Many interesting interesting and important techniques center around the ideas of embedding of a set of points in a higher-dimensional space, or projecting a set of points into a lower-dimensional space. We'llĬlose the lecture with an example of this type. In some cases the results of the kind of analysis we are considering here mightīe to introduce unpleasant artifacts that obscure what is really going on. Structure is not revealed by choosing a new set of basis vectors, and indeed
It's easy, all the same, to create or find data sets whose instrinsic There are many ways to do this, and often the techniques are spectacularlyĮffective. Or "sphering" the data to remove correlations). Work better, such as subtracting averages to center the data on the origin, (There areĪlso a few pre-processing steps that can sometimes help to make such techniques Given a body of observations describedĪs vectors, they can provide a new descriptions in terms of vectors whose elementsĪre linear combinations of the elements of the original vectors. One warning: although these techniques are widely applicable and can be veryĮffective, they are also very limited. This lecture presents a general framework for thinking about data-dependent In the new coordinate system, we can represent the data more succinctly, throwingĪway many or even most of the apparent degrees of freedom in the original representation. To "see" patterns in the data more clearly we might also do it because We mightĭo this because the result allows us (or more likely, a computer algorithm) Way, so as to optimally match some structure found in the data itself.
"Wavelets" have similar properties, adding theīenefit of locality in time (and/or space) as well as frequency.Īlternatively, we might choose our new coordinate system in a data-dependent Vectors reduces the effect of an LSI to element-wise multiplication rather than Of linear shift-invariant systems, a set of coordinates with sinusoids as basis This is the case for Fourier coefficients: because sinusoids are eigenfunctions Specific properties of the data set being analyzed. Vectors have particularly useful mathematical properties, independent of the This new coordinate system might be chosen because its basis In dealing with any sorts of multi-dimensional data, it is often helpful toįind a new coordinate system, different from the one in which the data initially The frequencies of visible light, but the spectral sensitivities of the retinal Of coordinates: the natural dimensions of early color vision are not Lights with very different spectra can be "metamers". Region of the spectrum) has many more degrees of freedom than that, so that
Only three dimensions, even though electromagnetic radiation (in the visible-light Reduction: human perception of colors "lives" in a space with We can focus on two different aspects of this situation. Linear way, at least when examined in classical color-matching experiments. Despite the fact that theĬell responses are highly non-linear, the overall system behaves in a remarkably Pigments can only span a three-dimensional space. Three light-sensitive pigments, each with a different spectral sensitivity,Īnd thus the encoding of colors in the retina by the responses of these three Human visual system into a three-dimension subspace. In the discussion of early color vision, weĮxplored the way that multi-dimensional optical spectra are projected by the three_d_figure (( 0, 0 ), fig_desc = 'H = Span\ B.Introduction to subspace methods Background
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