74,155 research outputs found
Quantum to Classical Transitions via Weak Measurements and Post-Selection
This work will incorporate a few related tools for addressing the conceptual
difficulties arising from sewing together classical and quantum mechanics:
deterministic operators, weak measurements and post-selection. Weak
Measurement, based on a very weak von Neumann coupling, is a unique kind of
quantum measurement with numerous theoretical and practical applications. In
contrast to other measurement techniques, it allows to gather a small amount of
information regarding the quantum system, with only a negligible probability of
collapsing it. A single weak measurement yields an almost random outcome, but
when performed repeatedly over a large ensemble, the averaged outcome becomes
increasingly robust and accurate. Importantly, a long sequence of weak
measurements can be thought of as a single projective measurement. I claim in
this work that classical variables appearing in the macro-world, such as centre
of mass, moment of inertia, pressure and average forces, result from a
multitude of quantum weak measurements performed in the micro-world. Here
again, the quantum outcomes are highly uncertain, but the law of large numbers
obliges their convergence to the definite quantities we know from our everyday
lives. By augmenting this description with a final boundary condition and
employing the notion of "classical robustness under time-reversal" I will draw
a quantitative borderline between the classical and quantum regimes. I will
conclude by analyzing the role of macroscopic systems in amplifying and
recording quantum outcomes.Comment: To be published as a book chapter in "Quantum Structural Studies:
Classical Emergence from the Quantum Level", R.E. Kastner, J. Jeknic-Dugic,
G. Jaroszkiewicz (Eds.), World Scientific Publishing Co. arXiv admin note:
substantial text overlap with arXiv:1406.638
A tutorial on conformal prediction
Conformal prediction uses past experience to determine precise levels of
confidence in new predictions. Given an error probability , together
with a method that makes a prediction of a label , it produces a
set of labels, typically containing , that also contains with
probability . Conformal prediction can be applied to any method for
producing : a nearest-neighbor method, a support-vector machine, ridge
regression, etc.
Conformal prediction is designed for an on-line setting in which labels are
predicted successively, each one being revealed before the next is predicted.
The most novel and valuable feature of conformal prediction is that if the
successive examples are sampled independently from the same distribution, then
the successive predictions will be right of the time, even though
they are based on an accumulating dataset rather than on independent datasets.
In addition to the model under which successive examples are sampled
independently, other on-line compression models can also use conformal
prediction. The widely used Gaussian linear model is one of these.
This tutorial presents a self-contained account of the theory of conformal
prediction and works through several numerical examples. A more comprehensive
treatment of the topic is provided in "Algorithmic Learning in a Random World",
by Vladimir Vovk, Alex Gammerman, and Glenn Shafer (Springer, 2005).Comment: 58 pages, 9 figure
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