# Interview about the Institute of Cognitive Science

Published:

I had a lot of fun working with the marketing team for the institute to explain the master’s program from the view of an international student. It is not apparent from a short clip, but there were many hours of raw footage required to produce each minute of video. I happened to be experimenting that day with mapping images to a Hilbert curve space to see if it improved classification of digits using a neural network.

The motivation for using a Hilbert curve is that when images are flattened into a one-dimensional vector, a technique used in training convolutional neural networks, some of the spatial relations between pixels are lost. Hilbert curves preserve the spatial relations by mapping each pixel onto a 1-dimensional curve:

In the end, mapping the image space onto a Hilbert curve didn’t improve image classification, but I enjoyed learning about the algorithm for mapping to Hilbert spaces.

### Math

I contribute to development of the linear algebra library Normaliz developed by the Institute of Mathematics at the University of Osnabrueck, where I made a visualization of Hilbert bases using D3JS and Python for the wikipedia article.

Play with me

What is the Hilbert basis?

Starting with the lattice $L \subset \mathbb{Z}^d$ (all the black dots in this 2-dimensional example) and a convex (bounded) polyhedral (many-sided) cone (imagine the gray shaded part extended infinitely) with generators (the yellow circles) $a_1,\ldots,a_n\in\mathbb{Z}^d$ and

$C={\lambda_1 a_1 + \ldots + \lambda_n a_n \mid \lambda_1,\ldots,\lambda_n \geq 0, \lambda_1,\ldots,\lambda_n \in\mathbb{R}\}\subset\mathbb{R}^d}$,

there is a finite set of generating integral vectors that can produce all the lattice points in the cone.

Simply put, the Hilbert basis of a convex cone is the unique minimal generating set (the small red circles) which can be extended or combined to produce the entire monoid $C\cap L$ (all the lattice points in the cone).

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