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Preprints, Working Papers, ... Year : 2022

## On the Multidimensional Random Subset Sum Problem

Arthur Carvalho Walraven da Cunha
Andrea Clementi
• Function : Author
Francesco d'Amore
Hicham Lesfari
• Function : Author
• PersonId : 1133457
Emanuele Natale
Luca Trevisan
• Function : Author

#### Abstract

In the Random Subset Sum Problem, given $n$ i.i.d. random variables $X_1, ..., X_n$, we wish to approximate any point $z \in [-1,1]$ as the sum of a suitable subset $X_{i_1(z)}, ..., X_{i_s(z)}$ of them, up to error $\varepsilon$. 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 $\mathbf{z} \in [-1,1]^d$. In 1998, G. S. Lueker showed that, in the one-dimensional setting, $n=\mathcal{O}(\log \frac 1\varepsilon)$ samples guarantee the approximation property with high probability. In this work, we prove that, in $d$ dimensions, $n = \mathcal{O}(d^3\log \frac 1\varepsilon \cdot (\log \frac 1\varepsilon + \log d))$ 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.

### Dates and versions

hal-03738204 , version 1 (25-07-2022)
hal-03738204 , version 2 (26-07-2022)
hal-03738204 , version 3 (15-11-2022)

### Identifiers

• HAL Id : hal-03738204 , version 3

### Cite

Luca Becchetti, Arthur Carvalho Walraven da Cunha, Andrea Clementi, Francesco d'Amore, Hicham Lesfari, et al.. On the Multidimensional Random Subset Sum Problem. 2022. ⟨hal-03738204v3⟩

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