Fixed-Parameter Complexity and Approximability of Norm Maximization

Knauer, Christian; König, Stefan; Werner, Daniel

doi:10.1007/s00454-015-9667-0

Cited by 6 publications

(3 citation statements)

References 24 publications

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“…Taking into account the hardness of approximating the circumradius of an H-presented polytope even around a fixed center (cf. [15,37]), the improvement of the bound by the computation of s 0 (P ) is quasi at no cost.…”

Section: Proofmentioning

confidence: 99%

Sharpening Geometric Inequalities Using Computable Symmetry Measures

Brandenberg

König

2014

Mathematika

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Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the convex body. Since these coefficients are bounded by the dimension but possibly smaller, our inequalities sharpen the original ones. Since they can often be computed efficiently, the improved bounds may also be used to obtain better bounds in approximation algorithms.

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Section: Proofmentioning

confidence: 99%

Sharpening Geometric Inequalities Using Computable Symmetry Measures

Brandenberg

König

2014

Mathematika

Self Cite

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“…Finally, note that the number of dimensions appears naturally in parameterized complexity studies for geometric problems (Giannopoulos, Knauer, & Rote, 2009;Knauer, König, & Werner, 2015); moreover, it occurs also in recent studies for principal component analysis (PCA) (Fomin, Golovach, & Simonov, 2020;Simonov, Fomin, Golovach, & Panolan, 2019) and in computer vision (Chin, Cai, & Neumann, 2020).…”

Section: Introductionmentioning

confidence: 90%

The Computational Complexity of ReLU Network Training Parameterized by Data Dimensionality

Froese¹,

Hertrich

Niedermeier

2022

jair

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Understanding the computational complexity of training simple neural networks with rectified linear units (ReLUs) has recently been a subject of intensive research. Closing gaps and complementing results from the literature, we present several results on the parameterized complexity of training two-layer ReLU networks with respect to various loss functions. After a brief discussion of other parameters, we focus on analyzing the influence of the dimension d of the training data on the computational complexity. We provide running time lower bounds in terms of W[1]-hardness for parameter d and prove that known brute-force strategies are essentially optimal (assuming the Exponential Time Hypothesis). In comparison with previous work, our results hold for a broad(er) range of loss functions, including lp-loss for all p ∈ [0, ∞]. In particular, we improve a known polynomial-time algorithm for constant d and convex loss functions to a more general class of loss functions, matching our running time lower bounds also in these cases.

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“…Finally, note that the number of dimensions appears naturally in parameterized complexity studies for geometric problems [Giannopoulos et al, 2009, Knauer et al, 2015; moreover, it occurs also in recent studies for principal component analysis (PCA) [Fomin et al, 2020, Simonov et al, 2019 and in computer vision [Chin et al, 2020].…”

Section: Introductionmentioning

confidence: 95%

The Computational Complexity of ReLU Network Training Parameterized by Data Dimensionality

Froese,

Hertrich,

Niedermeier

2021

Preprint

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Understanding the computational complexity of training simple neural networks with rectified linear units (ReLUs) has recently been a subject of intensive research. Closing gaps and complementing results from the literature, we present several results on the parameterized complexity of training two-layer ReLU networks with respect to various loss functions. After a brief discussion of other parameters, we focus on analyzing the influence of the dimension d of the training data on the computational complexity. We provide running time lower bounds in terms of W[1]-hardness for parameter d and prove that known brute-force strategies are essentially optimal (assuming the Exponential Time Hypothesis). In comparison with previous work, our results hold for a broad(er) range of loss functions, including ℓ p -loss for all p ∈ [0, ∞]. In particular, we extend a known polynomialtime algorithm for constant d and convex loss functions to a more general class of loss functions, matching our running time lower bounds also in these cases.Preprint. Under review.

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Fixed-Parameter Complexity and Approximability of Norm Maximization

Cited by 6 publications

References 24 publications

Sharpening Geometric Inequalities Using Computable Symmetry Measures

Sharpening Geometric Inequalities Using Computable Symmetry Measures

The Computational Complexity of ReLU Network Training Parameterized by Data Dimensionality

The Computational Complexity of ReLU Network Training Parameterized by Data Dimensionality

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