Intrinsic versus extrinsic dimensionality of ground truths
Dennis Elbrächter  1@  
1 : ETH Zurich

Neural networks tend to be particularly successful in scenarios where the data points are high-dimensional.The learned neural network then, by it's nature, represents a function defined for any input in a high-dimensionalspace. Thus it is natural to also think of the ground truth as a function on a high-dimensional domain whenconsidering it's approximation by neural networks. However, looking at, e.g., image data we can be quite surethat most potential inputs will never actually appear in any relevant task (e.g. in the case of images, only atiny subset of all possible configurations of pixel values will produce a humanly meaningful picture).In particular one might argue that interesting ground truths need to be of significantly lower complexity, i.e.intrinsic dimensionality, than the dimensionality of their input allows (even under other assumptions on theirsimplicity as, e.g., some kind of smoothness). One could now be so bold to go further and conjecture that, infact, this combination of low intrinsic and high extrinsic dimensionality is a key prior which allows for successfullearning by neural networks.I will present my considerations on a formal notion of intrinsic dimensionality, which is designed to be particularlysuitable for the study of approximation by neural networks, and moreover, has an empirical proxy which canbe efficiently computed for finite sets of data points.

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