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You seem knowledgeable so maybe you could give me some advice.

I developed a new method using SVD/PCA with some basis rotation method. I can potentially characterize or estimate errors on my input data, but I have no idea how to propagate these errors to my basis vectors. I'm at a bit of a loss as to where to look from here. My only idea is to bootstrap... But that's a bit lame :) and not very rigorous

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I am not sure exactly what you mean, but I will try with the following advice.

See if you are able to define each element of your input data as a stochastic variable that you can sample from. Then make a function to generate a set of fixed value input data elements by sampling from the stochastic variables. Then run this many times to generate resulting basis vectors. Calculate relevant measures of variability from these vectors.

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