By James R. Kirkwood
Offers advent to research of real-valued capabilities of 1 variable. this article is for a student's first summary arithmetic path. Writing kind is much less formal and fabric offered in a manner such that the coed can boost an instinct for the topic and procure a few adventure in developing proofs. The slower velocity of the topic and the eye given to examples are supposed to ease the student's transition from computational to theoretical arithmetic.
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19) is extensively studied . Let us mention here a cheap way to obtain uniqueness. This can be achieved by an arbitrarily small generic perturbation of the dictionary elements, such as adding a small independent Gaussian vector to each di . Then one can see that the solution ˛O (and therefore xO as well) are unique almost surely. 2 we see that xO is m-sparse in the perturbed dictionary. 2. 1 then reads as follows. 3 (Sparse recovery). 3 form the core of the area of compressed sensing, see [15, 19, 26, 39].
We will use one such result due to Y. Seginer . 6 (Seginer’s bound for general random matrices). , mean zero random variables. Then Á E kGk Ä C E max kGi k2 C E max kGj k2 i j where the maxima are taken over all rows Gi and over all columns Gj of G, respectively. 5. We shall first control the error in the operator norm. 38) is bounded by the second term. ıij p/Xij : So we can apply Y. 40) It remains to bound the `2 norms of rows and columns of Y pX. Let us do this for rows; a similar argument would control the columns.
Heuristically, the spherical mean width measures the size of the bulk of K. n/, the M bound states that the size of the random section K \ E is bounded by the spherical mean width of K. 3 illustrates. But when the dimension of the subspace E grows toward n (so the p codimension m becomes small), the diameter of K \ E also grows by a factor of n=m. This gives a precise control of how E in this case interferes with the outliers of K. 2. d. observations of x. So far, we have not been clear about possible distributions of the observations yi .