7,007 research outputs found
Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs
Laplacian mixture models identify overlapping regions of influence in
unlabeled graph and network data in a scalable and computationally efficient
way, yielding useful low-dimensional representations. By combining Laplacian
eigenspace and finite mixture modeling methods, they provide probabilistic or
fuzzy dimensionality reductions or domain decompositions for a variety of input
data types, including mixture distributions, feature vectors, and graphs or
networks. Provable optimal recovery using the algorithm is analytically shown
for a nontrivial class of cluster graphs. Heuristic approximations for scalable
high-performance implementations are described and empirically tested.
Connections to PageRank and community detection in network analysis demonstrate
the wide applicability of this approach. The origins of fuzzy spectral methods,
beginning with generalized heat or diffusion equations in physics, are reviewed
and summarized. Comparisons to other dimensionality reduction and clustering
methods for challenging unsupervised machine learning problems are also
discussed.Comment: 13 figures, 35 reference
Binary search in graphs revisited
In the classical binary search in a path the aim is to detect an unknown target by asking as few queries as possible, where each query reveals the direction to the target. This binary search algorithm has been recently extended by Emamjomeh-Zadeh et al. (in: Proceedings of the 48th annual ACM SIGACT symposium on theory of computing, STOC 2016, Cambridge, pp. 519–532, 2016) to the problem of detecting a target in an arbitrary graph. Similarly to the classical case in the path, the algorithm of Emamjomeh-Zadeh et al. maintains a candidates’ set for the target, while each query asks an appropriately chosen vertex—the “median”—which minimises a potential Φ among the vertices of the candidates’ set. In this paper we address three open questions posed by Emamjomeh-Zadeh et al., namely (a) detecting a target when the query response is a direction to an approximately shortest path to the target, (b) detecting a target when querying a vertex that is an approximate median of the current candidates’ set (instead of an exact one), and (c) detecting multiple targets, for which to the best of our knowledge no progress has been made so far. We resolve questions (a) and (b) by providing appropriate upper and lower bounds, as well as a new potential Γ that guarantees efficient target detection even by querying an approximate median each time. With respect to (c), we initiate a systematic study for detecting two targets in graphs and we identify sufficient conditions on the queries that allow for strong (linear) lower bounds and strong (polylogarithmic) upper bounds for the number of queries. All of our positive results can be derived using our new potential Γ that allows querying approximate medians
Learning mutational graphs of individual tumour evolution from single-cell and multi-region sequencing data
Background. A large number of algorithms is being developed to reconstruct
evolutionary models of individual tumours from genome sequencing data. Most
methods can analyze multiple samples collected either through bulk multi-region
sequencing experiments or the sequencing of individual cancer cells. However,
rarely the same method can support both data types.
Results. We introduce TRaIT, a computational framework to infer mutational
graphs that model the accumulation of multiple types of somatic alterations
driving tumour evolution. Compared to other tools, TRaIT supports multi-region
and single-cell sequencing data within the same statistical framework, and
delivers expressive models that capture many complex evolutionary phenomena.
TRaIT improves accuracy, robustness to data-specific errors and computational
complexity compared to competing methods.
Conclusions. We show that the application of TRaIT to single-cell and
multi-region cancer datasets can produce accurate and reliable models of
single-tumour evolution, quantify the extent of intra-tumour heterogeneity and
generate new testable experimental hypotheses
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