Foundations of Quantization for Probability Distributions

By S. Graf, Harald Luschgy. 2000

923 citations recorded by the source corpus.

Subfields: Computer Networks and Communications, Computer Vision and Pattern Recognition, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/4c9915936ee46e3e2c8331c823d44b3b741e6ac8

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In Computer Networks and Communications

In Computer Vision and Pattern Recognition

In Electrical and Electronic Engineering

Citations in the corpus

A QUANTIZATION TREE METHOD FOR PRICING AND HEDGING MULTIDIMENSIONAL AMERICAN OPTIONS (2005, methodology): “When PX no longer has a compact support, this is still true: one shows by induction on N (see Graf and Luschgy 2000) that x →‖X − X̂ x‖p reaches an absolute minimum on (Rd )N at some x∗ ∈ (Rd )N. PROPOSITION 2.2.”
Functional quantization rate and mean regularity of processes with an application to Lévy processes (2006, positive): “The minimal N th quantization error of X is then defined by eN,r(X,E) := inf {( Emin a∈α ‖X − a‖r )1/r :α⊂E, card(α)≤N } .”
AN OPTIMAL MARKOVIAN QUANTIZATION ALGORITHM FOR MULTI-DIMENSIONAL STOCHASTIC CONTROL PROBLEMS (2004, methodology): “Furthermore, one shows the following simple facts (see [21] or [15] and references therein): – If suppPY has an infinite support, any optimal grid Γ∗ has N pairwise distinct elements.”
Stabilizability of Stochastic Linear Systems with Finite Feedback Data Rates (2004, background): “6 of [13], an approach which does not permit recursive application.”
Clustering with Bregman Divergences: an Asymptotic Analysis (2016, methodology): “Graf and Luschgy proved that asymptotic centroid's distribution is like P r,EU (z) ∼ P d/(d+2r) (z).”
Approximations for Probability Distributions and Stochastic Optimization Problems (2011, methodology): “As for a proof we refer the reader to [7], Theorem 6.”
Sharp asymptotics of the functional quantization problem for Gaussian processes (2004, methodology): “Consequently, By Corollary 2.4, this implies and thus Letting c → 1 yields the assertion.”
Random projection trees and low dimensional manifolds (2008, positive): “Lemmas 9 and 10 then tell us that pk ≤ P[top split cleanly separates B from B′] · 0 + P[top split intersects both B and B′] · 2pk−1 + P[all other split configurations] · pk−1 ≤ 1 192 · 0 + 1 384 · 2pk−1 + „ 1− 1 192 − 1 384 « · pk−1 = „ 1− 1 384 « pk−1.”
Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance (2017, background): “The following proposition, which appears in [22], shows that many well behaved sets are regular, and so implies the existence of many examples for which our results are tight.”
Quantization (2019, positive): “We refer to Graf and Luschgy [9] for rigorous mathematical foundations of quantization theory.”
Optimal quantization methods and applications to numerical problems in finance (2004, background): “However, one shows that J2,2 = 5 18 √ 3 and that Jp,d ∼ ( d 2πe ) p 2 as d goes to infinity (see [13] for some proofs and other results using non Euclidean norms).”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “It is linked to quantization problems for the measures μ and ν (see [188] for a survey of the theory from a signal processing point of view, and [74] for a variational asymptotical approach).”
Scenario tree modeling for multistage stochastic programs (2009, methodology): “These scenarios might be obtained by quantization techniques [15], by sampling or resampling techniques based on parametric or nonparametric stochastic models of ξ .”
DISTORTION MISMATCH IN THE QUANTIZATION OF PROBABILITY MEASURES (2006, methodology): “There exists a constant C1 ∈ (0,∞) such that λd(B(x, ρ) ∩B(0,M) ∩A) ≥ C1ρ for every x ∈ B(0,M) ∩ A and ρ ≤ bC since B(0,M) ∩ A is finite union of convex sets (see [6]).”