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RE: Dimensionality reduction with SVD
I wish I understood better exactly what SVD does. I've used it many times, and it works great, but I'm a math user, not a real mathematician. The two applications where I've found it helpful is when fitting an over determined set of equations and PCA.
The funny thing is that PCA makes complete sense to me when I'm thinking in terms of diagonalizing a covariance matrix to find the eigenvalue and vectors, but the SVD approach to PCA is still a black box to me.
I have to admit that my mathematics background is almost entirely in abstract algebra, so I have to work hard on numerical methods and linear-algebra based ML techniques. :)
I minored in Math and thought Abstract Algebra was absolutely fantastic! It was like learning a whole new way of thinking. My professor at the time (1999) taught by having the homework due the day before the lecture. His claim was that it generated questions during class. I very much enjoyed this approach, although at times it was very frustrating.
At this point the only aspects of Abstract Algebra that I still use is group theory as it applies to symmetry in quantum mechanics, and even then, this is already worked out so it is just applying the established irreducible representations. (OK... to be more honest, I haven't been active in the lab in years, but this is because I spend my time working with my graduate students and writing proposals to support them.)