Siegfried Graf's Foundations of Quantization for Probability Distributions PDF

By Siegfried Graf

Due to the speedily expanding want for tactics of knowledge compression, quantization has turn into a flourishing box in sign and photo processing and data conception. an analogous concepts also are utilized in data (cluster analysis), trend attractiveness, and operations learn (optimal place of provider centers). The e-book supplies the 1st mathematically rigorous account of the basic idea underlying those functions. The emphasis is at the asymptotics of quantization blunders for totally non-stop and targeted sessions of singular chances (surface measures, self-similar measures) offering a few new effects for the 1st time. Written for researchers and graduate scholars in likelihood conception the monograph is of capability curiosity to everyone operating within the disciplines pointed out above.

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Therefore, cl = -SIX1] and c2 = E]X1]. This yields c2,z(x) = s&,2(x) = ((-a,a) : a c R ~, IJall = EIX~]). 6(c) that v2,~(x) = elJxll ~ - (EIX1 I)2 = d E X ~ - ( E I X d ) 2. = ( d - 1 ) E X ~ + V2,2(X1). 54 I. 19 E x a m p l e ( U n i f o r m d i s t r i b u t i o n o n a n e u c l i d e a n b a l l ) Let the underlying norm be t h e / s - n o r m on R d and let P -- U(B(0, 1)). Then P is a spherical distribution. Consider the case n = 2 and r = 2. (0, 1)) l~T~--y2))dy -1 P = )~d-l(Bd_l(O , t 1))/(1 - yS)(d-W: dy ~d(B~(O, 1)) -1 t r(i+3) f - ¢-~r(½ + 3) J(1 - ys)(d-,)/s dy, ttl <_ 1.

Proof Given f E ~'~, let # I denote the image measure of P under the m a p R ~ -+ ]R~ x R d, x ~-> (x, f(x)). Then EIIX - f(X)lF = f IIx - yll~ d#f(x,y) > p;(P, Pf). d This implies V,~,~(P) > inf p~(P, pf) > inf p~(P, Q). ~IIx - all r d#(x, y) Rd×¢~ : f 2tlx- all"dP(x), hence py(P,Q) >_E ~ n l l X - alff. 34 I. 1, this yields inf pr(p, Q) > v,,r(P). QET~,~ [] A measure Q E P , is called n - o p t i m a l q u a n t i z i n g m e a s u r e for P o f o r d e r r if Vn,r(P) = prr(p, Q). If f E 9vn is an n-optimal quantizer, then p I E P , is an n-optimal quantizing measure.

17 E x a m p l e ( U n i f o r m d i s t r i b u t i o n on a c u b e a n d t h e c u b e q u a n t i z e r ) Let P = U([0,1]) d) and consider a tesselation of [0,1] d consisting of n = k d translates C 1 , . •. , C ~ of the cube [0,1d ~] . Denote by ai the midpoint of Ci. Then • , L 2~ : i-1 , . . tCi)). 2. From scale i=l and translation invariance of Mr it follows that II /* E]]X - fn(X)]l r -- Z i=1 ] ]Ix - ai[I~ dx J Ci = y ~ Mr(C~)P(C~)(d÷r)/~ i=1 = n-r/~Mr([O, 1]d). 2: Square quantizer for U([0,1] 2) We see that n-level quantization for P reduces Vr(P) = Mr(J0, n -rId.

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