Quick hack: visualizing RBF bandwidth

A few weeks ago, I was explaining the general concepts behind support vector machine classifiers. and kernels.  The majority of the audience has no background in linear algebra, so I had to rely on a lot of analogies and pictures.  I had previously introduced the notions of decision function and decision boundary (the zero-crossing of the decision function), and described dot products, projections, and linear equations, as simply as possible.

The overview of SVMs was centered around the observations that the decision function is, eventually, a weighted additive superposition (linear combination) of evaluations of “things that behave like projections in a higher-dimensional space via a non-linear mapping” (kernel functions) over the support vectors (a subset of the training samples, chosen based on the idea of “fat margins”).

Most of the explanations and pictures were based on linear functions, but I wanted to give an idea of what these kernels look like, how their “superposition” looks like, and how kernel parameters vary the picture (and may relate to overfitting).   For that I chose radial basis functions. I found myself doing a lot of handwaving in the process, until I realized that I could whip up an animation.  Following that class, I had 1.5 hours during another midterm, so I did that (Python with Matplotlib animations, FTW!!).  The result follows.

Here is how the decision boundary changes as the bandwidth becomes narrower:

For large radii, there are a fewer support vectors and kernel evaluations cover a large swath of the space.  As the radii shrink, all points become support vectors, and the SVM essentially devolves into a “table model” (i.e., the “model” is the data, and only the data, with no generalization ability whatsoever).

This decision boundary is the zero-crossing of the decision function, which can also be fully visualized in this case.  One way to understand this is that the non-linear feature mapping “deforms” the 2D-plane into a more complex surface (where, however, we can still talk about “projections”, in a way), in such a way that I can still use a plane (z=0) to separate the two classes.  Here is how that surface changes, again as the bandwidth becomes narrower:

Finally, in order to justify that, for this dataset, a really large radius is the appropriate choice, I ran the same experiments with multiple random subsets of the training data and showed that, for large radii, the decision boundaries are almost the same across all subsets, but for smaller radii, they start to diverge significantly.

Here is the source code I used (warning: this is raw and uncut, no cleanup for release!).  One of these days (or, at least, by next year’s class) I’ll get around to making multiple concurrent, alpha-blended animations for different subsets of the training set, to illustrate the last point better (I used static snapshots instead) and also give a nice visual illustration of model testing and ideas behind cross-validation; of course, feel free to play with the code. ;)


Thingiverse remix graph: visualizing the net of physical things

I recently became a happy owner of a Solidoodle 2 3D printer. This has been the start of a beautiful addiction, but more on the hardware hacking aspects in another post.

If you haven’t heard of it before, 3D printing refers to a family of manufacturing methods, originally developed for rapid prototyping, the first of which appeared almost three decades ago. Much like mainframe computers in the 1960s, professional 3D printers cost up to hundred thousands of dollars. Starting with the RepRap project a few years ago, home 3D printers are now becoming available, in the few hundred to a couple of thousand dollar price range.  For now, these are targeted mostly to tinkerers, much closer to an Altair or, at best, an Apple II, than a MacBook. Despite the hype that currently surrounds 3D printing, empowering average users to turn bits into atoms (and vice versa) will likely have profound effects, similar to those witnessed when content (music, news, books, etc) went digital, as Chris Anderson eloquently argues with his usual, captivating dramatic flair. Personally, I’m similarly excited about this as I was about “big data” (for lack of a better term) around 2006 and mobile around 2008, so I’ll take this as a good sign. :)

One of the key challenges, however, is finding things to print!  This is crucial for 3D printing to really take off. Learning CAD software and successfully designing 3D objects takes substantial, time, effort, and skill. Affordable 3D scanners (like the ones from MatterformCADscan, and Makerbot) are beginning to appear. However, the most common way to find things is via online sharing of designs. Thingiverse is the most popular online community for “thing” sharing. Thingiverse items are freely available (usually under Creative Commons licenses), but there is also commercial potential: companies like Shapeways offer both manufacturing (using industrial 3D printers and manual post-processing) and marketing services for “thing” designs.

I’ve become a huge fan of Thingiverse.  You can check out my own user profile to find things that I’ve designed myself, or things that I’ve virtually “collected” because I thought they were really cool or useful (or both). Thingiverse is run by MakerBot, which manufactures and sells 3D printers, and needs to help people find things to print. It’s a social networking site centered around “thing” designs. Consequently, the main entities are people (users) and things, and links/relationships revolve around people creating things, people liking things, people downloading and making things, people virtually collecting things, and so on. Other than people-thing relationships, links can also represent people following other people (a-la Twitter or Facebook), and things remixing other things (more on this soon). Each thing also has a number of associated files (polygon meshes for 3D printing, vector paths for lasercutting, original CAD files—anything that’s needed to make the thing).

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