Dynamic Linear Discriminant Analysis in High Dimensional Space

High-dimensional data that evolve dynamically feature predominantly in the modern data era. As a partial response to this, recent years have seen increasing emphasis to address the dimensionality challenge. However, the non-static nature of these datasets is largely ignored. This paper addresses both challenges by proposing a novel yet simple dynamic linear programming discriminant (DLPD) rule for binary classification. Different from the usual static linear discriminant analysis, the new method is able to capture the changing distributions of the underlying populations by modeling their means and covariances as smooth functions of covariates of interest. Under an approximate sparse condition, we show that the conditional misclassification rate of the DLPD rule converges to the Bayes risk in probability uniformly over the range of the variables used for modeling the dynamics, when the dimensionality is allowed to grow exponentially with the sample size. The minimax lower bound of the estimation of the Bayes risk is also established, implying that the misclassification rate of our proposed rule is minimax-rate optimal. The promising performance of the DLPD rule is illustrated via extensive simulation studies and the analysis of a breast cancer dataset.Comment: 34 pages; 3 figure

Non-linear Redshift-Space Power Spectra

Distances in cosmology are usually inferred from observed redshifts - an estimate that is dependent on the local peculiar motion - giving a distorted view of the three dimensional structure and affecting basic observables such as the correlation function and power spectrum. We calculate the full non-linear redshift-space power spectrum for Gaussian fields, giving results for both the standard flat sky approximation and the directly-observable angular correlation function and angular power spectrum. Coupling between large and small scale modes boosts the power on small scales when the perturbations are small. On larger scales power is slightly suppressed by the velocities perturbations on smaller scales. The analysis is general, but we comment specifically on the implications for future high-redshift observations, and show that the non-linear spectrum has significantly more complicated angular structure than in linear theory. We comment on the implications for using the angular structure to separate cosmological and astrophysical components of 21 cm observations.Comment: 22 pages, 6 figures, changed to version accepted in Physics Review

Non-Linear Relativity in Position Space

We propose two methods for obtaining the dual of non-linear relativity as previously formulated in momentum space. In the first we allow for the (dual) position space to acquire a non-linear representation of the Lorentz group independently of the chosen representation in momentum space. This requires a non-linear definition for the invariant contraction between momentum and position spaces. The second approach, instead, respects the linearity of the invariant contraction. This fully fixes the dual of momentum space and dictates a set of energy-dependent space-time Lorentz transformations. We discuss a variety of physical implications that would distinguish these two strategies. We also show how they point to two rather distinct formulations of theories of gravity with an invariant energy and/or length scale.Comment: 7 pages, revised versio

LINEAR CONNECTIONS ON EXTENDED SPACE-TIME

A modification of Kaluza-Klein theory is proposed which is general enough to admit an arbitrary finite noncommutative internal geometry. It is shown that the existence of a non-trival extension to the total geometry of a linear connection on space-time places severe restrictions on the structure of the noncommutative factor. A counter-example is given.Comment: 15 pages, plain Te

Kernel principal component analysis (KPCA) for the de-noising of communication signals

This paper is concerned with the problem of de-noising for non-linear signals. Principal Component Analysis (PCA) cannot be applied to non-linear signals however it is known that using kernel functions, a non-linear signal can be transformed into a linear signal in a higher dimensional space. In that feature space, a linear algorithm can be applied to a non-linear problem. It is proposed that using the principal components extracted from this feature space, the signal can be de-noised in its input space

The space of linear anti-symplectic involutions is a homogenous space

In this note we prove that the space of linear anti-symplectic involutions is the homogenous space Gl(n,\R)\Sp(n). This result is motivated by the study of symmetric periodic orbits in the restricted 3-body problem.Comment: 5 page