Wednesday, February 20, 2008

Nice Gaussian hubs

In trying to prove that anti-hubs are the root of all CBR evil (at least in my world), I've mainly looked at the fact that they have strange distributions, which typical modeling methods (i.e. GMMs!) tend to model in a way that isn't exactly helpful.
Recently, I've been looking at non-parametric modeling and through this got a nice visualization of the distributions of MFCC frames (probably something I should have done from the beginning!). Below I show the distributions for the first 6 MFCC dimensions (by row, so first row has MFCC's 1 through 3), first for a prototypical hub (Carly Simon's "We Have No Secrets", 368 100-occurrences) and then for a prototypical anti-hub (Bloodhound Gang's "Your Only Friends Are Make Believe", 2 100-occurrences).

We see that, indeed, a hub has nice, relatively Gaussian distributions, while the anti-hub's are nasty and multi-modal. This further vindicates the rationale for homogenization: modes exist in the distribution of anti-hubs' frames that are perhaps not relevant to a good timbral model and we'd like to get rid of them. Homogenization pushed to its extreme, after all, would lead to a nice single Gaussian.
To see if this idea really generalizes, I looked at how well each distribution fit a single Gaussian distribution, parameterized to the distribution's mean and variance. Below is the scatter plot of hubness vs. the log-likelihood.
It's not as strong a correlation (rho = 0.0939, p-val = 0.0196) as I was expecting from just looking at the histograms. We do see that the top most and least likely single Gaussians are anti-hubs. I think this could be explained by songs with lots of a single timbre (i.e. silence). This would mean lots of samples fall near the mean and there would be a very small variance, leading to a high likelihood values, while all of the relevant frames are far from this mean (i.e. music) and receive low likelihood values. This is the case with our favorite subset of tracks: those with "hidden" tracks, like the previously mentioned Jamiroquai track and "Chris Cayton" by Goldfinger (2 100-occurences, MFCC histograms below) (check out the comments on its page).

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