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Quantization for Infinite Affine Transformations

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Quantization for Infinite Affine Transformations Quantization for infinite affine transformations Do˘ganC¸¨omez NDSU Department of Mathematics, Analysis Seminar January 26, 2021 Outline 1. Main problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems Outline: 1. Main Problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems 6. Main results (i): optimal sets and associated errors 7. Main results (ii): quantization dimension and coefficients Do˘ganC¸¨omez Quantization for infinite affine transformations Quantization: The process of approximating a given probability measure µ with a discrete probability measure of finite support. It concerns determining an appropriate partitioning of the underlying space for the discrete measure, and error analysis. Hence, main goals of the theory are: 1. find the optimal discrete measure that yields \good" approximation of the given probability to within an allowable margin of error, and 2. estimate the rate at which some specified measure of error goes to 0 as n ! 1: d (Throughout, all the measures considered will be on R with Euclidean norm k k:) Outline 1. Main problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems Introduction Do˘ganC¸¨omez Quantization for infinite affine transformations It concerns determining an appropriate partitioning of the underlying space for the discrete measure, and error analysis. Hence, main goals of the theory are: 1. find the optimal discrete measure that yields \good" approximation of the given probability to within an allowable margin of error, and 2. estimate the rate at which some specified measure of error goes to 0 as n ! 1: d (Throughout, all the measures considered will be on R with Euclidean norm k k:) Outline 1. Main problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems Introduction Quantization: The process of approximating a given probability measure µ with a discrete probability measure of finite support. Do˘ganC¸¨omez Quantization for infinite affine transformations Hence, main goals of the theory are: 1. find the optimal discrete measure that yields \good" approximation of the given probability to within an allowable margin of error, and 2. estimate the rate at which some specified measure of error goes to 0 as n ! 1: d (Throughout, all the measures considered will be on R with Euclidean norm k k:) Outline 1. Main problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems Introduction Quantization: The process of approximating a given probability measure µ with a discrete probability measure of finite support. It concerns determining an appropriate partitioning of the underlying space for the discrete measure, and error analysis. Do˘ganC¸¨omez Quantization for infinite affine transformations d (Throughout, all the measures considered will be on R with Euclidean norm k k:) Outline 1. Main problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems Introduction Quantization: The process of approximating a given probability measure µ with a discrete probability measure of finite support. It concerns determining an appropriate partitioning of the underlying space for the discrete measure, and error analysis. Hence, main goals of the theory are: 1. find the optimal discrete measure that yields \good" approximation of the given probability to within an allowable margin of error, and 2. estimate the rate at which some specified measure of error goes to 0 as n ! 1: Do˘ganC¸¨omez Quantization for infinite affine transformations Outline 1. Main problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems Introduction Quantization: The process of approximating a given probability measure µ with a discrete probability measure of finite support. It concerns determining an appropriate partitioning of the underlying space for the discrete measure, and error analysis. Hence, main goals of the theory are: 1. find the optimal discrete measure that yields \good" approximation of the given probability to within an allowable margin of error, and 2. estimate the rate at which some specified measure of error goes to 0 as n ! 1: d (Throughout, all the measures considered will be on R with Euclidean norm k k:) Do˘ganC¸¨omez Quantization for infinite affine transformations Outline 1. Main problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems Introduction (cont.) Since the approximating measure is finite and discrete; d a) its support must partition R b) on each element of this partition it must take constant value. Hence, it must be a (discrete probability) measure µ whose distribution function is of Pn the form f (x) = i=1 ai χAi (x): In order to achieve the best approximation (if exists) the set of points α = fai g and the partition fAi g; where ai 2 Ai ; must be carefully chosen. Once α is determined, then it turns out that fAi g is the Voronoi partition w.r.t α. Namely, Ai is the set of d all points in R which are closest to ai : The points ai are also called the centers of the Voronoi regions Ai : Determining the centers suffice! Do˘ganC¸¨omez Quantization for infinite affine transformations Outline 1. Main problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems Introduction (cont.) If P is the probability measure being approximated by µ, then there are various metrics to determine the error involved. The most suitable metric for our purposes is Kantorovich metric (or Wasserstein metric): Z ρ(µ, ν) = inf[ kx − yk2dη(x; y)]1=2; η where the infimum is taken over all probability measures η with marginals µ and ν; resp. Do˘ganC¸¨omez Quantization for infinite affine transformations If R kxk2dµ(x) < 1 then there is some set α for which the infimum is attained. Outline 1. Main problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems Quantization error Quantization error d Let µ be a Borel probability measure on R ; and n ≥ 1: The n-th quantization error for µ is Z 2 d Vn := Vn(µ) = inff d(x; α) dµ(x): α ⊂ R ; jαj ≤ ng; where d(x; α) is the distance from x to the set α. Do˘ganC¸¨omez Quantization for infinite affine transformations Outline 1. Main problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems Quantization error Quantization error d Let µ be a Borel probability measure on R ; and n ≥ 1: The n-th quantization error for µ is Z 2 d Vn := Vn(µ) = inff d(x; α) dµ(x): α ⊂ R ; jαj ≤ ng; where d(x; α) is the distance from x to the set α. If R kxk2dµ(x) < 1 then there is some set α for which the infimum is attained. Do˘ganC¸¨omez Quantization for infinite affine transformations If α is an optimal set of n-quantizers for a continuous measure, then the centroids of the associated Voronoi partition consists of elements of α. Furthermore, if a 2 α, then it is the expected value of its Voronoi region. There may be more than one optimal set of n-means for µ; even in the case that it is absolutely continuous w.r.t. Lebesgue measure m: Outline 1. Main problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems Optimal sets (cont.) Optimal set of n-means A set α, jαj ≤ n; for which the infimum is attained is an optimal set of n-means, or optimal set of n-quantizers. The elements of an optimal set of n-means are called optimal n-quantizers. Do˘ganC¸¨omez Quantization for infinite affine transformations Furthermore, if a 2 α, then it is the expected value of its Voronoi region. There may be more than one optimal set of n-means for µ; even in the case that it is absolutely continuous w.r.t. Lebesgue measure m: Outline 1. Main problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems Optimal sets (cont.) Optimal set of n-means A set α, jαj ≤ n; for which the infimum is attained is an optimal set of n-means, or optimal set of n-quantizers. The elements of an optimal set of n-means are called optimal n-quantizers. If α is an optimal set of n-quantizers for a continuous measure, then the centroids of the associated Voronoi partition consists of elements of α. Do˘ganC¸¨omez Quantization for infinite affine transformations There may be more than one optimal set of n-means for µ; even in the case that it is absolutely continuous w.r.t. Lebesgue measure m: Outline 1. Main problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems Optimal sets (cont.) Optimal set of n-means A set α, jαj ≤ n; for which the infimum is attained is an optimal set of n-means, or optimal set of n-quantizers.
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