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Dear diary (all readers welcome) by Vincent Randal
These notes describe my effort to learn something about Machine Learning.. Very smart people with background in linear transform techniques (Fourier transform, wavelets, etc) are curious what?s going on inside neural networks. Ingrid Daubechies comments on that at 2:50 (t=170s) into this video:
https://youtu.be/vlGYETlHivQ&t=170s
Because Daubechies and Mallat comment on neural networks in their 2017 Abel prize lectures, I recommend watching them (including Condes? lecture) here:
https://abelprize.no/abel-prize-laureates/2017
Also, Ole Christensen?s lectures on Real Analysis are interesting for their coverage of Fourier and wavelet transforms.
In these notes I will write about an example using PWM (pulse width modulation) that illustrates the use of the Fourier transform in Machine Learning. As Ole Christensen says, electrical engineers are quite happy to talk about time varying ?signals? which are merely functions. We can sample an analog signal by multiplying it by a sampling function. We will use the fact that time domain multiplication is equivalent to frequency domain convolution *AND* the faster we sample the more we spread the features apart in the frequency domain.
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By the way that last point is the reason for 8x, 16x, or more oversampling found in audio CD players that were common 30 years ago. Substantial oversampling allows the signal to be recovered via low cost first-order low-pass filter implemented with just a capacitor and resistor.
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Gratitude: It was a very brief time some 30+ years ago when I had the privilege to learn something from mathematicians, computer scientists, and engineers at the University of Texas at San Antonio. It was a brief and marvelous time for which I am grateful to many people including Michael Humphries, Larry Williams, Mary Lou Zeeman, Dmitry Gokhman, Kay Robbins, Steve Robbins, John Schmalzel, Parimal Patel, Marvin Chatkoff, Benham Kamali, and probably more whose names will be added as I remember them.
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