluster and refine approach similar to GRACLUS and METIS that removes an additional order of magnitude from the runtime of our algorithm while still maintaining competitive accuracy

Multiway Spectral Clustering: A Margin-Based Perspective

Spectral clustering is a broad class of clustering procedures in which an intractable combinatorial optimization formulation of clustering is "relaxed" into a tractable eigenvector problem, and in which the relaxed solution is subsequently "rounded" into an approximate discrete solution to the original problem. In this paper we present a novel margin-based perspective on multiway spectral clustering. We show that the margin-based perspective illuminates both the relaxation and rounding aspects of spectral clustering, providing a unified analysis of existing algorithms and guiding the design of new algorithms. We also present connections between spectral clustering and several other topics in statistics, specifically minimum-variance clustering, Procrustes analysis and Gaussian intrinsic autoregression.Comment: Published in at http://dx.doi.org/10.1214/08-STS266 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Segmentation of striatal brain structures from high resolution pet images

Dissertation presented at the Faculty of Science and Technology of the New University of Lisbon in fulfillment of the requirements for the Masters degree in Electrical Engineering and ComputersWe propose and evaluate fully automatic segmentation methods for the extraction of striatal brain surfaces (caudate, putamen, ventral striatum and white matter), from high resolution positron emission tomography (PET) images. In the preprocessing steps, both the right and the left striata were segmented from the high resolution PET images. This segmentation was achieved by delineating the brain surface, finding the plane that maximizes the reflective symmetry of the brain (mid-sagittal plane) and, finally, extracting the right and left striata from both hemisphere images. The delineation of the brain surface and the extraction of the striata were achieved using the DSM-OS (Surface Minimization – Outer Surface) algorithm. The segmentation of striatal brain surfaces from the striatal images can be separated into two sub-processes: the construction of a graph (named “voxel affinity matrix”) and the graph clustering. The voxel affinity matrix was built using a set of image features that accurately informs the clustering method on the relationship between image voxels. The features defining the similarity of pairwise voxels were spatial connectivity, intensity values, and Euclidean distances. The clustering process is treated as a graph partition problem using two methods, a spectral (multiway normalized cuts) and a non-spectral (weighted kernel k-means). The normalized cuts algorithm relies on the computation of the graph eigenvalues to partition the graph into connected regions. However, this method fails when applied to high resolution PET images due to the high computational requirements arising from the image size. On the other hand, the weighted kernel k-means classifies iteratively, with the aid of the image features, a given data set into a predefined number of clusters. The weighted kernel k-means and the normalized cuts algorithm are mathematically similar. After finding the optimal initial parameters for the weighted kernel k-means for this type of images, no further tuning is necessary for subsequent images. Our results showed that the putamen and ventral striatum were accurately segmented, while the caudate and white matter appeared to be merged in the same cluster. The putamen was divided in anterior and posterior areas. All the experiments resulted in the same type of segmentation, validating the reproducibility of our results

Generalized Network Dismantling

Finding the set of nodes, which removed or (de)activated can stop the spread of (dis)information, contain an epidemic or disrupt the functioning of a corrupt/criminal organization is still one of the key challenges in network science. In this paper, we introduce the generalized network dismantling problem, which aims to find the set of nodes that, when removed from a network, results in a network fragmentation into subcritical network components at minimum cost. For unit costs, our formulation becomes equivalent to the standard network dismantling problem. Our non-unit cost generalization allows for the inclusion of topological cost functions related to node centrality and non-topological features such as the price, protection level or even social value of a node. In order to solve this optimization problem, we propose a method, which is based on the spectral properties of a novel node-weighted Laplacian operator. The proposed method is applicable to large-scale networks with millions of nodes. It outperforms current state-of-the-art methods and opens new directions in understanding the vulnerability and robustness of complex systems.Comment: 6 pages, 5 figure