Stephen DeSalvo

Probabilistic Divide-and-Conquer: A New Exact Simulation Method, With Integer Partitions as an Example

We propose a new method, probabilistic divide-and-conquer, for improving the success probability in rejection sampling. For the example of integer partitions, there is an ideal recursive scheme which improves the rejection cost from asymptotically order

The Shannon entropy of Sudoku matrices

n^{3/4}

Probabilistic divide-and-conquer: Deterministic second half

to a constant. We show other examples for which a non--recursive, one--time application of probabilistic divide-and-conquer removes a substantial fraction of the rejection sampling cost. We also present a variation of probabilistic divide-and-conquer for generating i.i.d. samples that exploits features of the coupon collectors problem, in order to obtain a cost that is sublinear in the number of samples.

The probability of avoiding consecutive patterns in the Mallows distribution

We study properties of an ensemble of Sudoku matrices (a special type of doubly stochastic matrix when normalized) using their statistically averaged singular values. The determinants are very nearly Cauchy distributed about the origin. The largest singular value is , while the others decrease approximately linearly. The normalized singular values (obtained by dividing each singular value by the sum of all nine singular values) are then used to calculate the average Shannon entropy of the ensemble, a measure of the distribution of ‘energy’ among the singular modes and interpreted as a measure of the disorder of a typical matrix. We show the Shannon entropy of the ensemble to be 1.7331±0.0002, which is slightly lower than an ensemble of 9×9 Latin squares, but higher than a certain collection of 9×9 random matrices used for comparison. Using the notion of relative entropy or Kullback–Leibler divergence, which gives a measure of how one distribution differs from another, we show that the relative entropy between the ensemble of Sudoku matrices and Latin squares is of the order of 10−5. By contrast, the relative entropy between Sudoku matrices and the collection of random matrices has the much higher value, being of the order of 10−3, with the Shannon entropy of the Sudoku matrices having better distribution among the modes. We finish by ‘reconstituting’ the ‘average’ Sudoku matrix from its averaged singular components.

Improvements to exact Boltzmann sampling using probabilistic divide-and-conquer and the recursive method

We present a probabilistic divide-and-conquer (PDC) method for \emph{exact} sampling of conditional distributions of the form

Exact Sampling Algorithms for Latin Squares and Sudoku Matrices via Probabilistic Divide-and-Conquer

\mathcal{L}( {\bf X}\, |\, {\bf X} \in E)

COMPLETELY EFFECTIVE ERROR BOUNDS FOR STIRLING NUMBERS OF THE FIRST AND SECOND KINDS VIA POISSON APPROXIMATION

, where

On the random sampling of pairs, with pedestrian examples

{\bf X}

LOG-CONCAVITY OF THE PARTITION FUNCTION

is a random variable on

On the Singularity of Random Bernoulli Matrices— Novel Integer Partitions and Lower Bound Expansions

\mathcal{X}