The Curse of Dimensionality

What has been called The curse of dimensionality is a phenomenon that has some relevance to Floyd Toole’s work and to many of the ideas that flourish in those audiophile forums that think of themselves as being at the scientific end of the spectrum. In a nutshell:

when the dimensionality increases, the size of the space increases so fast that the available data become sparse. This sparsity is problematic for any method that requires statistical significance.

Having followed a few online discussions in recent days, it seems clear to me that many people think that Floyd Toole’s experiments were/are controlling only a single variable: the speakers’ ‘directionality’. This is far from the truth*. And even if the tests had been genuinely controlling a single variable, the fixing of the other variables would have restricted the experiments to such a tiny subset of the overall problem space as to make them potentially meaningless**. In reality, the experiments created a few sparse pieces of data within a small fraction of the overall problem space.

In this particular case the research was attempting to confirm the answers to questions which, without doing any experiments at all, most people would have given the ‘correct’ answers to anyway:

“Which do you think is better? Flat on-axis frequency response or not? Flat off-axis response or not? Smooth off-axis response or not?”

Which is not to say that identifying the questions in the first place was not a significant achievement. It is just the pseudo-science of the experiments, the implication that the methodology can be extended to ‘solve’ all audio problems, and the conclusions that people draw from the results that I object to.


* Directionality was not a direct variable. Instead, a selection of existing speakers with varying characteristics were listened to, and their directionality measured. A ‘directionality index’ was created ‘heuristically’. Many other incidental variables related to the speakers were not controlled in the experiments.

** For example, if we restrict all the speakers to the same position in the room regardless of whether they work best there or not, restrict the choice of music to a few pieces of audiophile music, restrict the listening to mono, restrict SPL to the capabilities of the smallest speaker in the selection etc. etc. To test the entire ‘problem space’ meaningfully would require thousands of experiments – the curse of dimensionality.

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