Wide partitions, Latin tableaux, and Rota's basis conjecture

Say that mu is a ``subpartition'' of an integer partition lambda if the multiset of parts of mu is a submultiset of the parts of lambda, and define an integer partition lambda to be ``wide'' if for every subpartition mu of lambda, mu >= mu' in dominance order (where mu' denotes the conjugate or transpose of mu). Then Brian Taylor and the first author have conjectured that an integer partition lambda is wide if and only if there exists a tableau of shape lambda such that (1) for all i, the entries in the ith row of the tableau are precisely the integers from 1 to lambda_i inclusive, and (2) for all j, the entries in the jth column of the tableau are pairwise distinct. This conjecture was originally motivated by Rota's basis conjecture and, if true, yields a new class of integer multiflow problems that satisfy max-flow min-cut and integrality. Wide partitions also yield a class of graphs that satisfy ``delta-conjugacy'' (in the sense of Greene and Kleitman), and the above conjecture implies that these graphs furthermore have a completely saturated stable set partition. We present several partial results, but the conjecture remains very much open.Comment: Joined forces with Goemans and Vondrak---several new partial results; 28 pages, submitted to Adv. Appl. Mat

Classification with imperfect training labels

We study the effect of imperfect training data labels on the performance of classification methods. In a general setting, where the probability that an observation in the training dataset is mislabelled may depend on both the feature vector and the true label, we bound the excess risk of an arbitrary classifier trained with imperfect labels in terms of its excess risk for predicting a noisy label. This reveals conditions under which a classifier trained with imperfect labels remains consistent for classifying uncorrupted test data points. Furthermore, under stronger conditions, we derive detailed asymptotic properties for the popular $k$-nearest neighbour ($k$nn), support vector machine (SVM) and linear discriminant analysis (LDA) classifiers. One consequence of these results is that the $k$nn and SVM classifiers are robust to imperfect training labels, in the sense that the rate of convergence of the excess risks of these classifiers remains unchanged; in fact, our theoretical and empirical results even show that in some cases, imperfect labels may improve the performance of these methods. On the other hand, the LDA classifier is shown to be typically inconsistent in the presence of label noise unless the prior probabilities of each class are equal. Our theoretical results are supported by a simulation study

The power of multifolds: Folding the algebraic closure of the rational numbers

It is well known that the usual Huzita-Hatori axioms for origami enable angle trisection but not angle quintisection. Using the concept of a multifold, Lang has achieved quintisection but not arbitrary algebraic numbers. We define the n-parameter multifold and show how to use one-parameter multifolds to obtain the algebraic closure of the rational numbers.Comment: 9 pages; 4th Int'l Conf. Origami Sci. Math. Educ. (4OSME

Increase in airway neutrophils after oral but not inhaled corticosteroid therapy in mild asthma

SummaryBackground: Neutrophils, in addition to eosinophils, are prominent in the airways of patients with severe asthma who are usually on long-term oral and inhaled corticosteroid treatment. We determined whether inhaled or oral corticosteroid therapy can induce airway neutrophilia.Methods: We performed two separate placebo-controlled studies in which patients with mild asthma were treated with either prednisolone (30mg per day for 7 days; n=9) or placebo tablets (n=8), or with either inhaled budesonide (800μg twice daily for 4 weeks; n=6) or inhaled placebo (n=6). Fiberoptic bronchoscopy was performed before treatment and at day 7 of oral treatment, and at day 28 of inhaled therapy. Bronchial sections were immunostained with an antibody to major basic protein for eosinophils, and with an antibody to neutrophil elastase for neutrophils. Induced sputum was obtained in the prednisolone study.Results: Neutrophils in airway submucosa increased after prednisolone from median 76 to 140/mm2 (P=0.05); this change was higher than that after placebo (P=0.04). Eosinophils decreased from 24 to 9/mm2 (P=0.03), but this was not significantly different from placebo. Eosinophils and neutrophils, and levels of IL-8 and myeloperoxidase in induced sputum did not change after prednisolone. There was no change in neutrophil counts after budesonide, but the reduction in eosinophils was greater than placebo (P=0.05). Budesonide improved bronchial responsiveness, but prednisolone did not.Conclusion: Corticosteroid therapy by the oral but not inhaled route can induce neutrophil recruitment into the airways of patients with mild asthma. This could explain the increase in airway neutrophils observed in severe asthmatics treated with oral corticosteroids

Trifocal Relative Pose from Lines at Points and its Efficient Solution

We present a new minimal problem for relative pose estimation mixing point features with lines incident at points observed in three views and its efficient homotopy continuation solver. We demonstrate the generality of the approach by analyzing and solving an additional problem with mixed point and line correspondences in three views. The minimal problems include correspondences of (i) three points and one line and (ii) three points and two lines through two of the points which is reported and analyzed here for the first time. These are difficult to solve, as they have 216 and - as shown here - 312 solutions, but cover important practical situations when line and point features appear together, e.g., in urban scenes or when observing curves. We demonstrate that even such difficult problems can be solved robustly using a suitable homotopy continuation technique and we provide an implementation optimized for minimal problems that can be integrated into engineering applications. Our simulated and real experiments demonstrate our solvers in the camera geometry computation task in structure from motion. We show that new solvers allow for reconstructing challenging scenes where the standard two-view initialization of structure from motion fails.Comment: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while most authors were in residence at Brown University's Institute for Computational and Experimental Research in Mathematics -- ICERM, in Providence, R

SECOM: A Novel Hash Seed and Community Detection Based-Approach for Genome-Scale Protein Domain Identification

With rapid advances in the development of DNA sequencing technologies, a plethora of high-throughput genome and proteome data from a diverse spectrum of organisms have been generated. The functional annotation and evolutionary history of proteins are usually inferred from domains predicted from the genome sequences. Traditional database-based domain prediction methods cannot identify novel domains, however, and alignment-based methods, which look for recurring segments in the proteome, are computationally demanding. Here, we propose a novel genome-wide domain prediction method, SECOM. Instead of conducting all-against-all sequence alignment, SECOM first indexes all the proteins in the genome by using a hash seed function. Local similarity can thus be detected and encoded into a graph structure, in which each node represents a protein sequence and each edge weight represents the shared hash seeds between the two nodes. SECOM then formulates the domain prediction problem as an overlapping community-finding problem in this graph. A backward graph percolation algorithm that efficiently identifies the domains is proposed. We tested SECOM on five recently sequenced genomes of aquatic animals. Our tests demonstrated that SECOM was able to identify most of the known domains identified by InterProScan. When compared with the alignment-based method, SECOM showed higher sensitivity in detecting putative novel domains, while it was also three orders of magnitude faster. For example, SECOM was able to predict a novel sponge-specific domain in nucleoside-triphosphatase (NTPases). Furthermore, SECOM discovered two novel domains, likely of bacterial origin, that are taxonomically restricted to sea anemone and hydra. SECOM is an open-source program and available at http://sfb.kaust.edu.sa/Pages/Software.aspx