High-performance Kernel Machines with Implicit Distributed Optimization and Randomization

In order to fully utilize "big data", it is often required to use "big models". Such models tend to grow with the complexity and size of the training data, and do not make strong parametric assumptions upfront on the nature of the underlying statistical dependencies. Kernel methods fit this need well, as they constitute a versatile and principled statistical methodology for solving a wide range of non-parametric modelling problems. However, their high computational costs (in storage and time) pose a significant barrier to their widespread adoption in big data applications. We propose an algorithmic framework and high-performance implementation for massive-scale training of kernel-based statistical models, based on combining two key technical ingredients: (i) distributed general purpose convex optimization, and (ii) the use of randomization to improve the scalability of kernel methods. Our approach is based on a block-splitting variant of the Alternating Directions Method of Multipliers, carefully reconfigured to handle very large random feature matrices, while exploiting hybrid parallelism typically found in modern clusters of multicore machines. Our implementation supports a variety of statistical learning tasks by enabling several loss functions, regularization schemes, kernels, and layers of randomized approximations for both dense and sparse datasets, in a highly extensible framework. We evaluate the ability of our framework to learn models on data from applications, and provide a comparison against existing sequential and parallel libraries.Comment: Work presented at MMDS 2014 (June 2014) and JSM 201

Braiding fluxes in Pauli Hamiltonian

Aharonov and Casher showed that Pauli Hamiltonians in two dimensions have gapless zero modes. We study the adiabatic evolution of these modes under the slow motion of $N$ fluxons with fluxes $\Phi_a\in\mathbb{R}$. The positions, $\mathbf{r}_a\in\mathbb{R}^2$, of the fluxons are viewed as controls. We are interested in the holonomies associated with closed paths in the space of controls. The holonomies can sometimes be abelian, but in general are not. They can sometimes be topological, but in general are not. We analyze some of the special cases and some of the general ones. Our most interesting results concern the cases where holonomy turns out to be topological which is the case when all the fluxons are subcritical, $\Phi_a<1$, and the number of zero modes is $D=N-1$. If $N\ge3$ it is also non-abelian. In the special case that the fluxons carry identical fluxes the resulting anyons satisfy the Burau representations of the braid group