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Nicolau Werneck (Nic)
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Engineer with a love for loud music and horror films.
Engineer with a love for loud music and horror films.

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Nicolau Werneck (Nic) commented on a post on Blogger.
This is some great investigation, too often people believe C++ can take you to a very different level of performance, but that is many times not the truth.

It is my experience that numerical programming like this can greatly benefit from "vector based" processing, where all those loops get done inside higher level functions that can operate with large vectors and arrays. On Scala you might achieve this using something like Breeze, for instance. Have you ever tried anything like that?

Also a good way to develop such a project is to actually have a separate core implemented in C or Java, but using a language like Scala or Python putting everything together. Usually a good way to have both high performance but all the practicality and reliability from higher level language... But I'm not sure in this project what you might be able to easily separate in a native library.

Anyway, great work!
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The Karpman drama triangle of deep learning

The Victim: MNIST digits and unclassified pictures of cats and teapots
The Persecutor: syllogisms and hand-crafted features
The Rescuer: Ten thousand layers of nonlinear functions furiously hill-climbing towards the next nine of your test set accuracy

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Automatic differentiation remains a terribly little used technique, and is maybe one of the reasons continuous optimizaiton methods based on gradient and Hessian calculations also share the same fate a bit. This is a little demo of a python package for AD, also illustrating the robust Tukey bisquare error function.

Although many AD tools are available out there, many actually have a problem that they share with tools for symbolic processing such as Wolfram Alpha: they cannot deal directly with summations, what is very important when you are using these tools to calculate statistics over samples...

https://gist.github.com/nlw0/f49ec407d8b2e0d98ba25f02f6174cae
robust_fit_ad.ipynb
robust_fit_ad.ipynb
gist.github.com
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I've been reading a little about "surpassed" structure-from-texture techniques from the 90s and 2000s, this is a little experiment to visualize what happens in the spectrum magnitude when you apply a perspective distortion to a sinusoidal image. What I really should have plotted is what happens when the distortion is in the direction orthogonal to the image equi-potential lines, or parallel. What I think the methods really relied on was on the orthogonal, but I am actually specifically interested in the other case.

https://gist.github.com/nlw0/da83343b1b83bc5a5afbb322f802fbf7
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New paper from +Alec Radford, +Luke Metz and me on stabilizing generative adversarial networks. Alec and Luke are quite awesome.

A summary of the paper: https://github.com/Newmu/dcgan_code
Arxiv link: http://arxiv.org/abs/1511.06434
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Keleti's Perimeter to Area Conjecture

It is clear that dividing the perimeter of a square of side 1 by its area results in a ratio of 4. Doing the same for two adjacent unit squares that share an edge results in a smaller ratio, in this case 3. So what can be said about this ratio in the case of an arbitrary union of (possibly overlapping) unit squares in the plane? Keleti's Perimeter to Area Conjecture was that this ratio never exceeds 4, although this is now known to be false. The picture shows a counterexample to the conjecture in which the ratio is approximately 4.28.

A problem related to this one appeared as Problem 6 on the famous Hungarian Schweitzer Competition in 1998. That problem asked for a proof that the perimeter-to-area ratio of a union of unit squares in the plane has an upper bound. Impressively, several Hungarian undergraduates were able to prove this for the competition. In the same year, Tamás Keleti published his Perimeter to Area Conjecture that the bound was exactly 4. This is now known to be incorrect; the best known bound is about 5.551 and was found by Keleti's student Zoltán Gyenes in his 2005 Master's thesis.

I found out about this problem from the paper Bounded – yes, but 4? (http://arxiv.org/abs/1507.08536) by Paul D. Humke, Cameron Marcott, Bjorn Mellem and Cole Stiegler. This paper was posted recently but is dated November 2013, and it does not mention progress on the problem that has been made since then. The 2014 paper Unions of regular polygons with large perimeter-to-area ratio (http://arxiv.org/abs/1402.5452) by Viktor Kiss and Zoltán Vidnyánszky proves that Keleti's conjecture is false. The picture here comes from Kiss and Vidnyánszky's paper.

The counterexample in the picture uses 25 unit squares, but Kiss and Vidnyánszky use a systematic method to construct (in Theorem 2.4) a counterexample using only five unit squares, and use an ad hoc method to construct (in Section 3) a counterexample using only four squares. They also (in Theorem 2.8) show that the analogue of Keleti's conjecture for equilateral triangles is false, by exhibiting a counterexample involving four triangles. The authors conjecture that similar constructions can be made for other regular polygons; more specifically, that their systematic method can be used to produce an analogous configuration of n+1 regular n-gons that has a greater perimeter-to-area ratio than a single regular n-gon. 

Kiss and Vidnyánszky pose a number of other interesting questions in their closing section. For example, is n the minimal possible number of n-gons in a counterexample? And does something analogous happen in three dimensions, with regular polyhedra?

Although Keleti's conjecture is false, it is known to be true under certain additional hypotheses. In his Master's thesis and in a 2011 paper, Gyenes proved that the conjecture holds (a) if the squares have a common centre, or (b) if all the squares have sides parallel to the x or y axes, or (c) if only two squares are involved.

Relevant links

Gyenes' Masters thesis: http://www.cs.elte.hu/~dom/z.pdf

A mathoverflow discussion of Keleti's Perimeter to Area Conjecture from 2010 (http://mathoverflow.net/questions/15188/) includes some interesting commentary on the problem from various people, including +Timothy Gowers.

The picture contains an oblique reference to Betteridge's law of headlines (https://en.wikipedia.org/wiki/Betteridge's_law_of_headlines) which states that Any headline that ends in a question mark can be answered by the word “no”. This principle also applies to the title of the paper by Humke et al. 

#mathematics #sciencesunday #spnetwork arXiv:1507.08536
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A classic-punk playlist I created some time ago. Trying to make it catch. http://open.spotify.com/user/nwerneck/playlist/3dJVQLNDoyp0GGhCbGL67q
Punk Klassix
Punk Klassix
open.spotify.com
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