All I need to break this logic is write down a model whose posterior doesn’t yield a linear predictor as optimal (or near optimal), right? Maybe the problems are too easy or too hopeless or our understand is too crude?
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Ok let me try to be precise. Let p(x_0) be a distribution within a limited space (e.g. the nD hypercube) as neural activities are bounded. Let y_i=f(x_{0i}) be a mapping into classes. Let x_i=x_{0i}+\eta where \eta is drawn from nD Gaussian of isotropic variance \sigma.
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Wouldn’t this relate to the statistical dimensionality of the problem (n vs. p) as well? fMRI classification is a notoriously high dimensional problem.
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I think technically it should depend on the noise per voxel and the number of voxels that can be combined to estimate the relevant dimensions. But for most estimation tasks, noise is still high on the relevant output dimensions. Even for large N.
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and everything was driven by
@schulz_maa , whose twitter handle i was missingPrikaži ovu nit -
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Nonparametric methods have a larger sample size requirement. They are less efficient because their convergence rate is always slower than parametric methods. So very useful to benchmark that current sample size of n=10,000 given current SNR of tools & dimensionality.
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Definitely. Just out of curiosity, do you know of a good paper that compares convergence rates across methods? It would seem that it should also depend on, e.g. smoothness assumptions, behind nonparametric methods.
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Also our experience with
@GaelVaroquaux@julienmairal@BertrandThirion https://arxiv.org/abs/1809.06035 , although we can use multi-layer linear networks -
...which we started to notice and published already as early as 2015: https://papers.nips.cc/paper/5646-semi-supervised-factored-logistic-regression-for-high-dimensional-neuroimaging-data …
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