After so many years in college, you’d think I’d learn by now: the best way to learn something is to teach it. But I’m still floored when it happens.
I recently had to teach my wife (a non-techie math student) SVD: Singular Value Decomposition. When she asked me, I was caught off guard. It’s kind of an esoteric thing to pull out of the blue.
My mind raced. I could start out by simply writing down the product of 3 matrices, listing their properties and giving her facts like which kinds of matrices have SVDs, but I don’t think of things as disjointed facts, and that’s not the way I want to teach them.
Because of my passion for video games, when I think of matrices I think of computer graphics. I pulled up Google, found a greyscale image, and explained how images are stored as matrices in computer memory.
I then related how SVD is to a matrix as a Taylor series (something she’s familiar with) is to a function. Without going into the details, I explained how we could keep all the terms to reproduce the image perfectly, but as with Taylor series, we don’t really care about perfect representations. Just as we only care for the (for example) 8th decimal place of a Taylor representation, we only care about keeping as many SVD terms that will keep the picture crisp and clear. That’s the trade-off: quality versus size. First you determine what quality means, and that determines the size (the number of terms to keep).
I then made things more concrete by showing what SVD terms look like (of course these are outer products):
and explained that the s_i decrease in magnitude as i increases. If we kept all r terms, the image would be perfect. If we kept r-1 terms, the picture would not be quite perfect, but our eyes couldn’t tell the difference.
It then occurred to me that I had the perfect demonstration. I don’t know for sure if jpg image files use SVD, but you can set their “quality level” and see the resulting image’s quality and file size.
I loaded the image into GIMP and explained all kinds of things, like why the image doesn’t degrade for a long, long time and then you hit a filesize where the image starts to degrade more and more rapidly. I related everything back to approximating functions with Taylor series. There comes a point where the terms you ignore contribute greatly to the approximation, and when you start to ignore even more terms, the approximation falls to pieces.
The great thing was that I was able to show the accelerated image degradation visually using image editing software.
At this point, she understood what SVD is used for and a forest-view understanding how it worked. We then went into the tree-view nitty-gritty details.
I still don’t know if jpg images use SVD, but even if they don’t, the idea is still the same so my lie would be a small pedagogical white lie.
Of course, our conversation was a bit more technical than what I’m relating in this story, but by the end, I was amazed at how much more concrete the whole thing was in my mind. I walked away from that tutoring session really understanding … really *feeling* the decomposition.
I love the excitement and sense of purpose of working in finance, but sometimes teaching gives me a sense of satisfaction that I can barely put into words. I feel myself improving every time I teach something.
Tags: calculus teaching, singular value decomposition, svd, taylor series
