A Note on the Erdös Distinct Subset Sums Problem

Dubroff, Quentin; Fox, Jacob; Xu, Max Wenqiang

doi:10.1137/20m1385883

Cited by 9 publications

(9 citation statements)

References 5 publications

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“…Erdős conjectured that the largest weight w ∈ W is upper bounded by c 0 2 n for some c 0 > 0 and therefore, choosing powers of two as the weight set is asymptotically optimal. The best known result for such weight sets yields 0.22002 • 2 n and currently, the best lower bound is Ω(2 n / √ n) [5,7,9]. Now, let us consider the following linear equation where the weights are fixed to the ascending powers of two but x i s are not necessarily binary.…”

Section: B Bijective Mappings From Finite Fields To Integersmentioning

confidence: 99%

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On Algebraic Constructions of Neural Networks with Small Weights

Kilic

Sima

Bruck

2022

2022 IEEE International Symposium on Information Theory (ISIT)

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Neural gates compute functions based on weighted sums of the input variables. The expressive power of neural gates (number of distinct functions it can compute) depends on the weight sizes and, in general, large weights (exponential in the number of inputs) are required. Studying the tradeoffs among the weight sizes, circuit size and depth is a wellstudied topic both in circuit complexity theory and the practice of neural computation. We propose a new approach for studying these complexity trade-offs by considering a related algebraic framework. Specifically, given a single linear equation with arbitrary coefficients, we would like to express it using a system of linear equations with smaller (even constant) coefficients. The techniques we developed are based on Siegel's Lemma for the bounds, anti-concentration inequalities for the existential results and extensions of Sylvester-type Hadamard matrices for the constructions.We explicitly construct a constant weight, optimal size matrix to compute the EQUALITY function (checking if two integers expressed in binary are equal). Computing EQUALITY with a single linear equation requires exponentially large weights. In addition, we prove the existence of the best-known weight size (linear) matrices to compute the COMPARISON function (comparing between two integers expressed in binary). In the context of the circuit complexity theory, our results improve the upper bounds on the weight sizes for the best-known circuit sizes for EQUALITY and COMPARISON.

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Section: B Bijective Mappings From Finite Fields To Integersmentioning

confidence: 99%

“…Since we can also encode Z by its binary expansion, the CRT gives a bijection between Z m and {0, 1} n as long as p 1 • • • p m > 2 n . By taking modulo p i of Equation (7), we can obtain the following matrix, defined as a CRT matrix:…”

Section: B Bijective Mappings From Finite Fields To Integersmentioning

confidence: 99%

On Algebraic Constructions of Neural Networks with Small Weights

Kilic

Sima

Bruck

2022

2022 IEEE International Symposium on Information Theory (ISIT)

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“…No advances have been made so far in removing the term n −1/2 from this lower bound, but there have been several improvements on the constant factor, including the work of Dubroff, Fox, and Xu [12], Guy [13], Elkies [10], Bae [4], and Aliev [3]. In particular, the best currently known lower bound states that…”

Section: Introductionmentioning

confidence: 99%

Variations on the Erdős distinct-sums problem

Costa

Fiore

Dalai

2023

Discrete Applied Mathematics

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“…No advances have been made so far in removing the term n −1/2 from this lower bound, but there have been several improvements on the constant factor, including the work of Dubroff, Fox and Xu [13], Guy [14], Elkies [11], Bae [5], and Aliev [1]. In particular, the best currently known lower bound states that…”

Section: Introductionmentioning

confidence: 99%

Variations on the Erdős distinct-sums problem

Costa¹,

Fiore²,

Dalai³

2021

Preprint

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Let {a 1 , ..., an} be a set of positive integers with a 1 < • • • < an such that all 2 n subset sums are distinct. A famous conjecture by Erdős states that an > c • 2 n for some constant c, while the best result known to date is of the form an > c • 2 n / √ n. In this paper, we weaken the condition by requiring that only sums corresponding to subsets of size smaller than or equal to λn be distinct. For this case, we derive lower and upper bounds on the smallest possible value of an.

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A Note on the Erdös Distinct Subset Sums Problem

Cited by 9 publications

References 5 publications

On Algebraic Constructions of Neural Networks with Small Weights

On Algebraic Constructions of Neural Networks with Small Weights

Variations on the Erdős distinct-sums problem

Variations on the Erdős distinct-sums problem

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