Finding Statistically Significant Interactions between Continuous Features

The search for higher-order feature interactions that are statistically significantly associated with a class variable is of high relevance in fields such as Genetics or Healthcare, but the combinatorial explosion of the candidate space makes this problem extremely challenging in terms of computational efficiency and proper correction for multiple testing. While recent progress has been made regarding this challenge for binary features, we here present the first solution for continuous features. We propose an algorithm which overcomes the combinatorial explosion of the search space of higher-order interactions by deriving a lower bound on the p-value for each interaction, which enables us to massively prune interactions that can never reach significance and to thereby gain more statistical power. In our experiments, our approach efficiently detects all significant interactions in a variety of synthetic and real-world datasets.Comment: 13 pages, 5 figures, 2 tables, accepted to the 28th International Joint Conference on Artificial Intelligence (IJCAI 2019

Significant Subgraph Mining with Multiple Testing Correction

The problem of finding itemsets that are statistically significantly enriched in a class of transactions is complicated by the need to correct for multiple hypothesis testing. Pruning untestable hypotheses was recently proposed as a strategy for this task of significant itemset mining. It was shown to lead to greater statistical power, the discovery of more truly significant itemsets, than the standard Bonferroni correction on real-world datasets. An open question, however, is whether this strategy of excluding untestable hypotheses also leads to greater statistical power in subgraph mining, in which the number of hypotheses is much larger than in itemset mining. Here we answer this question by an empirical investigation on eight popular graph benchmark datasets. We propose a new efficient search strategy, which always returns the same solution as the state-of-the-art approach and is approximately two orders of magnitude faster. Moreover, we exploit the dependence between subgraphs by considering the effective number of tests and thereby further increase the statistical power.Comment: 18 pages, 5 figure, accepted to the 2015 SIAM International Conference on Data Mining (SDM15

Graph Kernels

We present a unified framework to study graph kernels, special cases of which include the random walk (Gärtner et al., 2003; Borgwardt et al., 2005) and marginalized (Kashima et al., 2003, 2004; Mahé et al., 2004) graph kernels. Through reduction to a Sylvester equation we improve the time complexity of kernel computation between unlabeled graphs with n vertices from O(n^6) to O(n^3). We find a spectral decomposition approach even more efficient when computing entire kernel matrices. For labeled graphs we develop conjugate gradient and fixed-point methods that take O(dn^3) time per iteration, where d is the size of the label set. By extending the necessary linear algebra to Reproducing Kernel Hilbert Spaces (RKHS) we obtain the same result for d-dimensional edge kernels, and O(n^4) in the infinite-dimensional case; on sparse graphs these algorithms only take O(n^2) time per iteration in all cases. Experiments on graphs from bioinformatics and other application domains show that these techniques can speed up computation of the kernel by an order of magnitude or more. We also show that certain rational kernels (Cortes et al., 2002, 2003, 2004) when specialized to graphs reduce to our random walk graph kernel. Finally, we relate our framework to R-convolution kernels (Haussler, 1999) and provide a kernel that is close to the optimal assignment kernel of Fröhlich et al. (2006) yet provably positive semi-definite

Efficient network-guided multi-locus association mapping with graph cuts

As an increasing number of genome-wide association studies reveal the limitations of attempting to explain phenotypic heritability by single genetic loci, there is growing interest for associating complex phenotypes with sets of genetic loci. While several methods for multi-locus mapping have been proposed, it is often unclear how to relate the detected loci to the growing knowledge about gene pathways and networks. The few methods that take biological pathways or networks into account are either restricted to investigating a limited number of predetermined sets of loci, or do not scale to genome-wide settings. We present SConES, a new efficient method to discover sets of genetic loci that are maximally associated with a phenotype, while being connected in an underlying network. Our approach is based on a minimum cut reformulation of the problem of selecting features under sparsity and connectivity constraints that can be solved exactly and rapidly. SConES outperforms state-of-the-art competitors in terms of runtime, scales to hundreds of thousands of genetic loci, and exhibits higher power in detecting causal SNPs in simulation studies than existing methods. On flowering time phenotypes and genotypes from Arabidopsis thaliana, SConES detects loci that enable accurate phenotype prediction and that are supported by the literature. Matlab code for SConES is available at http://webdav.tuebingen.mpg.de/u/karsten/Forschung/scones/Comment: 20 pages, 6 figures, accepted at ISMB (International Conference on Intelligent Systems for Molecular Biology) 201

Statistical Tests for Detecting Differential RNA-Transcript Expression from Read Counts

As a fruit of the current revolution in sequencing technology, transcriptomes can now be analyzed at an unprecedented level of detail. These advances have been exploited for detecting differential expressed genes across biological samples and for quantifying the abundances of various RNA transcripts within one gene. However, explicit strategies for detecting the hidden differential abundances of RNA transcripts in biological samples have not been defined. In this work, we present two novel statistical tests to address this issue: a 'gene structure sensitive' Poisson test for detecting differential expression when the transcript structure of the gene is known, and a kernel-based test called Maximum Mean Discrepancy when it is unknown. We analyzed the proposed approaches on simulated read data for two artificial samples as well as on factual reads generated by the Illumina Genome Analyzer for two _C. elegans_ samples. Our analysis shows that the Poisson test identifies genes with differential transcript expression considerably better that previously proposed RNA transcript quantification approaches for this task. The MMD test is able to detect a large fraction (75%) of such differential cases without the knowledge of the annotated transcripts. It is therefore well-suited to analyze RNA-Seq experiments when the genome annotations are incomplete or not available, where other approaches have to fail

Graph Kernels

As new graph structured data is constantly being generated, learning and data mining on graphs have become a challenge in application areas such as molecular biology, telecommunications, chemoinformatics, and social network analysis. The central algorithmic problem in these areas, measuring similarity of graphs, has therefore received extensive attention in the recent past. Unfortunately, existing approaches are slow, lacking in expressivity, or hard to parameterize. Graph kernels have recently been proposed as a theoretically sound and promising approach to the problem of graph comparison. Their attractivity stems from the fact that by defining a kernel on graphs, a whole family of data mining and machine learning algorithms becomes applicable to graphs. These kernels on graphs must respect both the information represented by the topology and the node and edge labels of the graphs, while being efficient to compute. Existing methods fall woefully short; they miss out on important topological information, are plagued by runtime issues, and do not scale to large graphs. Hence the primary goal of this thesis is to make learning and data mining with graph kernels feasible. In the first half of this thesis, we review and analyze the shortcomings of state-of-the-art graph kernels. We then propose solutions to overcome these weaknesses. As highlights of our research, we - speed up the classic random walk graph kernel from O(n^6) to O(n^3), where n is the number of nodes in the larger graph, and by a factor of up to 1,000 in CPU runtime, by extending concepts from Linear Algebra to Reproducing Kernel Hilbert Spaces, - define novel graph kernels based on shortest paths that avoid tottering and outperform random walk kernels in accuracy, - define novel graph kernels that estimate the frequency of small subgraphs within a large graph and that work on large graphs hitherto not handled by existing graph kernels. In the second half of this thesis, we present algorithmic solutions to two novel problems in graph mining. First, we define a two-sample test on graphs. Given two sets of graphs, or a pair of graphs, this test lets us decide whether these graphs are likely to originate from the same underlying distribution. To solve this so-called two-sample-problem, we define the first kernel-based two-sample test. Combined with graph kernels, this results in the first two-sample test on graphs described in the literature. Second, we propose a principled approach to supervised feature selection on graphs. As in feature selection on vectors, feature selection on graphs aims at finding features that are correlated with the class membership of a graph. Towards this goal, we first define a family of supervised feature selection algorithms based on kernels and the Hilbert-Schmidt Independence Criterion. We then show how to extend this principle of feature selection to graphs, and how to combine it with gSpan, the state-of-the-art method for frequent subgraph mining. On several benchmark datasets, our novel procedure manages to select a small subset of dozens of informative features among thousands and millions of subgraphs detected by gSpan. In classification experiments, the features selected by our method outperform those chosen by other feature selectors in terms of classification accuracy. Along the way, we also solve several problems that can be deemed contributions in their own right: - We define a unifying framework for describing both variants of random walk graph kernels proposed in the literature. - We present the first theoretical connection between graph kernels and molecular descriptors from chemoinformatics. - We show how to determine sample sizes for estimating the frequency of certain subgraphs within a large graph with a given precision and confidence, which promises to be a key to the solution of important problems in data mining and bioinformatics. Three branches of computer science immediately benefit from our findings: data mining, machine learning, and bioinformatics. For data mining, our efficient graph kernels allow us to bring to bear the large family of kernel methods to mining problems on real-world graph data. For machine learning, we open the door to extend strong theoretical results on learning on graphs into useful practical applications. For bioinformatics, we make a number of principled kernel methods and efficient kernel functions available for biological network comparison, and structural comparisons of proteins. Apart from these three areas, other fields may also benefit from our findings, as our algorithms are general in nature and not restricted to a particular type of application