Why skewing works: learning difficult Boolean functions with greedy tree learners
ICML '05 Proceedings of the 22nd international conference on Machine learning
Exploiting Product Distributions to Identify Relevant Variables of Correlation Immune Functions
The Journal of Machine Learning Research
On balanced colorings of the n-cube
Journal of Algebraic Combinatorics: An International Journal
Equivalence classes of Boolean functions for first-order correlation
IEEE Transactions on Information Theory
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Weights of 1 or 0 are assigned to the vertices of the n-cube in n-dimensional Euclidean space. Such an n-cube is called balanced if its center of mass coincides precisely with its geometric center. The seldom-used n-variable form of Pólya's enumeration theorem is applied to express the number Nn, 2k of balanced configurations with 2k vertices of weight 1 in terms of certain partitions of 2k. A system of linear equations of Vandermonde type is obtained, from which recurrence relations are derived which are computationally efficient for fixed k. It is shown how the numbers Nn, 2k depend on the numbers An, 2k of specially restricted configurations. A table of values of Nn, 2k and An, 2k is provided for n = 3, 4, 5, and 6. The case in which arbitrary, nonnegative, integral weights are allowed is also treated. Finally, alternative derivations of the main results are developed from the perspective of superposition.