3,869 research outputs found
Finite sample performance of linear least squares estimators under sub-Gaussian martingale difference noise
Linear Least Squares is a very well known technique for parameter estimation,
which is used even when sub-optimal, because of its very low computational
requirements and the fact that exact knowledge of the noise statistics is not
required. Surprisingly, bounding the probability of large errors with finitely
many samples has been left open, especially when dealing with correlated noise
with unknown covariance. In this paper we analyze the finite sample performance
of the linear least squares estimator under sub-Gaussian martingale difference
noise. In order to analyze this important question we used concentration of
measure bounds. When applying these bounds we obtained tight bounds on the tail
of the estimator's distribution. We show the fast exponential convergence of
the number of samples required to ensure a given accuracy with high
probability. We provide probability tail bounds on the estimation error's norm.
Our analysis method is simple and uses simple type bounds on the
estimation error. The tightness of the bounds is tested through simulation. The
proposed bounds make it possible to predict the number of samples required for
least squares estimation even when least squares is sub-optimal and used for
computational simplicity. The finite sample analysis of least squares models
with this general noise model is novel
Extremal families of cubic Thue equations
We exactly determine the integral solutions to a previously untreated
infinite family of cubic Thue equations of the form with at least
such solutions. Our approach combines elementary arguments, with lower
bounds for linear forms in logarithms and lattice-basis reduction
Anomalous transport from holography: Part I
We revisit the transport properties induced by the chiral anomaly in a
charged plasma holographically dual to anomalous Maxwell
theory in Schwarzschild-. Off-shell constitutive relations for vector
and axial currents are derived using various approximations generalising most
of known in the literature anomaly-induced phenomena and revealing some new
ones. In a weak external field approximation, the constitutive relations have
all-order derivatives resummed into six momenta-dependent transport coefficient
functions: the diffusion, the electric/magnetic conductivity, and three anomaly
induced functions. The latter generalise the chiral magnetic and chiral
separation effects. Nonlinear transport is studied assuming presence of
constant background external fields. The chiral magnetic effect, including all
order nonlinearity in magnetic field, is proven to be exact when the magnetic
field is the only external field that is turned on. Non-linear corrections to
the constitutive relations due to electric and axial external fields are
computed.Comment: v1: 42 pages, 10 multi-figures, 3 appendices; v2: minor corrections
introduced and a few refs adde
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