Approximating Loops in a Shortest Homology Basis from Point Data

Inference of topological and geometric attributes of a hidden manifold from its point data is a fundamental problem arising in many scientific studies and engineering applications. In this paper we present an algorithm to compute a set of loops from a point data that presumably sample a smooth manifold $M\subset \mathbb{R}^d$. These loops approximate a {\em shortest} basis of the one dimensional homology group $H_1(M)$ over coefficients in finite field $\mathbb{Z}_2$. Previous results addressed the issue of computing the rank of the homology groups from point data, but there is no result on approximating the shortest basis of a manifold from its point sample. In arriving our result, we also present a polynomial time algorithm for computing a shortest basis of $H_1(K)$ for any finite {\em simplicial complex} $K$ whose edges have non-negative weights

Canonical, Stable, General Mapping using Context Schemes

Motivation: Sequence mapping is the cornerstone of modern genomics. However, most existing sequence mapping algorithms are insufficiently general. Results: We introduce context schemes: a method that allows the unambiguous recognition of a reference base in a query sequence by testing the query for substrings from an algorithmically defined set. Context schemes only map when there is a unique best mapping, and define this criterion uniformly for all reference bases. Mappings under context schemes can also be made stable, so that extension of the query string (e.g. by increasing read length) will not alter the mapping of previously mapped positions. Context schemes are general in several senses. They natively support the detection of arbitrary complex, novel rearrangements relative to the reference. They can scale over orders of magnitude in query sequence length. Finally, they are trivially extensible to more complex reference structures, such as graphs, that incorporate additional variation. We demonstrate empirically the existence of high performance context schemes, and present efficient context scheme mapping algorithms. Availability and Implementation: The software test framework created for this work is available from https://registry.hub.docker.com/u/adamnovak/sequence-graphs/. Contact: [email protected] Supplementary Information: Six supplementary figures and one supplementary section are available with the online version of this article.Comment: Submission for Bioinformatic

ConSole: using modularity of contact maps to locate solenoid domains in protein structures.

BackgroundPeriodic proteins, characterized by the presence of multiple repeats of short motifs, form an interesting and seldom-studied group. Due to often extreme divergence in sequence, detection and analysis of such motifs is performed more reliably on the structural level. Yet, few algorithms have been developed for the detection and analysis of structures of periodic proteins.ResultsConSole recognizes modularity in protein contact maps, allowing for precise identification of repeats in solenoid protein structures, an important subgroup of periodic proteins. Tests on benchmarks show that ConSole has higher recognition accuracy as compared to Raphael, the only other publicly available solenoid structure detection tool. As a next step of ConSole analysis, we show how detection of solenoid repeats in structures can be used to improve sequence recognition of these motifs and to detect subtle irregularities of repeat lengths in three solenoid protein families.ConclusionsThe ConSole algorithm provides a fast and accurate tool to recognize solenoid protein structures as a whole and to identify individual solenoid repeat units from a structure. ConSole is available as a web-based, interactive server and is available for download at http://console.sanfordburnham.org

Vertex Sparsifiers: New Results from Old Techniques

Given a capacitated graph $G = (V,E)$ and a set of terminals $K \subseteq V$, how should we produce a graph $H$ only on the terminals $K$ so that every (multicommodity) flow between the terminals in $G$ could be supported in $H$ with low congestion, and vice versa? (Such a graph $H$ is called a flow-sparsifier for $G$.) What if we want $H$ to be a "simple" graph? What if we allow $H$ to be a convex combination of simple graphs? Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC 2010], we give efficient algorithms for constructing: (a) a flow-sparsifier $H$ that maintains congestion up to a factor of $O(\log k/\log \log k)$, where $k = |K|$, (b) a convex combination of trees over the terminals $K$ that maintains congestion up to a factor of $O(\log k)$, and (c) for a planar graph $G$, a convex combination of planar graphs that maintains congestion up to a constant factor. This requires us to give a new algorithm for the 0-extension problem, the first one in which the preimages of each terminal are connected in $G$. Moreover, this result extends to minor-closed families of graphs. Our improved bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2010. Final version to appear in SIAM J. Computin

Neural Network Memory Architectures for Autonomous Robot Navigation

This paper highlights the significance of including memory structures in neural networks when the latter are used to learn perception-action loops for autonomous robot navigation. Traditional navigation approaches rely on global maps of the environment to overcome cul-de-sacs and plan feasible motions. Yet, maintaining an accurate global map may be challenging in real-world settings. A possible way to mitigate this limitation is to use learning techniques that forgo hand-engineered map representations and infer appropriate control responses directly from sensed information. An important but unexplored aspect of such approaches is the effect of memory on their performance. This work is a first thorough study of memory structures for deep-neural-network-based robot navigation, and offers novel tools to train such networks from supervision and quantify their ability to generalize to unseen scenarios. We analyze the separation and generalization abilities of feedforward, long short-term memory, and differentiable neural computer networks. We introduce a new method to evaluate the generalization ability by estimating the VC-dimension of networks with a final linear readout layer. We validate that the VC estimates are good predictors of actual test performance. The reported method can be applied to deep learning problems beyond robotics