13,442 research outputs found
Stabilized Nearest Neighbor Classifier and Its Statistical Properties
The stability of statistical analysis is an important indicator for
reproducibility, which is one main principle of scientific method. It entails
that similar statistical conclusions can be reached based on independent
samples from the same underlying population. In this paper, we introduce a
general measure of classification instability (CIS) to quantify the sampling
variability of the prediction made by a classification method. Interestingly,
the asymptotic CIS of any weighted nearest neighbor classifier turns out to be
proportional to the Euclidean norm of its weight vector. Based on this concise
form, we propose a stabilized nearest neighbor (SNN) classifier, which
distinguishes itself from other nearest neighbor classifiers, by taking the
stability into consideration. In theory, we prove that SNN attains the minimax
optimal convergence rate in risk, and a sharp convergence rate in CIS. The
latter rate result is established for general plug-in classifiers under a
low-noise condition. Extensive simulated and real examples demonstrate that SNN
achieves a considerable improvement in CIS over existing nearest neighbor
classifiers, with comparable classification accuracy. We implement the
algorithm in a publicly available R package snn.Comment: 48 Pages, 11 Figures. To Appear in JASA--T&
Single-particle machine for quantum thermalization
The long time accumulation of the \textit{random} actions of a single
particle "reservoir" on its coupled system can transfer some temperature
information of its initial state to the coupled system. This dynamic process
can be referred to as a quantum thermalization in the sense that the coupled
system can reach a stable thermal equilibrium with a temperature equal to that
of the reservoir. We illustrate this idea based on the usual micromaser model,
in which a series of initially prepared two-level atoms randomly pass through
an electromagnetic cavity. It is found that, when the randomly injected atoms
are initially prepared in a thermal equilibrium state with a given temperature,
the cavity field will reach a thermal equilibrium state with the same
temperature as that of the injected atoms. As in two limit cases, the cavity
field can be cooled and "coherently heated" as a maser process, respectively,
when the injected atoms are initially prepared in ground and excited states.
Especially, when the atoms in equilibrium are driven to possess some coherence,
the cavity field may reach a higher temperature in comparison with the injected
atoms. We also point out a possible experimental test for our theoretical
prediction based on a superconducting circuit QED system.Comment: 9 pages,4 figures
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