On the Multidimensional Random Subset Sum Problem
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In the Random Subset Sum Problem, given n i.i.d. random variables X_1, ..., X_n, we wish to approximate any point z ∈ [-1,1] as the sum of a suitable subset X_i_1(z), ..., X_i_s(z) of them, up to error ε. Despite its simple statement, this problem is of fundamental interest to both theoretical computer science and statistical mechanics. More recently, it gained renewed attention for its implications in the theory of Artificial Neural Networks. An obvious multidimensional generalisation of the problem is to consider n i.i.d. d-dimensional random vectors, with the objective of approximating every point 𝐳∈ [-1,1]^d. Rather surprisingly, after Lueker's 1998 proof that, in the one-dimensional setting, n=O(log1/ε) samples guarantee the approximation property with high probability, little progress has been made on achieving the above generalisation. In this work, we prove that, in d dimensions, n = O(d^3log1/ε· (log1/ε+ log d)) samples suffice for the approximation property to hold with high probability. As an application highlighting the potential interest of this result, we prove that a recently proposed neural network model exhibits universality: with high probability, the model can approximate any neural network within a polynomial overhead in the number of parameters.
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