Diagonale

Résumé

vignette|Le segment [D'B'] est une diagonale du carré A'B'C'D'.[D'B'] et [A'C] sont tous deux des diagonales du cube ci-dessus. Dans le plan On appelle diagonale d'un polygone tout segment reliant deux sommets non consécutifs (non reliés par un côté). Un polygone à n côtés possède donc diagonales. Un quadrilatère est un parallélogramme si, et seulement si, ses diagonales se croisent en leur milieu. Dans l'espace On appelle diagonale de l'espace une diagonale d'un polytope, diagonale de l'espace principale une diagonale principale d'un polytope, diagonale de l'espace brisée une diagonale brisée d'un hypercube. On appelle triagonale une diagonale d'un polyèdre, triagonale principale une diagonale principale d'un polyèdre, triagonale brisée une diagonale brisée d'un cube. On appelle quadragonale une diagonale d'un polytope à quatre dimensions, quadragonale principale une diagonale principale d'un polytope à quatre dimensions, quadragonale brisée une diagonale b

Source officielle

https://fr.wikipedia.org/wiki/Diagonale

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Efficient second-order methods for model compression

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Second-order information, in the form of Hessian- or Inverse-Hessian-vector products, is a fundamental tool for solving optimization problems. Recently, there has been a tremendous amount of work on utilizing this information for the current compute and memory-intensive deep neural networks, usually via coarse-grained approximations (such as diagonal, blockwise, or Kronecker-factorization). However, not much is known about the quality of these approximations. Our work addresses this question, and in particular, we propose a method called ‘WoodFisher’ that leverages the structure of the empirical Fisher information matrix, along with the Woodbury matrix identity, to compute a faithful and efficient estimate of the inverse Hessian. Our main application is to the task of compressing neural networks, where we build on the classical Optimal Brain Damage/Surgeon framework (LeCun et al., 1990; Hassibi and Stork, 1993). We demonstrate that WoodFisher significantly outperforms magnitude pruning (isotropic Hessian), as well as methods that maintain other diagonal estimates. Further, even when gradual pruning is considered, our method results in a gain in test accuracy over the state-of-the-art approaches, for standard image classification datasets such as CIFAR-10, ImageNet. We also propose a variant called ‘WoodTaylor’, which takes into account the first-order gradient term, and can lead to additional improvements. An important advantage of our methods is that they allow us to automatically set the layer-wise pruning thresholds, avoiding the need for any manual tuning or sensitivity analysis.

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Atomic-scale insights into the origin of rectangular lattice in nanographene probed by scanning tunneling microscopy

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We conducted atomic-scale scanning tunneling microscopy of a graphene nanosheet on graphite. In addition to a rhombus lattice representing the (root 3x root 3)R30 degrees superstructure, we resolved another quadrangle lattice similar to a rectangle in the sheet. Its lattice size was approximately 0.37 x 0.22 nm(2) . To clarify the origin of this unique rectangular lattice, the overlap of the root 3x root 3 superstructures along the direction of their long diagonals was theoretically examined using a simple model. The electron distribution with high energy in the occupied states of armchair-edged graphene nanoribbons (AGNRs) was calculated based on first principles. A rectangular lattice, resembling the one observed experimentally, was found to form on the AGNR under a specific width condition. This finding was also analyzed in terms of Clar's theory and the scattering of electron waves. We propose that wrinkles and adsorbates in graphene play a role similar to an armchair edge, resulting in the root 3x root 3 phase. If these local defects are in close proximity, the rhombus phases interact to generate electronic structures predicted for AGNRs. This is probably the reason why a rectangular lattice was imaged on the graphene sheet that is not an ideal AGNR.

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The double exponential runtime is tight for 2-stage stochastic ILPs

Klaus Jansen, Kim-Manuel Klein, Alexandra Anna Lassota

We consider fundamental algorithmic number theoretic problems and their relation to a class of block structured Integer Linear Programs (ILPs) called 2-stage stochastic. A 2-stage stochastic ILP is an integer program of the form min{c(T)x vertical bar Ax = b, l

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