118 research outputs found
Empirical processes, typical sequences and coordinated actions in standard Borel spaces
This paper proposes a new notion of typical sequences on a wide class of
abstract alphabets (so-called standard Borel spaces), which is based on
approximations of memoryless sources by empirical distributions uniformly over
a class of measurable "test functions." In the finite-alphabet case, we can
take all uniformly bounded functions and recover the usual notion of strong
typicality (or typicality under the total variation distance). For a general
alphabet, however, this function class turns out to be too large, and must be
restricted. With this in mind, we define typicality with respect to any
Glivenko-Cantelli function class (i.e., a function class that admits a Uniform
Law of Large Numbers) and demonstrate its power by giving simple derivations of
the fundamental limits on the achievable rates in several source coding
scenarios, in which the relevant operational criteria pertain to reproducing
empirical averages of a general-alphabet stationary memoryless source with
respect to a suitable function class.Comment: 14 pages, 3 pdf figures; accepted to IEEE Transactions on Information
Theor
Learning from compressed observations
The problem of statistical learning is to construct a predictor of a random
variable as a function of a related random variable on the basis of an
i.i.d. training sample from the joint distribution of . Allowable
predictors are drawn from some specified class, and the goal is to approach
asymptotically the performance (expected loss) of the best predictor in the
class. We consider the setting in which one has perfect observation of the
-part of the sample, while the -part has to be communicated at some
finite bit rate. The encoding of the -values is allowed to depend on the
-values. Under suitable regularity conditions on the admissible predictors,
the underlying family of probability distributions and the loss function, we
give an information-theoretic characterization of achievable predictor
performance in terms of conditional distortion-rate functions. The ideas are
illustrated on the example of nonparametric regression in Gaussian noise.Comment: 6 pages; submitted to the 2007 IEEE Information Theory Workshop (ITW
2007
Scaling and renormalization in fault-tolerant quantum computers
This work is concerned with phrasing the concepts of fault-tolerant quantum
computation within the framework of disordered systems, Bernoulli site
percolation in particular. We show how the so-called "threshold theorems" on
the possibility of fault-tolerant quantum computation with constant error rate
can be cast as a renormalization (coarse-graining) of the site percolation
process describing the occurrence of errors during computation. We also use
percolation techniques to derive a trade-off between the complexity overhead of
the fault-tolerant circuit and the threshold error rate.Comment: 4 pages, 2 eps figures; revtex4; based on talk given at the Simons
Conference on Quantum and Reversible Computation, Stony Brook NY, May 28-31;
minor typographical change
- …