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Hubness in 3D

August 13th, 2011

As I was delving deeper and deeper into the hubness-related topics, I also started making some visualizations which would enrich our intuition about the problems at hand. This was not, however, an easy task. Hubness is a property of inherently high dimensional data – where influential points (hubs) emerge as a consequence of distance concentration and other peculiar geometric phenomena which plague such high dimensional spaces. As humans, we can not, however, visualize these things in their original dimensionality, we have to significantly reduce it and represent it in some other way. This being as it is – I have either been applying multidimensional scaling onto high dimensional data OR playing around with 2D data (even though hubness as such does not and can not exist in 2D) to see if I could gain some more insight into the related problems. After all, the hubness-aware classification algorithms that I have been developing (and still am) are also applicable to lower dimensional data and can lead to significant improvements even in such restricted cases.

So, some time ago – I thought about this – 3D data is still not high dimensional and no proper hubness can be found there – but it has to be better than 2D, so maybe I could see some more interesting things if I dare taking a peek into 3D nearest-neighbor structures. And so I did. Long story short – I took some UCI data, performed some dimensionality reduction to reduce it to 3D – and then determined the class affiliation probabilities of each voxel by applying the above-mentioned hubness-aware algorithms (most of which you can find in my publication list). Also, I compared the results to the basic k-nearest neighbor methods. The Binn-Phong shading model was used for lighting the images (so that the 3D structures can be more easily perceived). The volume of each cube was projected onto each of its sides, so 6 images were created for each cube.

I am not going to discuss any of the comparisons nor discuss the few selected images which I’ve given here. I have generated many such images, but what I wanted to do here was to select some of them which simply – looked cool 🙂 – and make a sort of a gallery of interesting/weird hubness-related 3D images. Not very scientific, I know, but there are always publications/conferences/journals for serious stuff. No reason why we shouldn’t have some fun every now and then. So, enjoy 😉

generated by HIKNN generated by HIKNN
generated by HIKNN generated by HIKNN
generated by kNN generated by kNN
generated by HIKNN generated by hFNN
generated by kNN generated by HIKNN
generated by HIKNN generated by HIKNN
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