The distortion-rate performance of certain randomly-designed scalar
quantizers is determined. The central results are the mean-squared error
distortion and output entropy for quantizing a uniform random variable with
thresholds drawn independently from a uniform distribution. The distortion is
at most 6 times that of an optimal (deterministically-designed) quantizer, and
for a large number of levels the output entropy is reduced by approximately
(1-gamma)/(ln 2) bits, where gamma is the Euler-Mascheroni constant.

Minimum mean squared error (MMSE) estimators of signals from samples
corrupted by jitter (timing noise) and additive noise are nonlinear, even when
the signal prior and additive noise have normal distributions. This paper
develops stochastic algorithms based on Gibbs sampling and slice sampling to
approximate optimal MMSE estimators in this Bayesian formulation. Simulations
demonstrate that these nonlinear algorithms can improve significantly upon the
linear MMSE estimator.

Frame permutation quantization (FPQ) is a new vector quantization technique
using finite frames. In FPQ, a vector is encoded using a permutation source
code to quantize its frame expansion. This means that the encoding is a partial
ordering of the frame expansion coefficients. Compared to ordinary permutation
source coding, FPQ produces a greater number of possible quantization rates and
a higher maximum rate. Various representations for the partitions induced by
FPQ are presented and reconstruction algorithms based on linear programming and
quadratic programming are derived.

The replica method is a non-rigorous but widely-accepted technique from
statistical physics used in the asymptotic analysis of large, random, nonlinear
problems. This paper applies the replica method to non-Gaussian maximum a
posteriori (MAP) estimation. It is shown that with random linear measurements
and Gaussian noise, the asymptotic behavior of the MAP estimate of an
n-dimensional vector decouples as n scalar MAP estimators. The result is a
counterpart to Guo and Verdu's replica analysis of minimum mean-squared error
estimation.