On the computation of Wasserstein barycenters

The Wasserstein barycenter is an important notion in the analysis of high dimensional data with a broad range of applications in applied probability, economics, statistics, and in particular to clustering and image processing. In this paper, we state a general version of the equivalence of the Wasserstein barycenter problem to the n-coupling problem. As a consequence, the coupling to the sum principle (characterizing solutions to the n-coupling problem) provides a novel criterion for the explicit characterization of barycenters. Based on this criterion, we provide as a main contribution the simple to implement iterative swapping algorithm (ISA) for computing barycenters. The ISA is a completely non-parametric algorithm which provides a sharp image of the support of the barycenter and has a quadratic time complexity which is comparable to other well established algorithms designed to compute barycenters. The algorithm can also be applied to more complex optimization problems like the k-barycenter problem

The nonlinear Bernstein-Schr\"odinger equation in Economics

In this paper we relate the Equilibrium Assignment Problem (EAP), which is underlying in several economics models, to a system of nonlinear equations that we call the "nonlinear Bernstein-Schr\"odinger system", which is well-known in the linear case, but whose nonlinear extension does not seem to have been studied. We apply this connection to derive an existence result for the EAP, and an efficient computational method.Comment: 8 pages, submitted to Lecture Notes in Computer Scienc

A COL17A1 Splice-Altering Mutation Is Prevalent in Inherited Recurrent Corneal Erosions

PurposeCorneal dystrophies are a genetically heterogeneous group of disorders. We previously described a family with an autosomal dominant epithelial recurrent erosion dystrophy (ERED). We aimed to identify the underlying genetic cause of ERED in this family and 3 additional ERED families. We sought to characterize the potential function of the candidate genes using the human and zebrafish cornea.DesignCase series study of 4 white families with a similar ERED. An experimental study was performed on human and zebrafish tissue to examine the putative biological function of candidate genes.ParticipantsFour ERED families, including 28 affected and 17 unaffected individuals.MethodsHumanLinkage-12 arrays (Illumina, San Diego, CA) were used to genotype 17 family members. Next-generation exome sequencing was performed on an uncle–niece pair. Segregation of potential causative mutations was confirmed using Sanger sequencing. Protein expression was determined using immunohistochemistry in human and zebrafish cornea. Gene expression in zebrafish was assessed using whole-mount in situ hybridization. Morpholino-induced transient gene knockdown was performed in zebrafish embryos.Main Outcome MeasuresLinkage microarray, exome analysis, DNA sequence analysis, immunohistochemistry, in situ hybridization, and morpholino-induced genetic knockdown results.ResultsLinkage microarray analysis identified a candidate region on chromosome chr10:12,576,562–112,763,135, and exploration of exome sequencing data identified 8 putative pathogenic variants in this linkage region. Two variants segregated in 06NZ–TRB1 with ERED: COL17A1 c.3156C→T and DNAJC9 c.334G→A. The COL17A1 c.3156C→T variant segregated in all 4 ERED families. We showed biologically relevant expression of these proteins in human cornea. Both proteins are expressed in the cornea of zebrafish embryos and adults. Zebrafish lacking Col17a1a and Dnajc9 during development show no gross corneal phenotype.ConclusionsThe COL17A1 c.3156C→T variant is the likely causative mutation in our recurrent corneal erosion families, and its presence in 4 independent families suggests that it is prevalent in ERED. This same COL17A1 c.3156C→T variant recently was identified in a separate pedigree with ERED. Our study expands the phenotypic spectrum of COL17A1 disease from autosomal recessive epidermolysis bullosa to autosomal dominant ERED and identifies COL17A1 as a key protein in maintaining integrity of the corneal epithelium

Linkage to chromosome 2q32.2-q33.3 in familial serrated neoplasia (Jass syndrome)

Causative genetic variants have to date been identified for only a small proportion of familial colorectal cancer (CRC). While conditions such as Familial Adenomatous Polyposis and Lynch syndrome have well defined genetic causes, the search for variants underlying the remainder of familial CRC is plagued by genetic heterogeneity. The recent identification of families with a heritable predisposition to malignancies arising through the serrated pathway (familial serrated neoplasia or Jass syndrome) provides an opportunity to study a subset of familial CRC in which heterogeneity may be greatly reduced. A genome-wide linkage screen was performed on a large family displaying a dominantly-inherited predisposition to serrated neoplasia genotyped using the Affymetrix GeneChip Human Mapping 10 K SNP Array. Parametric and nonparametric analyses were performed and resulting regions of interest, as well as previously reported CRC susceptibility loci at 3q22, 7q31 and 9q22, were followed up by finemapping in 10 serrated neoplasia families. Genome-wide linkage analysis revealed regions of interest at 2p25.2-p25.1, 2q24.3-q37.1 and 8p21.2-q12.1. Finemapping linkage and haplotype analyses identified 2q32.2-q33.3 as the region most likely to harbour linkage, with heterogeneity logarithm of the odds (HLOD) 2.09 and nonparametric linkage (NPL) score 2.36 (P = 0.004). Five primary candidate genes (CFLAR, CASP10, CASP8, FZD7 and BMPR2) were sequenced and no segregating variants identified. There was no evidence of linkage to previously reported loci on chromosomes 3, 7 and 9

ON REGRESSION REPRESENTATIONS OF STOCHASTIC-PROCESSES

We construct a.s. nonlinear regression representations of general stochastic processes (X(n))n is-an-element-of N. As a consequence we obtain in particular special regression representations of Markov chains and of certain m-dependent sequences. For m-dependent sequences we obtain a constructive method to check, whether these sequences have a monotone (m + 1)-block factor representation

ON REGRESSION REPRESENTATIONS OF STOCHASTIC-PROCESSES

AbstractWe construct a.s. nonlinear regression representations of general stochastic processes (Xn)nϵN. As a consequence we obtain in particular special regression representations of Markov chains and of certain m-dependent sequences. For m-dependent sequences we obtain a constructive method to check, whether these sequences have a monotone (m+1)-block factor representation

Optimal Coupling of Multivariate Distributions and Stochastic Processes

Same explicit optimal coupling results are derived with respect to minimal metrics of lp-type. In particular, the optimality of radial transformations, positive transformations, and monotone transformations of the components is established.