Unbiased Matrix Rounding

Abstract

We show several ways to round a real matrix to an integer one such that the rounding errors in all rows and columns as well as the whole matrix are less than one. This is a classical problem with applications in many fields, in particular, statistics.We improve earlier solutions of different authors in two ways. For rounding matrices of size m × n, we reduce the runtime from O((mn) 2 ) to O(mn log(mn)). Second, our roundings also have a rounding error of less than one in all initial intervals of rows and columns. Consequently, arbitrary intervals have an error of at most two. This is particularly useful in the statistics application of controlled rounding.The same result can be obtained via (dependent) randomized rounding. This has the additional advantage that the rounding is unbiased, that is, for all entries yij of our rounding, we have E(yij) = xij , where xij is the corresponding entry of the input matrix.

Construction of Low-Discrepancy Point Sets of Small Size by Bracketing Covers and Dependent Randomized Rounding

“…However, we can achieve much smaller rounding errors by using a very recent result of Güntürk, Yılmaz and the first author [DGY06]. Let us remark first that there are higher-dimensional matrices…”

“…The surprising result of [DGY06] is that by allowing larger deviations in the variables we can guarantee much better errors in the subarrays. In particular, we are able to remove any dependence on the grid size k. To ease reading, we reformulate and prove their result in our language.…”

Unbiased Rounding of Rational Matrices

“…1 q Z m×n a rounding Y ∈ Z m×n such that the inequalities (1), (2) and (3) Previous results on this particular rounding problem were given by Doerr et al in [7]. Theorem 2 extends their result to arbitrary rational matrices.…”

“…Otherwise, simply subtract the integral part of X before rounding and add it again afterwards. Furthermore, we assume X to have integral row and column sums, as justified by the following lemma from [7].…”

“…Using this observation and choosing q = 2 , this gives the result from [7] for unbiased rounding of matrices of -bit numbers. …”