Feature request: supporting MAE loss

[XaXaXa12345]

Dear NNLM Team, Your package is great. Actually, at the moment this is only package supporting missing values. It would be nice to have absolute loss |x| supported as well. Kind Regards. Andrey.

[zdebruine]

I'm not sure why you'd like to use absolute loss rather than mean squared loss. NMF is not convex in absolute error, because NNLS is not convex in absolute error, only in squared error (hint, we are solving least SQUARES equations). Thus, absolute error can be deceiving because the exact absolute error solution can be different than the convex NNLS solution solved by least squares.

[XaXaXa12345]

Squared error loss for matrix factorization isn't convex either, unless regularized, depending on regularization parameter. Take a hessian and see that it's not PSD in general case. Absolute loss is resistant to outliers, since it's median.