12,207 research outputs found
Generalized Bhattacharyya and Chernoff upper bounds on Bayes error using quasi-arithmetic means
Bayesian classification labels observations based on given prior information,
namely class-a priori and class-conditional probabilities. Bayes' risk is the
minimum expected classification cost that is achieved by the Bayes' test, the
optimal decision rule. When no cost incurs for correct classification and unit
cost is charged for misclassification, Bayes' test reduces to the maximum a
posteriori decision rule, and Bayes risk simplifies to Bayes' error, the
probability of error. Since calculating this probability of error is often
intractable, several techniques have been devised to bound it with closed-form
formula, introducing thereby measures of similarity and divergence between
distributions like the Bhattacharyya coefficient and its associated
Bhattacharyya distance. The Bhattacharyya upper bound can further be tightened
using the Chernoff information that relies on the notion of best error
exponent. In this paper, we first express Bayes' risk using the total variation
distance on scaled distributions. We then elucidate and extend the
Bhattacharyya and the Chernoff upper bound mechanisms using generalized
weighted means. We provide as a byproduct novel notions of statistical
divergences and affinity coefficients. We illustrate our technique by deriving
new upper bounds for the univariate Cauchy and the multivariate
-distributions, and show experimentally that those bounds are not too
distant to the computationally intractable Bayes' error.Comment: 22 pages, include R code. To appear in Pattern Recognition Letter
The effect of noise correlations on randomized benchmarking
Among the most popular and well studied quantum characterization,
verification and validation techniques is randomized benchmarking (RB), an
important statistical tool used to characterize the performance of physical
logic operations useful in quantum information processing. In this work we
provide a detailed mathematical treatment of the effect of temporal noise
correlations on the outcomes of RB protocols. We provide a fully analytic
framework capturing the accumulation of error in RB expressed in terms of a
three-dimensional random walk in "Pauli space." Using this framework we derive
the probability density function describing RB outcomes (averaged over noise)
for both Markovian and correlated errors, which we show is generally described
by a gamma distribution with shape and scale parameters depending on the
correlation structure. Long temporal correlations impart large nonvanishing
variance and skew in the distribution towards high-fidelity outcomes --
consistent with existing experimental data -- highlighting potential
finite-sampling pitfalls and the divergence of the mean RB outcome from
worst-case errors in the presence of noise correlations. We use the
Filter-transfer function formalism to reveal the underlying reason for these
differences in terms of effective coherent averaging of correlated errors in
certain random sequences. We conclude by commenting on the impact of these
calculations on the utility of single-metric approaches to quantum
characterization, verification, and validation.Comment: Updated and expanded to include full derivation. Related papers
available from http://www.physics.usyd.edu.au/~mbiercuk/Publications.htm
Heegaard Floer homology and integer surgeries on links
Let L be a link in an integral homology three-sphere. We give a description
of the Heegaard Floer homology of integral surgeries on L in terms of some data
associated to L, which we call a complete system of hyperboxes for L. Roughly,
a complete systems of hyperboxes consists of chain complexes for (some versions
of) the link Floer homology of L and all its sublinks, together with several
chain maps between these complexes. Further, we introduce a way of presenting
closed four-manifolds with b_2^+ > 1 by four-colored framed links in the
three-sphere. Given a link presentation of this kind for a four-manifold X, we
then describe the Ozsvath-Szabo mixed invariants of X in terms of a complete
system of hyperboxes for the link. Finally, we explain how a grid diagram
produces a particular complete system of hyperboxes for the corresponding link.Comment: 231 pages, 54 figures; major revision: we now work with one U
variable for each w basepoint, rather than one per link component; we also
added Section 4, with an overview of the main resul
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