We develop algorithms for sampling from a probability distribution on a submanifold embedded in Rn. Applications are given to the evaluation of algorithms in 'Topological Statistics'; to goodness of fit tests in exponential families and to Neyman's smooth test. This article is partially expository, giving an introduction to the tools of geometric measure theory.

Channel and source coding theories are discussed. The following subject areas are covered: large constraint length convolutional codes (the Galileo code); decoder design (the big Viterbi decoder); Voyager's and Galileo's data compression scheme; current research in data compression for images; neural networks for soft decoding; neural networks for source decoding; finite-state codes; and fractals for data compression.

Let M be a random matrix chosen from Haar measure on the unitary group Un. Let Z = X + iY be a standard complex normal random variable with X and Y independent, mean 0 and variance ½ normal variables. We show that for j = 1, 2, …, Tr(Mj) are independent and distributed as √jZ asymptotically as n →∞. This result is used to study the set of eigenvalues of M. Similar results are given for the orthogonal and symplectic and symmetric groups.

We suggest a simple algorithm for Monte Carlo generation of uniformly distributed variables on a compact group. Example include random permutations, Rubik's cube positions, orthogonal, unitary, and symplectic matrices, and elements of GL n over a finite field. the algorithm reduces to the “standard” fast algorithm when there is one, but many new example are included.

Let G G be a compact separable topological group. When does there exist a probability P P such that P ∗ P = U P * P = U , where U U is Haar measure and P ≠ U P \ne U ? We show that such square roots exist if and only if G G is not abelian, nor the product of the quaternions and a product of two element groups. In the course of proving this we classify compact groups with the property that every closed subgroup is normal.

: We present some mathematical analysis for a class of curve fitting algorithms labeled 'projection pursuit' algorithms. These algorithms approximate a general function of p variables by a sum of non-linear functions of linear combinations. The approximation is computationally feasible and performs well in examples of nonparametric regression with noisy data, high dimensional density estimation, and multidimensional spline approximation. This note treats the algorithms from the point of view of approximation theory. It is easy to show that approximation is always possible.