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Concept# Diagonal

Summary

In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word diagonal derives from the ancient Greek διαγώνιος diagonios, "from angle to angle" (from διά- dia-, "through", "across" and γωνία gonia, "angle", related to gony "knee"); it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as diagonus ("slanting line").
In matrix algebra, the diagonal of a square matrix consists of the entries on the line from the top left corner to the bottom right corner.
There are also many other non-mathematical uses.
Non-mathematical uses
In engineering, a diagonal brace is a beam used to brace a rectangular structure (such as scaffolding) to withstand strong forces pushing into it; although called a diagonal, due to practical considerations diagonal braces are often not conn

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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

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.

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.

2020