Bounds for the number of threshold functions

Hassett constructed a class of modular compactifications of Mg,n by adding weights to the marked points. This leads to a natural wall and chamber decomposition of the domain of admissible weights Dg,n, where the moduli space and universal family remain constant inside a chamber, and may change upon crossing a wall. The goal of this paper is to count the number of chambers in this decomposition. We relate these chambers to a class of boolean functions known as linear threshold functions (LTFs), and discover a subclass of LTFs which are in bijection with the chambers. Using this relation, we prove an asymptotic formula for the number of chambers, and compute the exact number of chambers for n ≤ 9. In addition, we provide an algorithm for the enumeration of chambers of Dg,n and prove results in computational complexity.Acknowledgements. for helpful discussions and suggestions. We thank Nicolle Gruzling for providing us with code used to enumerate linear threshold functions for n ≤ 9. Finally, we thank the referee for many usual suggestions that helped improve this paper.2. Hassett's chamber decomposition D g,n

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“…Theorem 6.1. [Irm96] The number of linear threshold functions on n variables satisfies the asymptotic formula…”

Section: Asymptotic Resultsmentioning

confidence: 99%

Enumerating Hassett’s Wall and Chamber Decomposition of the Moduli Space of Weighted Stable Curves

Ascher

Dubé

Gershenson

et al. 2018

Experimental Mathematics

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“…Although the dimensionality of W is merely M × n, which is much smaller than the 2 n -dimensional space of probabilities, (10) can generate O(2 n 2 ) distinct perceptrons [11]. By including an appropriate threshold, a perceptron can assign any individual state x a positive response and assign a negative response to every other state.…”

Section: Measurements By Random Perceptronsmentioning

confidence: 99%

“…In a Boltzmann Machine, binary states x occur with probabilities given by the Boltzmann distribution p(x) ∝ e −E(x) (11) for an energy function…”

Section: Fidelity Of Compressed Sparse Distributionsmentioning

confidence: 99%

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Compressive neural representation of sparse, high-dimensional probabilities

Pitkow

2012

Preprint

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This paper shows how sparse, high-dimensional probability distributions could be represented by neurons with exponential compression. The representation is a novel application of compressive sensing to sparse probability distributions rather than to the usual sparse signals. The compressive measurements correspond to expected values of nonlinear functions of the probabilistically distributed variables. When these expected values are estimated by sampling, the quality of the compressed representation is limited only by the quality of sampling. Since the compression preserves the geometric structure of the space of sparse probability distributions, probabilistic computation can be performed in the compressed domain. Interestingly, functions satisfying the requirements of compressive sensing can be implemented as simple perceptrons. If we use perceptrons as a simple model of feedforward computation by neurons, these results show that the mean activity of a relatively small number of neurons can accurately represent a highdimensional joint distribution implicitly, even without accounting for any noise correlations. This comprises a novel hypothesis for how neurons could encode probabilities in the brain.

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“…The generalization of the inequality (7) for the number of threshold klogic functions was obtained in [9]. Asymptotics of logarithm of the number of polinomial threshold functions has been recently obtained in [1].…”

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confidence: 99%

A Lower Bound of the Number of Threshold Functions in Terms of Combinatorial Flags on the Boolean Cube

Irmatov

2018

Preprint

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and q W l := |span(w i n−l+1 , . . . , w in ) ∩ E|. Then for any weights p = (p 1 , . . . , p 2 n ), p i ∈ R, 2 n i=1 p i = 1 we have for the number of threshold functions P (2, n) the following lower boundand the right side of the inequality doesn't depend on the choice of p. Here the indices used in the numerator correspond to vectors from span(w i 1 , . . . , w in ) ∩ E = w i 1 , . . . , w in , . . . w i q W n .

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Bounds for the number of threshold functions

Cited by 9 publications

References 8 publications

Enumerating Hassett’s Wall and Chamber Decomposition of the Moduli Space of Weighted Stable Curves

Enumerating Hassett’s Wall and Chamber Decomposition of the Moduli Space of Weighted Stable Curves

Compressive neural representation of sparse, high-dimensional probabilities

A Lower Bound of the Number of Threshold Functions in Terms of Combinatorial Flags on the Boolean Cube

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