Efficient estimation of the ANOVA mean dimension, with an application to neural net classification
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The mean dimension of a black box function of d variables is a convenient way to summarize the extent to which it is dominated by high or low order interactions. It is expressed in terms of 2^d-1 variance components but it can be written as the sum of d Sobol' indices that can be estimated by leave one out methods. We compare the variance of these leave one out methods: a Gibbs sampler called winding stairs, a radial sampler that changes each variable one at a time from a baseline, and a naive sampler that never reuses function evaluations and so costs about double the other methods. For an additive function the radial and winding stairs are most efficient. For a multiplicative function the naive method can easily be most efficient if the factors have high kurtosis. As an illustration we consider the mean dimension of a neural network classifier of digits from the MNIST data set. The classifier is a function of 784 pixels. For that problem, winding stairs is the best algorithm. We find that inputs to the final softmax layer have mean dimensions ranging from 1.35 to 2.0.
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