Subtraction-Free Complexity, Cluster Transformations, and Spanning Trees

Fomin, Sergey; Grigoriev, Dima; Koshevoy, Gleb A.

doi:10.1007/s10208-014-9231-y

Cited by 12 publications

(20 citation statements)

References 26 publications

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“…The complexity of computation in this restricted model can be compared to that in other computational models which allow additional operations beyond addition and multiplication, or allow free reuse of already computed polynomials (straight-line computation). It is much easier to compute polynomials in models with subtraction [21] or division [10] than with only addition and multiplication [6,12,16,19]. Indeed, similar phenomena occur in the computation of integers as well as that of polynomials [5].…”

Section: Computability Questionsmentioning

confidence: 99%

Integer complexity and well-ordering

Altman¹

2015

Michigan Math. J.

123

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Abstract. Define n to be the complexity of n, the smallest number of ones needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that n ≥ 3 log 3 n for all n. Define the defect of n, denoted δ(n), to be n − 3 log 3 n. In this paper, we consider the set D := {δ(n) : n ≥ 1} of all defects. We show that as a subset of the real numbers, the set D is well-ordered, of order type ω ω . More specifically, for k ≥ 1 an integer, D ∩ [0, k) has order type ω k . We also consider some other sets related to D, and show that these too are well-ordered and have order type ω ω .

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Section: Computability Questionsmentioning

confidence: 99%

Integer complexity and well-ordering

Altman¹

2015

Michigan Math. J.

123

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“…By using a logarithmic transformation, DLSE T networks map to a family of ratios of generalized posynomials functions, which we show to be subtraction free universal approximators of positive functions over compact subsets of the positive orthant. Subtraction free expressions are fundamental objects in algebraic complexity, studied in particular in [8]. It is a result of independent interest that subtraction free expressions provide universal approximators.…”

Section: B Contributionmentioning

confidence: 99%

“…We next derive from Theorem 3 an approximation result by subtraction-free expressions. The latter are an important subclass of rational expressions, studied in [8]. Subtractionfree expressions are well formed expressions in several commutative variables x 1 , .…”

Section: B Universal Approximation By Subtraction-free Expressionsmentioning

confidence: 99%

A Universal Approximation Result for Difference of Log-Sum-Exp Neural Networks

Calafiore

Gaubert

Possieri

2020

IEEE Trans. Neural Netw. Learning Syst.

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We show that a neural network whose output is obtained as the difference of the outputs of two feedforward networks with exponential activation function in the hidden layer and logarithmic activation function in the output node (LSE networks) is a smooth universal approximator of continuous functions over convex, compact sets. By using a logarithmic transform, this class of networks maps to a family of subtractionfree ratios of generalized posynomials, which we also show to be universal approximators of positive functions over log-convex, compact subsets of the positive orthant. The main advantage of Difference-LSE networks with respect to classical feedforward neural networks is that, after a standard training phase, they provide surrogate models for design that possess a specific differenceof-convex-functions form, which makes them optimizable via relatively efficient numerical methods. In particular, by adapting an existing difference-of-convex algorithm to these models, we obtain an algorithm for performing effective optimization-based design. We illustrate the proposed approach by applying it to data-driven design of a diet for a patient with type-2 diabetes.

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“…In the case of the approximation factor = 1 (exact solution), we already know the answer: subtraction gates can then even exponentially decrease the circuit size. Namely, we already know that both directed and undirected versions of the MST problem (minimum weight spanning tree problem) on -vertex graphs require tropical (min, +) circuits of size 2 Ω( √ ) [15,17] but, as shown by Fomin, Grigoriev and Koshevoy [10], both these problems are solvable by tropical (min, +, −) circuits of size only ( 3 ). Unfortunately, no non-trivial lower bounds for (min, +, −) circuits are known so far.…”

Section: Problem 4 Is 2 (M ) Polynomial In ?mentioning

confidence: 99%

Approximation Limitations of Pure Dynamic Programming

Jukna¹,

Seiwert²

2020

SIAM J. Comput.

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We prove the first, even super-polynomial, lower bounds on the size of tropical (min,+) and (max,+) circuits approximating given optimization problems. Many classical dynamic programming (DP) algorithms for optimization problems are pure in that they only use the basic min, max, + operations in their recursion equations. Tropical circuits constitute a rigorous mathematical model for this class of algorithms. An algorithmic consequence of our lower bounds for tropical circuits is that the approximation powers of pure DP algorithms and greedy algorithms are incomparable. That pure DP algorithms can hardly beat greedy in approximation, is long known. New in this consequence is that also the converse holds.

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Subtraction-Free Complexity, Cluster Transformations, and Spanning Trees

Cited by 12 publications

References 26 publications

Integer complexity and well-ordering

Integer complexity and well-ordering

A Universal Approximation Result for Difference of Log-Sum-Exp Neural Networks

Approximation Limitations of Pure Dynamic Programming

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