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Old 01-16-2013, 09:36 AM
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magdon magdon is offline
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Join Date: Aug 2009
Location: Troy, NY, USA.
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Default Re: Exercises and Problems

Good example top clarify an important issue. In your example, the target function is not random. It assigns to a signal that is white noise a -1 and to a signal that is not white noise (say, containing a real image) a +1. A random target would assign random +1,-1 to the signal.

What you have described is what we call feature construction and you will study it later in the book. Specifically, given the input signal construct the fourrier transform and take (for example) the variance of this fourier signal. This variance is a good feature of the input for determining \pm1 (small variance would indicate a noise signal). Good feature construction is a very important part of learning from data successfully.



Quote:
Originally Posted by cygnids View Post
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 not-noisy 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 band-limited). 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 "not-noisy" 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 near-constant (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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