Kernel methods in machine learning

We review machine learning methods employing positive definite kernels. These methods formulate learning and estimation problems in a reproducing kernel Hilbert space (RKHS) of functions defined on the data domain, expanded in terms of a kernel. Working in linear spaces of function has the benefit of facilitating the construction and analysis of learning algorithms while at the same time allowing large classes of functions. The latter include nonlinear functions as well as functions defined on nonvectorial data. We cover a wide range of methods, ranging from binary classifiers to sophisticated methods for estimation with structured data.Comment: Published in at http://dx.doi.org/10.1214/009053607000000677 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Experimental model of the interfacial instability in aluminium reduction cells

A solution has been found to the long-standing problem of experimental modelling of the interfacial instability in aluminium reduction cells. The idea is to replace the electrolyte overlaying molten aluminium with a mesh of thin rods supplying current down directly into the liquid metal layer. This eliminates electrolysis altogether and all the problems associated with it, such as high temperature, chemical aggressiveness of media, products of electrolysis, the necessity for electrolyte renewal, high power demands, etc. The result is a room temperature, versatile laboratory model which simulates Sele-type, rolling pad interfacial instability. Our new, safe laboratory model enables detailed experimental investigations to test the existing theoretical models for the first time

Long and short range multi-locus QTL interactions in a complex trait of yeast

We analyse interactions of Quantitative Trait Loci (QTL) in heat selected yeast by comparing them to an unselected pool of random individuals. Here we re-examine data on individual F12 progeny selected for heat tolerance, which have been genotyped at 25 locations identified by sequencing a selected pool [Parts, L., Cubillos, F. A., Warringer, J., Jain, K., Salinas, F., Bumpstead, S. J., Molin, M., Zia, A., Simpson, J. T., Quail, M. A., Moses, A., Louis, E. J., Durbin, R., and Liti, G. (2011). Genome research, 21(7), 1131-1138]. 960 individuals were genotyped at these locations and multi-locus genotype frequencies were compared to 172 sequenced individuals from the original unselected pool (a control group). Various non-random associations were found across the genome, both within chromosomes and between chromosomes. Some of the non-random associations are likely due to retention of linkage disequilibrium in the F12 population, however many, including the inter-chromosomal interactions, must be due to genetic interactions in heat tolerance. One region of particular interest involves 3 linked loci on chromosome IV where the central variant responsible for heat tolerance is antagonistic, coming from the heat sensitive parent and the flanking ones are from the more heat tolerant parent. The 3-locus haplotypes in the selected individuals represent a highly biased sample of the population haplotypes with rare double recombinants in high frequency. These were missed in the original analysis and would never be seen without the multigenerational approach. We show that a statistical analysis of entropy and information gain in genotypes of a selected population can reveal further interactions than previously seen. Importantly this must be done in comparison to the unselected population's genotypes to account for inherent biases in the original population

An analytic solution to the Busemann-Petty problem on sections of convex bodies

We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R^n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n-1)-dimensional X-ray) gives the ((n-1)-dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in R^n and leads to a unified analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies in R^n such that the ((n-1)-dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (n-dimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the Busemann-Petty problem, intersection bodies, and positive definite distributions, our formula shows that the answer to the problem depends on the behavior of the (n-2)-nd derivative of the parallel section functions. The affirmative answer to the Busemann-Petty problem for n\le 4 and the negative answer for n\ge 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.Comment: 13 pages, published versio

Archimedes' law and its corrections for an active particle in a granular sea

We study the origin of buoyancy forces acting on a larger particle moving in a granular medium subject to horizontal shaking and its corrections before fluidization. In the fluid limit Archimedes' law is verified; before the limit memory effects counteract buoyancy, as also found experimentally. The origin of the friction is an excluded volume effect between active particles, which we study more exactly for a random walker in a random environment. The same excluded volume effect is also responsible for the mutual attraction between bodies moving in the granular medium. Our theoretical modeling proceeds via an asymmetric exclusion process, i.e., via a dissipative lattice gas dynamics simulating the position degrees of freedom of a low density granular sea.Comment: 22 pages,5 figure