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I'm working on a convolutional neural network and I've found various methods of building filters to convolve the image with. What are the advantages and disadvantages of each?

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In some cases, autoencoders may enhance your performance, produce biologically plausible filters, and more importantly, give you a model based on your data instead of predefined filters. Autoencoders provide you a filter that may fit your data better, in general. The only downside is, of course, the additional computation time. However, in many cases, it doesn't hinder you from learning online.

For more information, you might have a look at http://www.idsia.ch/~masci/papers/2011_icann.pdf

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