Re: Exercises and Problems
Thank you for your response and explanation. Mea culpa. I was thinking a bit too generally, perhaps loosely too, when I should have thought through an example such as the one you note.
In passing, let me restate what I had in mind in my earlier post. To begin with, I shouldn't have used the word frequency, but rather, I should have said wavelength since I used images as my example. I thought, if I had perfectly noisy set of images as input, and my end goal was to classify noisy vs notnoisy images, I would have a case where I attempt to learn the problem after a 2d Fourier transform of the images. For sake of discussion, in the transformed space, we'd see a flatish surface corresponding to various (spatial wavelengths, ie equal energy in all wavelengths since it is a Fourier transformed Gaussian), and basically a constant/flat function (glossing over a lot of practical points such as tapering to keep it bandlimited). I would use this smooth *characterization of noise* as an input to the learning algorithm, and eventually classify new images. Ie, if in the wavelength space the transformed image is flattish => it is "noisy" image, else => it is a "notnoisy" image.
In my mind, I started out with a target function which was completely random, ie noisy, and I used the Fourier transformed space to turn it into a nearconstant (smooth) function which I could use for classification. And presumably, I one managed to learn using a perfectly noisy image set. This thinking prompted my question, which questioned why random functions can't be learned. Needless adding, I did not map my thinking properly enough; the target function for this binary classification isn't the 2d Fourier transform; and further, I also mapped the input space (ie image pixel location) onto a wavelength space.
I like your simple example. Thank you.
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The whole is simpler than the sum of its parts.  Gibbs
