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 → ∞.
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 → ∞.
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 → ∞.
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 → ∞.
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 → ∞.
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 = {ai } and the partition {Ai }, where ai ∈ Ai , must be carefully chosen. Once α is determined, then it turns out that {Ai } 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) < ∞ 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) = inf{ d(x, α) dµ(x): α ⊂ R , |α| ≤ n},
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) = inf{ d(x, α) dµ(x): α ⊂ R , |α| ≤ n},
where d(x, α) is the distance from x to the set α.
If R kxk2dµ(x) < ∞ 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 ∈ 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, |α| ≤ 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 ∈ 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, |α| ≤ 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, |α| ≤ 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. Furthermore, if a ∈ α, then it is the expected value of its Voronoi region.
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 Optimal sets (cont.)
Optimal set of n-means A set α, |α| ≤ 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. Furthermore, if a ∈ 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.
Do˘ganC¸¨omez Quantization for infinite affine transformations R 2+δ if µ ⊥ m and kxk dµ(x) < ∞, for some δ > 0, then limn→∞ Vn = 0.
What is the asymptotic behavior of Vn?
Quantization dimension The lower and upper quantization dimension of µ:
2 log n 2 log n D := D(µ) = lim inf , and D := D(µ) = lim sup . n→∞ − log Vn n→∞ − log Vn
If D = D =: D = D(µ), then D is the quantization dimension of µ.
D does not depend on the underlying metric, it depends only on the support of µ.
1 2/D If D exists, then log Vn ∼ log( n ) . Hence, the quantization dimension measures the (logarithmic) asymptotic rate at which Vn goes to zero.
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 Asymptotics
For a probability measure µ, R 2 if µ << m and kxk dµ(x) < ∞, then limn→∞ Vn = 0
Do˘ganC¸¨omez Quantization for infinite affine transformations What is the asymptotic behavior of Vn?
Quantization dimension The lower and upper quantization dimension of µ:
2 log n 2 log n D := D(µ) = lim inf , and D := D(µ) = lim sup . n→∞ − log Vn n→∞ − log Vn
If D = D =: D = D(µ), then D is the quantization dimension of µ.
D does not depend on the underlying metric, it depends only on the support of µ.
1 2/D If D exists, then log Vn ∼ log( n ) . Hence, the quantization dimension measures the (logarithmic) asymptotic rate at which Vn goes to zero.
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 Asymptotics
For a probability measure µ, R 2 if µ << m and kxk dµ(x) < ∞, then limn→∞ Vn = 0 R 2+δ if µ ⊥ m and kxk dµ(x) < ∞, for some δ > 0, then limn→∞ Vn = 0.
Do˘ganC¸¨omez Quantization for infinite affine transformations Quantization dimension The lower and upper quantization dimension of µ:
2 log n 2 log n D := D(µ) = lim inf , and D := D(µ) = lim sup . n→∞ − log Vn n→∞ − log Vn
If D = D =: D = D(µ), then D is the quantization dimension of µ.
D does not depend on the underlying metric, it depends only on the support of µ.
1 2/D If D exists, then log Vn ∼ log( n ) . Hence, the quantization dimension measures the (logarithmic) asymptotic rate at which Vn goes to zero.
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 Asymptotics
For a probability measure µ, R 2 if µ << m and kxk dµ(x) < ∞, then limn→∞ Vn = 0 R 2+δ if µ ⊥ m and kxk dµ(x) < ∞, for some δ > 0, then limn→∞ Vn = 0.
What is the asymptotic behavior of Vn?
Do˘ganC¸¨omez Quantization for infinite affine transformations D does not depend on the underlying metric, it depends only on the support of µ.
1 2/D If D exists, then log Vn ∼ log( n ) . Hence, the quantization dimension measures the (logarithmic) asymptotic rate at which Vn goes to zero.
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 Asymptotics
For a probability measure µ, R 2 if µ << m and kxk dµ(x) < ∞, then limn→∞ Vn = 0 R 2+δ if µ ⊥ m and kxk dµ(x) < ∞, for some δ > 0, then limn→∞ Vn = 0.
What is the asymptotic behavior of Vn?
Quantization dimension The lower and upper quantization dimension of µ:
2 log n 2 log n D := D(µ) = lim inf , and D := D(µ) = lim sup . n→∞ − log Vn n→∞ − log Vn
If D = D =: D = D(µ), then D is the quantization dimension of µ.
Do˘ganC¸¨omez Quantization for infinite affine transformations 1 2/D If D exists, then log Vn ∼ log( n ) . Hence, the quantization dimension measures the (logarithmic) asymptotic rate at which Vn goes to zero.
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 Asymptotics
For a probability measure µ, R 2 if µ << m and kxk dµ(x) < ∞, then limn→∞ Vn = 0 R 2+δ if µ ⊥ m and kxk dµ(x) < ∞, for some δ > 0, then limn→∞ Vn = 0.
What is the asymptotic behavior of Vn?
Quantization dimension The lower and upper quantization dimension of µ:
2 log n 2 log n D := D(µ) = lim inf , and D := D(µ) = lim sup . n→∞ − log Vn n→∞ − log Vn
If D = D =: D = D(µ), then D is the quantization dimension of µ.
D does not depend on the underlying metric, it depends only on the support of µ.
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 Asymptotics
For a probability measure µ, R 2 if µ << m and kxk dµ(x) < ∞, then limn→∞ Vn = 0 R 2+δ if µ ⊥ m and kxk dµ(x) < ∞, for some δ > 0, then limn→∞ Vn = 0.
What is the asymptotic behavior of Vn?
Quantization dimension The lower and upper quantization dimension of µ:
2 log n 2 log n D := D(µ) = lim inf , and D := D(µ) = lim sup . n→∞ − log Vn n→∞ − log Vn
If D = D =: D = D(µ), then D is the quantization dimension of µ.
D does not depend on the underlying metric, it depends only on the support of µ.
1 2/D If D exists, then log Vn ∼ log( n ) . Hence, the quantization dimension measures the (logarithmic) asymptotic rate at which Vn goes to zero.
Do˘ganC¸¨omez Quantization for infinite affine transformations Quantization coefficients For any κ > 0, the κ-dimensional upper and lower quantization coefficients for µ are the numbers
κ/2 κ/2 cκ(µ) = lim sup nVn (µ) and cκ(µ) = lim inf nVn (µ), respectively. n n
If cκ(µ) = cκ(µ) = cκ(µ), then cκ(µ) is the quantization coefficients for µ.
Good News: The quantization coefficients provide us with more accurate information about the asymptotic behavior of the quantization error than the quantization dimension.
Bad News: Compared to the calculation of quantization dimension, it is usually much more difficult to determine whether the lower and the upper quantization coefficients exist, and finite.
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 Asymptotics (cont.)
For more accurate information on the behaviour of Vn, one looks at the actual asymptotic rate at which Vn goes to zero.
Do˘ganC¸¨omez Quantization for infinite affine transformations Good News: The quantization coefficients provide us with more accurate information about the asymptotic behavior of the quantization error than the quantization dimension.
Bad News: Compared to the calculation of quantization dimension, it is usually much more difficult to determine whether the lower and the upper quantization coefficients exist, and finite.
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 Asymptotics (cont.)
For more accurate information on the behaviour of Vn, one looks at the actual asymptotic rate at which Vn goes to zero.
Quantization coefficients For any κ > 0, the κ-dimensional upper and lower quantization coefficients for µ are the numbers
κ/2 κ/2 cκ(µ) = lim sup nVn (µ) and cκ(µ) = lim inf nVn (µ), respectively. n n
If cκ(µ) = cκ(µ) = cκ(µ), then cκ(µ) is the quantization coefficients for µ.
Do˘ganC¸¨omez Quantization for infinite affine transformations Bad News: Compared to the calculation of quantization dimension, it is usually much more difficult to determine whether the lower and the upper quantization coefficients exist, and finite.
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 Asymptotics (cont.)
For more accurate information on the behaviour of Vn, one looks at the actual asymptotic rate at which Vn goes to zero.
Quantization coefficients For any κ > 0, the κ-dimensional upper and lower quantization coefficients for µ are the numbers
κ/2 κ/2 cκ(µ) = lim sup nVn (µ) and cκ(µ) = lim inf nVn (µ), respectively. n n
If cκ(µ) = cκ(µ) = cκ(µ), then cκ(µ) is the quantization coefficients for µ.
Good News: The quantization coefficients provide us with more accurate information about the asymptotic behavior of the quantization error than the quantization dimension.
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 Asymptotics (cont.)
For more accurate information on the behaviour of Vn, one looks at the actual asymptotic rate at which Vn goes to zero.
Quantization coefficients For any κ > 0, the κ-dimensional upper and lower quantization coefficients for µ are the numbers
κ/2 κ/2 cκ(µ) = lim sup nVn (µ) and cκ(µ) = lim inf nVn (µ), respectively. n n
If cκ(µ) = cκ(µ) = cκ(µ), then cκ(µ) is the quantization coefficients for µ.
Good News: The quantization coefficients provide us with more accurate information about the asymptotic behavior of the quantization error than the quantization dimension.
Bad News: Compared to the calculation of quantization dimension, it is usually much more difficult to determine whether the lower and the upper quantization coefficients exist, and finite.
Do˘ganC¸¨omez Quantization for infinite affine transformations If µ has non-vanishing absolutely continuous part, then D(µ) exists and D = d. Also, the d-dimensional quantization coefficient cd (µ) exists and is finite.
If µ is singular, then D(µ) (or cκ(µ)) may not exist.
If µ has bounded support, then
D ∈ [dimH (µ), dimB (µ)] and D ∈ [dimP (µ), dimB (µ)].
If supp(µ) = K is compact,
dimH (µ) ≤ D(µ) ≤ dimB (K), and D(µ) ≤ dimB (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 Asymptotics (cont.)
Some folklore.
Do˘ganC¸¨omez Quantization for infinite affine transformations If µ is singular, then D(µ) (or cκ(µ)) may not exist.
If µ has bounded support, then
D ∈ [dimH (µ), dimB (µ)] and D ∈ [dimP (µ), dimB (µ)].
If supp(µ) = K is compact,
dimH (µ) ≤ D(µ) ≤ dimB (K), and D(µ) ≤ dimB (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 Asymptotics (cont.)
Some folklore.
If µ has non-vanishing absolutely continuous part, then D(µ) exists and D = d. Also, the d-dimensional quantization coefficient cd (µ) exists and is finite.
Do˘ganC¸¨omez Quantization for infinite affine transformations If µ has bounded support, then
D ∈ [dimH (µ), dimB (µ)] and D ∈ [dimP (µ), dimB (µ)].
If supp(µ) = K is compact,
dimH (µ) ≤ D(µ) ≤ dimB (K), and D(µ) ≤ dimB (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 Asymptotics (cont.)
Some folklore.
If µ has non-vanishing absolutely continuous part, then D(µ) exists and D = d. Also, the d-dimensional quantization coefficient cd (µ) exists and is finite.
If µ is singular, then D(µ) (or cκ(µ)) may not exist.
Do˘ganC¸¨omez Quantization for infinite affine transformations If supp(µ) = K is compact,
dimH (µ) ≤ D(µ) ≤ dimB (K), and D(µ) ≤ dimB (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 Asymptotics (cont.)
Some folklore.
If µ has non-vanishing absolutely continuous part, then D(µ) exists and D = d. Also, the d-dimensional quantization coefficient cd (µ) exists and is finite.
If µ is singular, then D(µ) (or cκ(µ)) may not exist.
If µ has bounded support, then
D ∈ [dimH (µ), dimB (µ)] and D ∈ [dimP (µ), dimB (µ)].
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 Asymptotics (cont.)
Some folklore.
If µ has non-vanishing absolutely continuous part, then D(µ) exists and D = d. Also, the d-dimensional quantization coefficient cd (µ) exists and is finite.
If µ is singular, then D(µ) (or cκ(µ)) may not exist.
If µ has bounded support, then
D ∈ [dimH (µ), dimB (µ)] and D ∈ [dimP (µ), dimB (µ)].
If supp(µ) = K is compact,
dimH (µ) ≤ D(µ) ≤ dimB (K), and D(µ) ≤ dimB (K).
Do˘ganC¸¨omez Quantization for infinite affine transformations This is due to the fact that F determines a probability measure PX , which in turn, determines all events involving X (i.e., the σ-algebra needed to study X ). So, all one needs to check is that there is a probability space on which X is well-defined. Hence, distribution functions (and their densities) play a crucial role in probability spaces. There exist only three types of pure probability distributions: discrete, absolutely continuous, and singularly continuous (w..r.t Lebesgue measure).
A r.v. X is called discrete if the set of values {xn} of it is countable. In this case the distribution function is a step function F with a discontinuity at each xn (with magnitude pn = PX ({xn}). F is also called discrete distribution. d A r.v. X is called continuous if its distribution function is continuous on R .
A r.v. X is called absolutely continuous if PX << m, where m is the Lebesgue measure. Equivalently, the associated distribution function F is absolutely continuous d on R . X is called singular if PX ⊥ 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 Types of probability distributions
In probability theory one often studies properties of a random variable X with a distribution function F without any reference to the underlying probability space.
Do˘ganC¸¨omez Quantization for infinite affine transformations So, all one needs to check is that there is a probability space on which X is well-defined. Hence, distribution functions (and their densities) play a crucial role in probability spaces. There exist only three types of pure probability distributions: discrete, absolutely continuous, and singularly continuous (w..r.t Lebesgue measure).
A r.v. X is called discrete if the set of values {xn} of it is countable. In this case the distribution function is a step function F with a discontinuity at each xn (with magnitude pn = PX ({xn}). F is also called discrete distribution. d A r.v. X is called continuous if its distribution function is continuous on R .
A r.v. X is called absolutely continuous if PX << m, where m is the Lebesgue measure. Equivalently, the associated distribution function F is absolutely continuous d on R . X is called singular if PX ⊥ 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 Types of probability distributions
In probability theory one often studies properties of a random variable X with a distribution function F without any reference to the underlying probability space. This is due to the fact that F determines a probability measure PX , which in turn, determines all events involving X (i.e., the σ-algebra needed to study X ).
Do˘ganC¸¨omez Quantization for infinite affine transformations There exist only three types of pure probability distributions: discrete, absolutely continuous, and singularly continuous (w..r.t Lebesgue measure).
A r.v. X is called discrete if the set of values {xn} of it is countable. In this case the distribution function is a step function F with a discontinuity at each xn (with magnitude pn = PX ({xn}). F is also called discrete distribution. d A r.v. X is called continuous if its distribution function is continuous on R .
A r.v. X is called absolutely continuous if PX << m, where m is the Lebesgue measure. Equivalently, the associated distribution function F is absolutely continuous d on R . X is called singular if PX ⊥ 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 Types of probability distributions
In probability theory one often studies properties of a random variable X with a distribution function F without any reference to the underlying probability space. This is due to the fact that F determines a probability measure PX , which in turn, determines all events involving X (i.e., the σ-algebra needed to study X ). So, all one needs to check is that there is a probability space on which X is well-defined. Hence, distribution functions (and their densities) play a crucial role in probability spaces.
Do˘ganC¸¨omez Quantization for infinite affine transformations A r.v. X is called discrete if the set of values {xn} of it is countable. In this case the distribution function is a step function F with a discontinuity at each xn (with magnitude pn = PX ({xn}). F is also called discrete distribution. d A r.v. X is called continuous if its distribution function is continuous on R .
A r.v. X is called absolutely continuous if PX << m, where m is the Lebesgue measure. Equivalently, the associated distribution function F is absolutely continuous d on R . X is called singular if PX ⊥ 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 Types of probability distributions
In probability theory one often studies properties of a random variable X with a distribution function F without any reference to the underlying probability space. This is due to the fact that F determines a probability measure PX , which in turn, determines all events involving X (i.e., the σ-algebra needed to study X ). So, all one needs to check is that there is a probability space on which X is well-defined. Hence, distribution functions (and their densities) play a crucial role in probability spaces. There exist only three types of pure probability distributions: discrete, absolutely continuous, and singularly continuous (w..r.t Lebesgue measure).
Do˘ganC¸¨omez Quantization for infinite affine transformations d A r.v. X is called continuous if its distribution function is continuous on R .
A r.v. X is called absolutely continuous if PX << m, where m is the Lebesgue measure. Equivalently, the associated distribution function F is absolutely continuous d on R . X is called singular if PX ⊥ 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 Types of probability distributions
In probability theory one often studies properties of a random variable X with a distribution function F without any reference to the underlying probability space. This is due to the fact that F determines a probability measure PX , which in turn, determines all events involving X (i.e., the σ-algebra needed to study X ). So, all one needs to check is that there is a probability space on which X is well-defined. Hence, distribution functions (and their densities) play a crucial role in probability spaces. There exist only three types of pure probability distributions: discrete, absolutely continuous, and singularly continuous (w..r.t Lebesgue measure).
A r.v. X is called discrete if the set of values {xn} of it is countable. In this case the distribution function is a step function F with a discontinuity at each xn (with magnitude pn = PX ({xn}). F is also called discrete distribution.
Do˘ganC¸¨omez Quantization for infinite affine transformations A r.v. X is called absolutely continuous if PX << m, where m is the Lebesgue measure. Equivalently, the associated distribution function F is absolutely continuous d on R . X is called singular if PX ⊥ 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 Types of probability distributions
In probability theory one often studies properties of a random variable X with a distribution function F without any reference to the underlying probability space. This is due to the fact that F determines a probability measure PX , which in turn, determines all events involving X (i.e., the σ-algebra needed to study X ). So, all one needs to check is that there is a probability space on which X is well-defined. Hence, distribution functions (and their densities) play a crucial role in probability spaces. There exist only three types of pure probability distributions: discrete, absolutely continuous, and singularly continuous (w..r.t Lebesgue measure).
A r.v. X is called discrete if the set of values {xn} of it is countable. In this case the distribution function is a step function F with a discontinuity at each xn (with magnitude pn = PX ({xn}). F is also called discrete distribution. d A r.v. X is called continuous if its distribution function is continuous on R .
Do˘ganC¸¨omez Quantization for infinite affine transformations X is called singular if PX ⊥ 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 Types of probability distributions
In probability theory one often studies properties of a random variable X with a distribution function F without any reference to the underlying probability space. This is due to the fact that F determines a probability measure PX , which in turn, determines all events involving X (i.e., the σ-algebra needed to study X ). So, all one needs to check is that there is a probability space on which X is well-defined. Hence, distribution functions (and their densities) play a crucial role in probability spaces. There exist only three types of pure probability distributions: discrete, absolutely continuous, and singularly continuous (w..r.t Lebesgue measure).
A r.v. X is called discrete if the set of values {xn} of it is countable. In this case the distribution function is a step function F with a discontinuity at each xn (with magnitude pn = PX ({xn}). F is also called discrete distribution. d A r.v. X is called continuous if its distribution function is continuous on R .
A r.v. X is called absolutely continuous if PX << m, where m is the Lebesgue measure. Equivalently, the associated distribution function F is absolutely continuous d on R .
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 Types of probability distributions
In probability theory one often studies properties of a random variable X with a distribution function F without any reference to the underlying probability space. This is due to the fact that F determines a probability measure PX , which in turn, determines all events involving X (i.e., the σ-algebra needed to study X ). So, all one needs to check is that there is a probability space on which X is well-defined. Hence, distribution functions (and their densities) play a crucial role in probability spaces. There exist only three types of pure probability distributions: discrete, absolutely continuous, and singularly continuous (w..r.t Lebesgue measure).
A r.v. X is called discrete if the set of values {xn} of it is countable. In this case the distribution function is a step function F with a discontinuity at each xn (with magnitude pn = PX ({xn}). F is also called discrete distribution. d A r.v. X is called continuous if its distribution function is continuous on R .
A r.v. X is called absolutely continuous if PX << m, where m is the Lebesgue measure. Equivalently, the associated distribution function F is absolutely continuous d on R . X is called singular if PX ⊥ m.
Do˘ganC¸¨omez Quantization for infinite affine transformations Cantor-Lebesgue function C(x) Note: F 0(x) = 0 a.e., F is continuous but not absolutely continuous, The measure ν defined by F is singular w.r.t. Lebesgue measure. Support of ν is the Cantor set.
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 Cantor distribution
The prime example of a singular continuous measure is defined by the distribution function 0 if x < 0 F (x) = C(x) if 0 ≤ x < 1, (Cantor-Lebesgue function) 1 if x ≥ 1,
Do˘ganC¸¨omez Quantization for infinite affine transformations Note: F 0(x) = 0 a.e., F is continuous but not absolutely continuous, The measure ν defined by F is singular w.r.t. Lebesgue measure. Support of ν is the Cantor set.
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 Cantor distribution
The prime example of a singular continuous measure is defined by the distribution function 0 if x < 0 F (x) = C(x) if 0 ≤ x < 1, (Cantor-Lebesgue function) 1 if x ≥ 1,
Cantor-Lebesgue function C(x)
Do˘ganC¸¨omez Quantization for infinite affine transformations F is continuous but not absolutely continuous, The measure ν defined by F is singular w.r.t. Lebesgue measure. Support of ν is the Cantor set.
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 Cantor distribution
The prime example of a singular continuous measure is defined by the distribution function 0 if x < 0 F (x) = C(x) if 0 ≤ x < 1, (Cantor-Lebesgue function) 1 if x ≥ 1,
Cantor-Lebesgue function C(x) Note: F 0(x) = 0 a.e.,
Do˘ganC¸¨omez Quantization for infinite affine transformations The measure ν defined by F is singular w.r.t. Lebesgue measure. Support of ν is the Cantor set.
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 Cantor distribution
The prime example of a singular continuous measure is defined by the distribution function 0 if x < 0 F (x) = C(x) if 0 ≤ x < 1, (Cantor-Lebesgue function) 1 if x ≥ 1,
Cantor-Lebesgue function C(x) Note: F 0(x) = 0 a.e., F is continuous but not absolutely continuous,
Do˘ganC¸¨omez Quantization for infinite affine transformations Support of ν is the Cantor set.
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 Cantor distribution
The prime example of a singular continuous measure is defined by the distribution function 0 if x < 0 F (x) = C(x) if 0 ≤ x < 1, (Cantor-Lebesgue function) 1 if x ≥ 1,
Cantor-Lebesgue function C(x) Note: F 0(x) = 0 a.e., F is continuous but not absolutely continuous, The measure ν defined by F is singular w.r.t. Lebesgue measure.
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 Cantor distribution
The prime example of a singular continuous measure is defined by the distribution function 0 if x < 0 F (x) = C(x) if 0 ≤ x < 1, (Cantor-Lebesgue function) 1 if x ≥ 1,
Cantor-Lebesgue function C(x) Note: F 0(x) = 0 a.e., F is continuous but not absolutely continuous, The measure ν defined by F is singular w.r.t. Lebesgue measure. Support of ν is the Cantor set.
Do˘ganC¸¨omez Quantization for infinite affine transformations Notice that the maps S1 and S2 that produce the Cantor set are affine maps. Indeed, n n they are similarity maps or similitudes. Recall that A map S : R → R is called a similarity transformation if ∃ a constant 0 < s < 1 (called contracting factor) such that
n kS(x) − S(y)k = skx − yk, ∀ x, y ∈ R . As in the case of the Cantor set, many well known fractals, such as Sierpinski Triangle, Sierpinski Carpet, Cantor Dust, Koch Curve, etc., are constructed as attractors of IFS’s.
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 Affine maps and similarities
Cantor set, which is a fractal, can also be constructed as the attractor of an “iterated d function system”. An iterated function system (IFS) is a pair (K, S), where K ⊂ R is n a compact set and S = {Si } is a collection of maps on R . When K = [0, 1] and 1 1 2 S = {S1, S2}, where S1x = 3 x, S2x = 3 x + 3 , then the attractor of this system is the ∞ 3k −1 i j set ∩n=0 ∪i,j=0 S1S2(K), which is the Cantor set C.
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 Affine maps and similarities
Cantor set, which is a fractal, can also be constructed as the attractor of an “iterated d function system”. An iterated function system (IFS) is a pair (K, S), where K ⊂ R is n a compact set and S = {Si } is a collection of maps on R . When K = [0, 1] and 1 1 2 S = {S1, S2}, where S1x = 3 x, S2x = 3 x + 3 , then the attractor of this system is the ∞ 3k −1 i j set ∩n=0 ∪i,j=0 S1S2(K), which is the Cantor set C.
Notice that the maps S1 and S2 that produce the Cantor set are affine maps. Indeed, n n they are similarity maps or similitudes. Recall that A map S : R → R is called a similarity transformation if ∃ a constant 0 < s < 1 (called contracting factor) such that
n kS(x) − S(y)k = skx − yk, ∀ x, y ∈ R . As in the case of the Cantor set, many well known fractals, such as Sierpinski Triangle, Sierpinski Carpet, Cantor Dust, Koch Curve, etc., are constructed as attractors of IFS’s.
Do˘ganC¸¨omez Quantization for infinite affine transformations N Let S = {Si }i=1 be similarity transformations with attractor A (a compact set). If p = (p1, p2,..., pN ) is a probability vector, then there is a unique probability measure d P on B(R ) with
N X −1 P = pi (P ◦ Si ). [Hutchinson, 1981] i=1
If pi > 0 for all 1 ≤ i ≤ N, then support(P) = A. This measure P is called the self-similar measure associated with (S, p). In the case that the set S of similarities satisfy some additional conditions (such as open set condition, or strong separation condition), then A is a fractal with d-dimensional Lebesgue measure 0. Hence, P is singular w.r.t. Lebesgue measure. Many standard fractals (Cantor Set, Sierpinski Triangle, Sierpinski Carpet, Koch Curve) satisfy OSC; furthermore, some also satisfy SSC (Cantor Set, Sierpinski Carpet).
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 Self-similar singular measures
Many IFS type fractals define continuous measures that are not necessarily absolutely continuous.
Do˘ganC¸¨omez Quantization for infinite affine transformations In the case that the set S of similarities satisfy some additional conditions (such as open set condition, or strong separation condition), then A is a fractal with d-dimensional Lebesgue measure 0. Hence, P is singular w.r.t. Lebesgue measure. Many standard fractals (Cantor Set, Sierpinski Triangle, Sierpinski Carpet, Koch Curve) satisfy OSC; furthermore, some also satisfy SSC (Cantor Set, Sierpinski Carpet).
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 Self-similar singular measures
Many IFS type fractals define continuous measures that are not necessarily absolutely continuous.
N Let S = {Si }i=1 be similarity transformations with attractor A (a compact set). If p = (p1, p2,..., pN ) is a probability vector, then there is a unique probability measure d P on B(R ) with
N X −1 P = pi (P ◦ Si ). [Hutchinson, 1981] i=1
If pi > 0 for all 1 ≤ i ≤ N, then support(P) = A. This measure P is called the self-similar measure associated with (S, p).
Do˘ganC¸¨omez Quantization for infinite affine transformations Many standard fractals (Cantor Set, Sierpinski Triangle, Sierpinski Carpet, Koch Curve) satisfy OSC; furthermore, some also satisfy SSC (Cantor Set, Sierpinski Carpet).
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 Self-similar singular measures
Many IFS type fractals define continuous measures that are not necessarily absolutely continuous.
N Let S = {Si }i=1 be similarity transformations with attractor A (a compact set). If p = (p1, p2,..., pN ) is a probability vector, then there is a unique probability measure d P on B(R ) with
N X −1 P = pi (P ◦ Si ). [Hutchinson, 1981] i=1
If pi > 0 for all 1 ≤ i ≤ N, then support(P) = A. This measure P is called the self-similar measure associated with (S, p). In the case that the set S of similarities satisfy some additional conditions (such as open set condition, or strong separation condition), then A is a fractal with d-dimensional Lebesgue measure 0. Hence, P is singular w.r.t. Lebesgue measure.
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 Self-similar singular measures
Many IFS type fractals define continuous measures that are not necessarily absolutely continuous.
N Let S = {Si }i=1 be similarity transformations with attractor A (a compact set). If p = (p1, p2,..., pN ) is a probability vector, then there is a unique probability measure d P on B(R ) with
N X −1 P = pi (P ◦ Si ). [Hutchinson, 1981] i=1
If pi > 0 for all 1 ≤ i ≤ N, then support(P) = A. This measure P is called the self-similar measure associated with (S, p). In the case that the set S of similarities satisfy some additional conditions (such as open set condition, or strong separation condition), then A is a fractal with d-dimensional Lebesgue measure 0. Hence, P is singular w.r.t. Lebesgue measure. Many standard fractals (Cantor Set, Sierpinski Triangle, Sierpinski Carpet, Koch Curve) satisfy OSC; furthermore, some also satisfy SSC (Cantor Set, Sierpinski Carpet).
Do˘ganC¸¨omez Quantization for infinite affine transformations In general, the existence of the quantization coefficient for singular measures is not known. However, under some very stringent conditions the quantization coefficient exists (Graf-Luschgy, 2000) Lindsay and Mauldin: If µ is a measure associated with a conformal finite IFS with OSC, then D(µ) exists. Roychowdhury: D(µ) exists for self-similar measures generated by infinite IFS on R (2011). In many cases the actual value of D(µ) is very difficult to compute or not known.
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 Some known results
Theorem [Graf-Luschgy, 2000] Let µ be a self-similar measure generated by an IFS S κ PN r κ+2 satisfying SSC and κ ∈ (0, ∞) satisfy i=1(pi si ) = 1. Then D(µ) exists and is equal to κ.
Do˘ganC¸¨omez Quantization for infinite affine transformations Lindsay and Mauldin: If µ is a measure associated with a conformal finite IFS with OSC, then D(µ) exists. Roychowdhury: D(µ) exists for self-similar measures generated by infinite IFS on R (2011). In many cases the actual value of D(µ) is very difficult to compute or not known.
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 Some known results
Theorem [Graf-Luschgy, 2000] Let µ be a self-similar measure generated by an IFS S κ PN r κ+2 satisfying SSC and κ ∈ (0, ∞) satisfy i=1(pi si ) = 1. Then D(µ) exists and is equal to κ. In general, the existence of the quantization coefficient for singular measures is not known. However, under some very stringent conditions the quantization coefficient exists (Graf-Luschgy, 2000)
Do˘ganC¸¨omez Quantization for infinite affine transformations Roychowdhury: D(µ) exists for self-similar measures generated by infinite IFS on R (2011). In many cases the actual value of D(µ) is very difficult to compute or not known.
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 Some known results
Theorem [Graf-Luschgy, 2000] Let µ be a self-similar measure generated by an IFS S κ PN r κ+2 satisfying SSC and κ ∈ (0, ∞) satisfy i=1(pi si ) = 1. Then D(µ) exists and is equal to κ. In general, the existence of the quantization coefficient for singular measures is not known. However, under some very stringent conditions the quantization coefficient exists (Graf-Luschgy, 2000) Lindsay and Mauldin: If µ is a measure associated with a conformal finite IFS with OSC, then D(µ) exists.
Do˘ganC¸¨omez Quantization for infinite affine transformations In many cases the actual value of D(µ) is very difficult to compute or not known.
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 Some known results
Theorem [Graf-Luschgy, 2000] Let µ be a self-similar measure generated by an IFS S κ PN r κ+2 satisfying SSC and κ ∈ (0, ∞) satisfy i=1(pi si ) = 1. Then D(µ) exists and is equal to κ. In general, the existence of the quantization coefficient for singular measures is not known. However, under some very stringent conditions the quantization coefficient exists (Graf-Luschgy, 2000) Lindsay and Mauldin: If µ is a measure associated with a conformal finite IFS with OSC, then D(µ) exists. Roychowdhury: D(µ) exists for self-similar measures generated by infinite IFS on R (2011).
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 Some known results
Theorem [Graf-Luschgy, 2000] Let µ be a self-similar measure generated by an IFS S κ PN r κ+2 satisfying SSC and κ ∈ (0, ∞) satisfy i=1(pi si ) = 1. Then D(µ) exists and is equal to κ. In general, the existence of the quantization coefficient for singular measures is not known. However, under some very stringent conditions the quantization coefficient exists (Graf-Luschgy, 2000) Lindsay and Mauldin: If µ is a measure associated with a conformal finite IFS with OSC, then D(µ) exists. Roychowdhury: D(µ) exists for self-similar measures generated by infinite IFS on R (2011). In many cases the actual value of D(µ) is very difficult to compute or not known.
Do˘ganC¸¨omez Quantization for infinite affine transformations The optimal sets of n-means for many probability measures is not known. The same is the case n-th quantization error Vn(µ). The problem is studied only in the case of some absolutely continuous measures and some special cases of singular measures.
Theorem [Graf & Luschgy (1997)] + Let P = PC be standard Cantor distribution and `(n) ∈ Z with 2`(n) ≤ n < 2`(n)+1, n ≥ 1.
For each n, αn(P) is determined. 1 1 `(n) `(n)+1 1 `(n) Vn(P) = 8 ( 18 ) [2 − n + 9 (n − 2 )]. The quantization coefficients do not exist.
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-Cantor distribution
Do˘ganC¸¨omez Quantization for infinite affine transformations Theorem [Graf & Luschgy (1997)] + Let P = PC be standard Cantor distribution and `(n) ∈ Z with 2`(n) ≤ n < 2`(n)+1, n ≥ 1.
For each n, αn(P) is determined. 1 1 `(n) `(n)+1 1 `(n) Vn(P) = 8 ( 18 ) [2 − n + 9 (n − 2 )]. The quantization coefficients do not exist.
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-Cantor distribution
The optimal sets of n-means for many probability measures is not known. The same is the case n-th quantization error Vn(µ). The problem is studied only in the case of some absolutely continuous measures and some special cases of singular measures.
Do˘ganC¸¨omez Quantization for infinite affine transformations 1 1 `(n) `(n)+1 1 `(n) Vn(P) = 8 ( 18 ) [2 − n + 9 (n − 2 )]. The quantization coefficients do not exist.
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-Cantor distribution
The optimal sets of n-means for many probability measures is not known. The same is the case n-th quantization error Vn(µ). The problem is studied only in the case of some absolutely continuous measures and some special cases of singular measures.
Theorem [Graf & Luschgy (1997)] + Let P = PC be standard Cantor distribution and `(n) ∈ Z with 2`(n) ≤ n < 2`(n)+1, n ≥ 1.
For each n, αn(P) is determined.
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 Optimal sets-Cantor distribution
The optimal sets of n-means for many probability measures is not known. The same is the case n-th quantization error Vn(µ). The problem is studied only in the case of some absolutely continuous measures and some special cases of singular measures.
Theorem [Graf & Luschgy (1997)] + Let P = PC be standard Cantor distribution and `(n) ∈ Z with 2`(n) ≤ n < 2`(n)+1, n ≥ 1.
For each n, αn(P) is determined. 1 1 `(n) `(n)+1 1 `(n) Vn(P) = 8 ( 18 ) [2 − n + 9 (n − 2 )]. The quantization coefficients do not exist.
Do˘ganC¸¨omez Quantization for infinite affine transformations 0 1
1/2 0 1
1/6 5/6 0 1
1/18 5/18 5/6 0 1
1/18 5/18 13/18 17/18
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-Cantor distribution
Do˘ganC¸¨omez Quantization for infinite affine transformations 0 1
1/6 5/6 0 1
1/18 5/18 5/6 0 1
1/18 5/18 13/18 17/18
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-Cantor distribution
0 1
1/2
Do˘ganC¸¨omez Quantization for infinite affine transformations 0 1
1/18 5/18 5/6 0 1
1/18 5/18 13/18 17/18
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-Cantor distribution
0 1
1/2 0 1
1/6 5/6
Do˘ganC¸¨omez Quantization for infinite affine transformations 0 1
1/18 5/18 13/18 17/18
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-Cantor distribution
0 1
1/2 0 1
1/6 5/6 0 1
1/18 5/18 5/6
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 Optimal sets-Cantor distribution
0 1
1/2 0 1
1/6 5/6 0 1
1/18 5/18 5/6 0 1
1/18 5/18 13/18 17/18
Do˘ganC¸¨omez Quantization for infinite affine transformations uniform probability distributions on equilateral triangles (Dettmann & Roychowdhury; 2017) probability distributions associated with the Cantor Dust, stretched Sierpinski Triangles, and condensation systems (C¸¨omez & Roychowdhury; 2017, 2019, and 2020, resp.). Note: All these systems satisfy SSC.
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 Analogous results
Optimal sets of n-means and the n-th quantization errors are also determined for
Do˘ganC¸¨omez Quantization for infinite affine transformations probability distributions associated with the Cantor Dust, stretched Sierpinski Triangles, and condensation systems (C¸¨omez & Roychowdhury; 2017, 2019, and 2020, resp.). Note: All these systems satisfy SSC.
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 Analogous results
Optimal sets of n-means and the n-th quantization errors are also determined for uniform probability distributions on equilateral triangles (Dettmann & Roychowdhury; 2017)
Do˘ganC¸¨omez Quantization for infinite affine transformations Note: All these systems satisfy SSC.
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 Analogous results
Optimal sets of n-means and the n-th quantization errors are also determined for uniform probability distributions on equilateral triangles (Dettmann & Roychowdhury; 2017) probability distributions associated with the Cantor Dust, stretched Sierpinski Triangles, and condensation systems (C¸¨omez & Roychowdhury; 2017, 2019, and 2020, resp.).
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 Analogous results
Optimal sets of n-means and the n-th quantization errors are also determined for uniform probability distributions on equilateral triangles (Dettmann & Roychowdhury; 2017) probability distributions associated with the Cantor Dust, stretched Sierpinski Triangles, and condensation systems (C¸¨omez & Roychowdhury; 2017, 2019, and 2020, resp.). Note: All these systems satisfy SSC.
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 Optimal sets-Cantor Dust
Figure: Optimal sets of n-means for 1, 2, 3, 4, 5, 9, 12 and 13 for Cantor Dust
Do˘ganC¸¨omez Quantization for infinite affine transformations To each S associate the probability that p = 1 , for all i, j ≥ 1. Then, by (i,j) (i,j) 2i+j Hutchinson’s theorem, associated to S and P = {p(i,j)}, there exists a unique Borel 2 probability measure P on R such that ∞ X −1 P = p(i,j)P ◦ S(i,j). i,j=1
The support of P lies in the unit square [0, 1]2. Such a measure is called an infinitely 2 generated affine measure on R .The pair (S, P) is called (infinite) affine system.
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 Infinite affine systems
Recall that maps that preserve lines and parallelism are affine maps. In the rest of the talk, I will consider a collection of countably infinite affine transformations 2 S = {S(i,j) : i, j ≥ 1} on R , where x 1 y 1 S (x, y) = ( + 1 − , + 1 − ). (i,j) 3i 3i−1 3j 3j−1 Clearly, these affine transformations are all contractive but are not similarity mappings.
Do˘ganC¸¨omez Quantization for infinite affine transformations The support of P lies in the unit square [0, 1]2. Such a measure is called an infinitely 2 generated affine measure on R .The pair (S, P) is called (infinite) affine system.
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 Infinite affine systems
Recall that maps that preserve lines and parallelism are affine maps. In the rest of the talk, I will consider a collection of countably infinite affine transformations 2 S = {S(i,j) : i, j ≥ 1} on R , where x 1 y 1 S (x, y) = ( + 1 − , + 1 − ). (i,j) 3i 3i−1 3j 3j−1 Clearly, these affine transformations are all contractive but are not similarity mappings.
To each S associate the probability that p = 1 , for all i, j ≥ 1. Then, by (i,j) (i,j) 2i+j Hutchinson’s theorem, associated to S and P = {p(i,j)}, there exists a unique Borel 2 probability measure P on R such that ∞ X −1 P = p(i,j)P ◦ S(i,j). i,j=1
Do˘ganC¸¨omez Quantization for infinite affine transformations The pair (S, P) is called (infinite) affine system.
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 Infinite affine systems
Recall that maps that preserve lines and parallelism are affine maps. In the rest of the talk, I will consider a collection of countably infinite affine transformations 2 S = {S(i,j) : i, j ≥ 1} on R , where x 1 y 1 S (x, y) = ( + 1 − , + 1 − ). (i,j) 3i 3i−1 3j 3j−1 Clearly, these affine transformations are all contractive but are not similarity mappings.
To each S associate the probability that p = 1 , for all i, j ≥ 1. Then, by (i,j) (i,j) 2i+j Hutchinson’s theorem, associated to S and P = {p(i,j)}, there exists a unique Borel 2 probability measure P on R such that ∞ X −1 P = p(i,j)P ◦ S(i,j). i,j=1
The support of P lies in the unit square [0, 1]2. Such a measure is called an infinitely 2 generated affine measure on R .
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 Infinite affine systems
Recall that maps that preserve lines and parallelism are affine maps. In the rest of the talk, I will consider a collection of countably infinite affine transformations 2 S = {S(i,j) : i, j ≥ 1} on R , where x 1 y 1 S (x, y) = ( + 1 − , + 1 − ). (i,j) 3i 3i−1 3j 3j−1 Clearly, these affine transformations are all contractive but are not similarity mappings.
To each S associate the probability that p = 1 , for all i, j ≥ 1. Then, by (i,j) (i,j) 2i+j Hutchinson’s theorem, associated to S and P = {p(i,j)}, there exists a unique Borel 2 probability measure P on R such that ∞ X −1 P = p(i,j)P ◦ S(i,j). i,j=1
The support of P lies in the unit square [0, 1]2. Such a measure is called an infinitely 2 generated affine measure on R .The pair (S, P) is called (infinite) affine system.
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
The behaviour of this affine system (S, P) is best observed by looking at the first few iterates (basic rectangles) of the transformations on [0, 1]2.
...... ··· (1,3) (2,3) (3,3)
··· (1,2)(1,1) (1,2) (2,2) (3,2)
(1,1) (2,1) (3,1)
(1,1)(1,2) (1,1)(2,2) ···
(1,1)(1,1) ···
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
End of Talk, Part 1
THANK YOU!
Do˘ganC¸¨omez Quantization for infinite affine transformations