Lp -mean quantization error Introduction to Optimal Quantization(s) L p -mean quantization error ⊲ What about “Optimal”? Is there an optimal way to select the grid/ N -quantizer to classify the data? In data analysis optimal clustering ? ⊲ The L p - mean quantization error Definition The L p -mean quantization error induced by a grid Γ ⊂ R d with size | Γ | ≤ N , N ∈ N ‚ ‚ ‚ ‚ ‚ ‚ ‚ dist( X , Γ) ‚ x ∈ Γ | X − x | e p ( X ; Γ) = p = ‚ min ‚ (1) p (only depends on the distribution µ = P X of X ). ⊲ The optimal L p -mean quantization problem consists in minimizing (1) over all grids of size | Γ | ≤ N . We define the L p -optimal mean quantization error at level N as n‚ ‚ o ‚ ‚ p : Γ ⊂ R d , | Γ | ≤ N x ∈ Γ | X − x | e p , N ( X ) := inf ‚ min ‚ . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 10 / 81
Lp -mean quantization error Introduction to Optimal Quantization(s) Voronoi Quantization ⊲ Noting that ` ´ = | X ( ω ) − b X Ξ(Ω) | | X ( ω ) − Ξ( ω ) | ≥ dist X ( ω ) , Ξ(Ω) one derives the more general optimality result ˘ ¯ � X − Ξ � p : Ξ ∈ L p ( R d ) , Card (Ξ(Ω)) ≤ N = W p ( P X , P N ) . e p , N ( X ) = inf Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 11 / 81
Lp -mean quantization error Introduction to Optimal Quantization(s) Voronoi Quantization ⊲ Noting that ` ´ = | X ( ω ) − b X Ξ(Ω) | | X ( ω ) − Ξ( ω ) | ≥ dist X ( ω ) , Ξ(Ω) one derives the more general optimality result ˘ ¯ � X − Ξ � p : Ξ ∈ L p ( R d ) , Card (Ξ(Ω)) ≤ N = W p ( P X , P N ) . e p , N ( X ) = inf X Γ provides an optimal L p -mean discretization of X by ⇒ Voronoi Quantization b Γ-valued random variables for every p ∈ (0 , + ∞ ). Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 11 / 81
Lp -mean quantization error Introduction to Optimal Quantization(s) Voronoi Quantization ⊲ Noting that ` ´ = | X ( ω ) − b X Ξ(Ω) | | X ( ω ) − Ξ( ω ) | ≥ dist X ( ω ) , Ξ(Ω) one derives the more general optimality result ˘ ¯ � X − Ξ � p : Ξ ∈ L p ( R d ) , Card (Ξ(Ω)) ≤ N = W p ( P X , P N ) . e p , N ( X ) = inf X Γ provides an optimal L p -mean discretization of X by ⇒ Voronoi Quantization b Γ-valued random variables for every p ∈ (0 , + ∞ ). Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 11 / 81
Lp -mean quantization error Introduction to Optimal Quantization(s) Voronoi Quantization ⊲ Noting that ` ´ = | X ( ω ) − b X Ξ(Ω) | | X ( ω ) − Ξ( ω ) | ≥ dist X ( ω ) , Ξ(Ω) one derives the more general optimality result ˘ ¯ � X − Ξ � p : Ξ ∈ L p ( R d ) , Card (Ξ(Ω)) ≤ N = W p ( P X , P N ) . e p , N ( X ) = inf X Γ provides an optimal L p -mean discretization of X by ⇒ Voronoi Quantization b Γ-valued random variables for every p ∈ (0 , + ∞ ). ⇒ The Nearest Neighbor projection is the coding rule, which yields the smallest L p -mean approximation error for X . Theorem (Kieffer, Cuesta-Albertos, (P.), Graf-Luschgy) ( a ) Let p ∈ (0 , + ∞ ) , X ∈ L p . For every level N ≥ 1 , there exists (at least) one L p -optimal quantization grid Γ ∗ , N at level N and N �− → e p , N ( X ) ↓ 0 (vanishes if supp ( X ) is finite, ↓ ↓ 0 otherwise) “ X Γ N , ∗ ” X Γ N , ∗ a . s . (stationarity/self-consistency). X | b = b ( b ) If p = 2 , E Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 11 / 81
Lp -mean quantization error Introduction to Optimal Quantization(s) Sketch of proof ( p ≥ 1) ( a ) We proceed by induction N = 1: ξ �→ � X − ξ � p is convex and coercive and atteins its minmum at an L p -median. ⇒ N + 1: Let ξ ∈ supp( X ) \ Γ ∗ , N , Γ ∗ , N L p -optimal at level N . N = N +1 := e p ( X , Γ ∗ , N ∪ { ξ } ) p < e p ( X , Γ ∗ , N ) p = e p , N ( X ) p ℓ ∗ so that n o K ∗ = Γ ⊂ R d : | Γ | = N + 1 , e p ( X , Γ) p ≤ ℓ ∗ � = ∅ , closed . . . N +1 . . . and bounded (send one component or more to infinity and use Fatou’s Lemma). → e p ( X , Γ) attains a global minimum over K ∗ . Then Γ �− “ X Γ N , ∗ ” ⊥ L 2 ` ´ X Γ N , ∗ − E X Γ N , ∗ ) ( b ) The random variable b X | b σ ( b . Hence “ X Γ N , ∗ ”‚ “ X Γ N , ∗ ”‚ ‚ X Γ N , ∗ ‚ ‚ ‚ X Γ N , ∗ − E ‚ X − b ‚ 2 ‚ X − E X | b ‚ 2 ‚b X | b ‚ 2 2 = 2 + 2 . Hence, uniqueness of conditional expectation yields “ X Γ N , ∗ ” X Γ N , ∗ X | b = b a . s . E Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 12 / 81
Lp -mean quantization error Introduction to Optimal Quantization(s) Applications ˘ ¯ Signal transmission: Let Γ ∗ , N = x ∗ 1 , . . . , x ∗ N X Γ ∗ , N = x ∗ Pre-processing I : re-ordering the labels i so that i �→ p ∗ i := P ( b i ) is decreasing. Pre-processing II : encoding i � Code ( i ) see [CT06]. A who emits and B who receives both share the one-to-one bible. x ∗ i ↔ Code ( i ) X is encoded, Code( i ) is transmitted, then decoded. Naive encoding : dyadic coding of the labels i N X p ∗ Complexity = i (1 + ⌊ log 2 i ⌋ ) ≤ 1 + ⌊ log 2 N ⌋ . i =1 ˘ 2 i − 1 ¯ Uniform signal X ∼ U ([0 , 1]) then Γ ∗ , N = and p ∗ i = 1 2 N , i = 1 : N N so that N X Complexity = 1 + 1 ⌊ log 2 i ⌋ ∼ log 2 ( N / e ) . N i =1 On the way to Shannon’s Source coding theorem (see e . g . [Dembo-Zeitouni]). . . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 13 / 81
Lp -mean quantization error Introduction to Optimal Quantization(s) Quantization for (Probability and) Numerics: What for? Cubature formulas for the computation of expectations. N X ` ´ X Γ ∗ , N ) F ( b p ∗ i F ( x ∗ E F ( X ) ≈ E = i ) . i =1 X Γ ∗ , N . i ) i =1 ,..., N of b What is needed? The distribution ( x ∗ i , p ∗ How to perform grid optimization? Lloyd I (Lloyd, 1982) and CLVQ (Mc Queen, further on). Conditional expectation approximation: ` Y Γ Y ´ F ( b X Γ X | b E ( F ( X ) | Y ) ≈ E . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 14 / 81
Lp -mean quantization error Introduction to Optimal Quantization(s) Quantization for (Probability and) Numerics: What for? Cubature formulas for the computation of expectations. N X ` ´ X Γ ∗ , N ) F ( b p ∗ i F ( x ∗ E F ( X ) ≈ E = i ) . i =1 X Γ ∗ , N . i ) i =1 ,..., N of b What is needed? The distribution ( x ∗ i , p ∗ How to perform grid optimization? Lloyd I (Lloyd, 1982) and CLVQ (Mc Queen, further on). Conditional expectation approximation: ` Y Γ Y ´ F ( b X Γ X | b E ( F ( X ) | Y ) ≈ E . Clustering (unsupervised learning): What for? Unsupervised classification Mc Queen, 1957; (up to improvements like Self-Organizing Kohonen Maps, Cottrell-Fort-P. 1998, among others). How to perform? Lloyd I (Lloyd, 1982) and CLVQ (Mc Queen, 1967, further on). A typical problem in progress: P n Distribution µ n ( ω, d ξ ) = 1 k =1 δ ξ k ( ω ) , ( ξ k ) k ≥ 1 i.i.d. n L 2 -Optimal quantization grid Γ ∗ n ( ω ) at a fixed level N ≥ 1. n ( ω ) = Γ ∗ , N optimal grid at level N for µ = L ( ξ 1 ). One has lim n → + ∞ Γ ∗ At which rate? Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 14 / 81
Lp -mean quantization error Introduction to Optimal Quantization(s) Extension and. . . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 15 / 81
Lp -mean quantization error Introduction to Optimal Quantization(s) Extension and. . . ⊲ Generalization to infinite dimension Still true in: a separable Hilbert space, even in a reflexive Banach space E (Cuesta-Albertos, PTRF , 1997) for a tight r.v. ‚ ‚ ‚ min ‚ is l.s.c. fro the product weak topology on E N ( x 1 , . . . , x N ) �− → 1 ≤ i ≤ N | X − x i | E p or even in a L 1 space (Graf-Luschgy-P., J. of Approx. , 2005) using τ -topology. . . ` ´ but. . . not in C ([0 , T ] , R ) , � · � sup . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 15 / 81
Lp -mean quantization error Introduction to Optimal Quantization(s) Extension and. . . ⊲ Generalization to infinite dimension Still true in: a separable Hilbert space, even in a reflexive Banach space E (Cuesta-Albertos, PTRF , 1997) for a tight r.v. ‚ ‚ ‚ min ‚ is l.s.c. fro the product weak topology on E N ( x 1 , . . . , x N ) �− → 1 ≤ i ≤ N | X − x i | E p or even in a L 1 space (Graf-Luschgy-P., J. of Approx. , 2005) using τ -topology. . . ` ´ but. . . not in C ([0 , T ] , R ) , � · � sup . ⊲ Convergence to 0 e p , N ( X ) ↓ 0 as N → + ∞ . Let ( z n ) n ≥ 1 be an everywhere dense sequence in R d » – ` ´ p = E e p , N ( X ) p ≤ e p 1 ≤ i ≤ N | X − z i | p X , { z 1 , . . . , z N } min ↓ 0 as N → + ∞ . by the Lebesgue dominated convergence theorem. ⊲ But. . . at which rate? At least for the finite dimensional vector space. Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 15 / 81
Introduction to Optimal Quantization(s) Quantization Rates/Zador’s Theorem Theorem (Zador’s Theorem, from 1963 (PhD) to 2000) ( a ) Sharp asymptotic (Zador, Kieffer, Bucklew & Wise, Graf & Luschgy in [GL00]): Let X ∈ L p + ( R d ) with distribution P X = ϕ.λ d ⊥ + ν . Then „Z « ( d + p ) / d R d ϕ d / ( d + p ) d λ d 1 d · e p , N ( X ) = Q p , |·| . N →∞ N lim ` ´ 1 d . e p , N U ([0 , 1] d ) where Q p , |·| = inf N ≥ 1 N . ( b ) Non-asymptotic (Pierce, Graf & Luschgy in [GL00], Luschgy-P. [LP08]): Let p ′ > p. There exists C p , p ′ , d ∈ (0 , + ∞ ) such that, for every R d -valued X r.v. e p , N ( X ) ≤ C p , p ′ , d σ p ′ ( X ) . N − 1 d . ∀ N ≥ 1 , Remarks. • σ p ′ ( X ) := inf a ∈ R d � X − a � p ′ ≤ + ∞ is the L p ′ -(pseudo-)standard deviation. • The rate N − 1 d is known as the curse of dimensionality . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 16 / 81
Introduction to Optimal Quantization(s) Quantization Rates/Zador’s Theorem Theorem (Zador’s Theorem, 2016) ( a ) Sharp asymptotic (Zador, Kieffer, Bucklew & Wise, Graf & Luschgy in [GL00], Luschgy-P., 2016): Let X ∈ L p ( R d ) with distribution P X = ϕ.λ d ⊥ + ν such that ϕ is essentially L p -radial and non-increasing [e.g. ϕ ( ξ ) ≍ g ( | ξ | 0 ) , g ↓ on ( a 0 , + ∞ ) &. . . ] Then „Z « ( d + p ) / d 1 R d ϕ d / ( d + p ) d λ d d · e p , N ( X ) = Q p , |·| · N →∞ N lim ` ´ 1 d · e p , N U ([0 , 1] d ) where Q p , |·| = inf N N . ( b ) Non-asymptotic (Pierce, Graf & Luschgy in [GL00], Luschgy-P. [LP08]): Let p ′ > p. There exists C p , p ′ , d ∈ (0 , + ∞ ) such that, for every R d -valued X r.v. e p , N ( X ) ≤ C p , p ′ , d σ p ′ ( X ) . N − 1 d . ∀ N ≥ 1 , Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 17 / 81
Introduction to Optimal Quantization(s) Numerical computation of quantizers Numerical computation of quantizers ⊲ Stationary quantizers Optimal grids Γ ∗ at level satisfy ` X Γ ∗ = E X Γ ∗ ) b X | b or equivalently if Γ ∗ = { x ∗ 1 , . . . , x ∗ N ` ´ x ∗ X | X ∈ C i (Γ ∗ ) i = E (Nearly) optimal grids can be computed by optimization algorithms : Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 18 / 81
Introduction to Optimal Quantization(s) Numerical computation of quantizers Numerical computation of quantizers ⊲ Stationary quantizers Optimal grids Γ ∗ at level satisfy ` X Γ ∗ = E X Γ ∗ ) b X | b or equivalently if Γ ∗ = { x ∗ 1 , . . . , x ∗ N ` ´ x ∗ X | X ∈ C i (Γ ∗ ) i = E (Nearly) optimal grids can be computed by optimization algorithms : ⊲ Lloyd’s I algorithm (Randomized) fixed-point method. n = 0 Initial grid Γ [0] = { x [0] 1 , . . . , x [0] N } ⇒ k + 1 Standard step : Let Γ [ k ] the current grid. k = ` ´ ` X Γ [ k ] = x [ k ] ´ x [ k +1] X | X ∈ C i (Γ [ k ] ) X | b = E = E i i and set Γ [ k +1] = { x [ k +1] , i = 1 : N } . i i Proposition (Lloyd I always makes the quantization error decrease) ‚ X Γ ( k +1) ‚ ‚ ‚ ‚ X Γ ( k ) ‚ ` X Γ ( k ) ´ ‚ X − b ‚ ‚ X − E X | b ‚ ‚ X − b ‚ 2 ≤ 2 ≤ 2 | {z } Γ ( k +1) - valued Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 18 / 81
Introduction to Optimal Quantization(s) Numerical computation of quantizers When d = 1 and L ( X ) is log-concave: exponetially fast convergence (Kieffer, 1982). Renewal of interest for 1-D quantization for quadrature formulas [Callegaro et al., 2017]. However . . . no general proof of convergence when L ( X ) has a non compact support and d ≥ 2. Splitting method : initialize Lloyd’s I procedure inductively on the size N by ` ´ Γ N , (0) = Γ N − 1 , ( ∞ ) ∪ { ξ N } , ξ N ∈ supp L ( X ) . (see P. -Yu, SICON, 2016). Then Γ N +1 , ( k ) → Γ N , ( ∞ ) (stationary quantizer of full size N . . . ) as k → + ∞ Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 19 / 81
Introduction to Optimal Quantization(s) Numerical computation of quantizers When d = 1 and L ( X ) is log-concave: exponetially fast convergence (Kieffer, 1982). Renewal of interest for 1-D quantization for quadrature formulas [Callegaro et al., 2017]. However . . . no general proof of convergence when L ( X ) has a non compact support and d ≥ 2. Splitting method : initialize Lloyd’s I procedure inductively on the size N by ` ´ Γ N , (0) = Γ N − 1 , ( ∞ ) ∪ { ξ N } , ξ N ∈ supp L ( X ) . (see P. -Yu, SICON, 2016). Then Γ N +1 , ( k ) → Γ N , ( ∞ ) (stationary quantizer of full size N . . . ) as k → + ∞ Practical implementation based on Monte Carlo simulations (or a dataset). P M ` ´ m =1 g ( X m ) 1 { X m ∈ C i (Γ) } X Γ = x i g ( X ) | b , ( X m ) m ≥ 1 i.i.d. ∼ X . E = lim P M M → + ∞ m =1 1 { X m ∈ C i (Γ) } Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 19 / 81
Introduction to Optimal Quantization(s) Numerical computation of quantizers ⊲ Competitive Learning Vector Quantization algorithm ( p = 2 ) “Simply” a Stochastic gradient descent Let D N : ( R d ) N → R + be the (quadratic) distortion function 1 ≤ i ≤ N � X − x i � 2 → D N ( x ) := E min x ∈ ( R d ) N . min Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 20 / 81
Introduction to Optimal Quantization(s) Numerical computation of quantizers ⊲ Competitive Learning Vector Quantization algorithm ( p = 2 ) “Simply” a Stochastic gradient descent Let D N : ( R d ) N → R + be the (quadratic) distortion function 1 ≤ i ≤ N � X − x i � 2 → D N ( x ) := E min x ∈ ( R d ) N . min As soon as | · | is smooth enough ⇒ D N is differentiable at grids of full size. and if Γ = { x 1 , . . . , x N } , ` ˆ` ´ ˜´ ∂ D N ∂ x i (Γ) = 2 E x i − X 1 { X ∈ C i (Γ) } i =1: N Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 20 / 81
Introduction to Optimal Quantization(s) Numerical computation of quantizers ⊲ Competitive Learning Vector Quantization algorithm ( p = 2 ) “Simply” a Stochastic gradient descent Let D N : ( R d ) N → R + be the (quadratic) distortion function 1 ≤ i ≤ N � X − x i � 2 → D N ( x ) := E min x ∈ ( R d ) N . min As soon as | · | is smooth enough ⇒ D N is differentiable at grids of full size. and if Γ = { x 1 , . . . , x N } , ` ˆ` ´ ˜´ ∂ D N ∂ x i (Γ) = 2 E x i − X 1 { X ∈ C i (Γ) } i =1: N Main point : ∇ D N (Γ) = 0 iff Γ is stationary. Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 20 / 81
Introduction to Optimal Quantization(s) Numerical computation of quantizers ⊲ Competitive Learning Vector Quantization algorithm ( p = 2 ) “Simply” a Stochastic gradient descent Let D N : ( R d ) N → R + be the (quadratic) distortion function 1 ≤ i ≤ N � X − x i � 2 → D N ( x ) := E min x ∈ ( R d ) N . min As soon as | · | is smooth enough ⇒ D N is differentiable at grids of full size. and if Γ = { x 1 , . . . , x N } , ` ˆ` ´ ˜´ ∂ D N ∂ x i (Γ) = 2 E x i − X 1 { X ∈ C i (Γ) } i =1: N Main point : ∇ D N (Γ) = 0 iff Γ is stationary. Hence we can implement a zero search (stochastic) gradient . . . known as Competitive Learning Vector Quantization Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 20 / 81
Introduction to Optimal Quantization(s) Numerical computation of quantizers • d = 1: Z x i +1 / 2 X N | ξ − x i | 2 d P X ( ξ ) D N ( x ) = x i − 1 / 2 i =1 ⇒ Evaluation of Voronoi-Cells, Gradient and Hessian is simple if f X , F X & E 1 X have closed form � Newton-Raphson. • d ≥ 2: Stochastic Gradient Method: CLVQ Simulate ξ 1 , ξ 2 , . . . independent copies of X Generate step sequence γ 1 , γ 2 , . . . A B + n ց 0 γ n = η ≈ 0 Usually: step γ n = or Grid updating n �→ n + 1: Selection: select winner index: i ∗ ∈ argmin i | x n i − ξ n | ( ` ´ x n +1 := x n i ∗ + γ n ( x n x n i ∗ − ξ n ) ≡ dilat ( ξ n ; 1 − γ n ) i ∗ i ∗ Learning: x n +1 := x n for j � = i ∗ . j , j Nearest neighbour search: Computational challenge of simulation based stochastic optimization methods : 1 { X ∈ C i (Γ) } ≡ NEAREST NEIGHBOUR SEARCH Highly challenging problem in higher dimension, say d ≥ 4 or 5. Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 21 / 81
Introduction to Optimal Quantization(s) Optimal Quantizers Figure: A random Quantizer for N (0 , I 2 ) of size N = 500 in ( R 2 , | · | 2 ). Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 22 / 81
Introduction to Optimal Quantization(s) Optimal Quantizers Figure: A Quantizer for N (0 , I 2 ) of size N = 500 in ( R 2 , | · | 2 ). Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 23 / 81
Introduction to Optimal Quantization(s) Optimal Quantizers Benett’s conjecture (1955): a coloured approach Figure: An N -quantization of X ∼ N (0; I 2 ) with coloured weights: P ( X ∈ C i (Γ ( ∗ , N ) )) ( with J. Printems ) Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 24 / 81
Introduction to Optimal Quantization(s) Optimal Quantizers Toward Benett’s conjecture: Γ ( ∗ , N ) = { x 1 . . . , x N } X ∼ N (0; I 2 ). Figure: x i �→ P ( X ∈ C i (Γ ( ∗ , N ) ) (green Gaussian line); x i �→ E | X − x i | 2 1 { X ∈ C i (Γ ( ∗ , N ) ) } (red flat line) ( with J.C. fort ) → E | X − x i | 2 1 X ∈ C i (Γ ∗ , N ) ≃ Constant. Local inertia: x i �− „ « 1 x 2 3 i Weights: x i �→ P ( X ∈ C i (Γ ( ∗ , N ) ) ≃ C . e − (fitting) 2 Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 25 / 81
Introduction to Optimal Quantization(s) Optimal Quantizers More on Benett’s conjecture ⊲ Benett’s conjecture (weak form): In any dimension d , L p -optimal quantizers satisfy → E | X − x i | 2 1 X ∈ C i (Γ ∗ , N ) ≃ e N ( X ) Local inertia: x i �− . N „ « d x 2 d + p i Weights: x i �→ P ( X ∈ C i (Γ ( ∗ , N ) ) ≃ C . e − . 2 When d = 1 is holds uniformly on compacts sets ([Fort-P.], ’03), when d ≥ 1 at least in a measure sense. ⊲ Strong Benett’s conjecture: Conjecture on the geometric form of Voronoi cells go U ([0 , 1] d ) ( d = 2: regular hexagon, d = 3 octaedron, d ≥ 4 ????). Generic form of Voronoi cells for A.C. distributions. Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 26 / 81
Introduction to Optimal Quantization(s) Optimal Quantizers Quantizing Non-Gaussian multivariate distributions Figure: A Quantizer for ( B 1 , sup t ∈ [0 , 1] B t , B std B.M. of size N = 500 in ( R 2 , | · | 2 ). Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 27 / 81
Back to learning Back to clustering If ( ξ k ) k ≥ i.i.d.d. ξ 1 ∼ µ = L ( ξ 1 ) on R d , consider its empirical measure X n µ n ( ω, d ξ ) = 1 , δ ξ k . n k =1 Assume that µ ( B (0; 1)) = 1. For every ω ∈ Ω, there exists (at least) an optimal quantizer Γ ( N ) ( ω, n ) for µ n ( ω, d ξ ). Then (Biau et al., 2008, see [BDL08]) 0 s 1 r “ ´” d N 1 − 2 ` d log n Nd Γ ( N ) ( ω, n ) , µ @ A − e 2 , N ( µ ) ≤ C min E e 2 n , n where C > 0 is a universal real constant. See also (Graf-Luschgy, AoP, 2002, [GL02]) for other results on empirical measures (bounded support). Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 28 / 81
Quantization and Cubature Cubature formulae Back to numerical Probability? Quantization for Cubature ⊲ Assume that we have access to L ( b X Γ ): both the grid and the Voronoi cell weights Γ = { x 1 , . . . , x N } and p Γ i = P ( X ∈ C i (Γ)) , i = 1 , . . . , N . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 29 / 81
Quantization and Cubature Cubature formulae Back to numerical Probability? Quantization for Cubature ⊲ Assume that we have access to L ( b X Γ ): both the grid and the Voronoi cell weights Γ = { x 1 , . . . , x N } and p Γ i = P ( X ∈ C i (Γ)) , i = 1 , . . . , N . X Γ ) for some Lipschitz continuous F : R d → R becomes ⇒ The computation of E F ( b = straightforward: N X E F ( b X Γ ) = p Γ i F ( x i ) . i =1 Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 29 / 81
Quantization and Cubature Cubature formulae Back to numerical Probability? Quantization for Cubature ⊲ Assume that we have access to L ( b X Γ ): both the grid and the Voronoi cell weights Γ = { x 1 , . . . , x N } and p Γ i = P ( X ∈ C i (Γ)) , i = 1 , . . . , N . X Γ ) for some Lipschitz continuous F : R d → R becomes ⇒ The computation of E F ( b = straightforward: N X E F ( b X Γ ) = p Γ i F ( x i ) . i =1 ⊲ As a first error estimate, we already know that | E F ( X ) − E F ( b X Γ ) | ≤ [ F ] Lip E | X − b X Γ | . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 29 / 81
Quantization and Cubature Error estimates Error Estimates ⊲ First order. Moreover, if Γ N , ∗ is L 1 -optimal at level N ≥ 1 n o | E F ( X ) − E F ( Y ) | , card ( Y (Ω)) ≤ N inf sup [ F ] Lip ≤ 1 ˛ X Γ N , ∗ ˛ X Γ N , ∗ ) | = E | E F ( X ) − E F ( b ˛ X − b ˛ = e 1 , N ( X ) = sup [ F ] Lip ≤ 1 i . e . Optimal Quantization is optimal for the class of Lipschitz functions or equivalently. ` ´ e 1 , N ( X ) = W 1 L ( X ) , P N . ˘ ¯ with P N = atomic distribution with at most N atoms . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 30 / 81
Quantization and Cubature Error estimates Error Estimates ⊲ First order. Moreover, if Γ N , ∗ is L 1 -optimal at level N ≥ 1 n o | E F ( X ) − E F ( Y ) | , card ( Y (Ω)) ≤ N inf sup [ F ] Lip ≤ 1 ˛ X Γ N , ∗ ˛ X Γ N , ∗ ) | = E | E F ( X ) − E F ( b ˛ X − b ˛ = e 1 , N ( X ) = sup [ F ] Lip ≤ 1 i . e . Optimal Quantization is optimal for the class of Lipschitz functions or equivalently. ` ´ e 1 , N ( X ) = W 1 L ( X ) , P N . ˘ ¯ with P N = atomic distribution with at most N atoms . ⊲ Second order. Proposition Second order cubature error bound Assume F ∈ C 1 Lip and the grid Γ is stationary (e . g . because it is L 2 -optimal), i . e . X Γ = E ( X | b b X Γ ) . Then a Taylor expansion yields ` ˛ X Γ ´ | E F ( X ) − E F ( b | E F ( X ) − E F ( b ∇ F ( b ˛ X − b X Γ ) | X Γ ) − E X Γ ) = | [ DF ] Lip · E | X − b X Γ | 2 . ≤ Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 30 / 81
Quantization and Cubature Error estimates ⊲ Convexity Furthermore, if F is convex, then Jensen’s inequality implies for stationary grids Γ E F ( b X Γ ) ≤ E F ( X ) . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 31 / 81
Quantization and Cubature Error estimates Quantization for Conditional expectation (Pythagoras’ Theorem) ⊲ Applications in Numerical Probability = conditional expectation approximation. b b X = q X ( X ) Y = q Y ( Y ) Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 32 / 81
Quantization and Cubature Error estimates Quantization for Conditional expectation (Pythagoras’ Theorem) ⊲ Applications in Numerical Probability = conditional expectation approximation. b b X = q X ( X ) Y = q Y ( Y ) Proposition (Pythagoras’ Theorem for conditional expectation) Let P ( y , du ) = L ( X | Y = y ) be a regular version of the conditional distribution of X given Y , so that ` ´ g ( X ) | Y = Pg ( Y ) a . s . E Then ‚ ` ´ ` ´‚ ‚ ‚ ‚ ‚ ‚ E g ( b X ) | b ‚ 2 ‚ X − b ‚ 2 ‚ Pg ( Y ) − Pg ( b ‚ 2 [ g ] 2 g ( X ) | Y − E ≤ Y X 2 + Y ) Lip 2 2 ‚ ‚ ‚ ‚ [ g ] 2 ‚ X − b ‚ 2 2 + [ Pg ] 2 ‚ Y − b ‚ 2 ≤ X Y 2 . Lip Lip If P propagates Lipschitz continuity: [ Pg ] Lip ≤ [ P ] Lip [ g ] Lip . then quantization produces a control of the error. Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 32 / 81
Quantization and Cubature Error estimates Quantization for Conditional expectation ⊲ Sketch of proof. As ´ L 2 ( P ) ` Pg ( Y ) | b ⊥ σ ( b Pg ( Y ) − E Y Y ) and “ ´” ⊥ “ ´” ` ´ ` ´ ` ´ ` ` ´ ` g ( b X ) | b Pg ( Y ) | b Pg ( Y ) | b g ( b X ) | b E g ( X ) | Y − E Y = E g ( X ) | Y − E Y + E Y − E Y so that by Pythagoras’ theorem ‚ ` ´ ` ´‚ ‚ ` ´‚ ‚ ` ´ ` ´‚ ‚ E g ( b X ) | b ‚ 2 ‚ Pg ( Y ) − E Pg ( Y ) | b ‚ 2 ‚ E Pg ( X ) | b g ( b X ) | b ‚ 2 g ( X ) | Y − E − E Y 2 = Y 2 + Y Y 2 ‚ ´‚ ‚ ‚ ‚ 2 ‚ 2 ‚ Pg ( Y ) − Pg ( b ‚ g ( X ) − g ( b ≤ Y ) 2 + X ) 2 . ‚ ‚ ‚ ‚ ‚ 2 ‚ 2 ≤ [ Pg ] 2 ‚ Y − b 2 + [ g ] 2 ‚ X − b Y X 2 . Lip Lip ⊲ If p � = 2, a Minkowski like control is preserved ‚ ´‚ ‚ ‚ ‚ ‚ ` ´ ` ‚ E g ( b X ) | b ‚ ‚ X − b ‚ ‚ Pg ( Y ) − Pg ( b ‚ g ( X ) | Y − E Y ≤ [ g ] Lip X p + Y ) p p ‚ ‚ ‚ ‚ ‚ X − b ‚ Y − b ‚ ‚ ≤ [ g ] Lip X p + [ Pg ] Lip Y p . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 33 / 81
Application to BSDE A typical result (BSDE) ⊲ We consider a “standard” BSDE: Z T Z T f ( s , X s , Y s , Z s ) ds − t ∈ [0 , T ] , Y t = h ( X T ) + Z s dW s , t t where the exogenous process ( X t ) t ∈ [0 , T ] is a diffusion Z t Z t x ∈ R d . X t = x + b ( s , X s ) ds + σ ( s , X s ) dW s , 0 0 with b , σ , h Lipschitz continuous in x , f Lipschitz in ( x , y , z ) uniformly in t ∈ [0 , T ]. . . ⊲ which is the probabilistic representation of the partially non-linear PDE ∂ t u ( t , x ) + Lu ( t , x ) + f ( t , x , u ( t , x ) , ( ∂ ∗ x u σ )( t , x )) = 0 on [0 , T ) × R d , u ( T , . ) = h ` ´ with Lg = ( ∇ b | g ) + 1 σ ∗ D 2 g σ 2 Tr . ⊲ . . . and its time discretization scheme with step ∆ n = T n recursively defined by ¯ h ( ¯ Y t n = X t n n ) , n ` ´ ¯ E ( ¯ t n k , ¯ k , E ( ¯ k ) , ¯ Y t n = Y t n k +1 |F t n k ) + ∆ n f X t n Y t n k +1 |F t n ζ t n , k k ` ¯ ´ ` ´ 1 1 ¯ ( ¯ k +1 − ¯ k +1 − W t n k ) |F t k k +1 − W t n k ) |F t k ζ t n = ∆ n E Y t n k +1 ( W t n = ∆ n E Y t n Y t n k )( W t n k where ¯ X is the Euler scheme of X defined by ¯ k +1 = ¯ k + b ( n k , ¯ k )∆ n + σ ( n k , ¯ X t n X t n X t n X t n k )( W t n k +1 − W t n k ) . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 34 / 81
Application to BSDE ⊲ . . . spatially discretized by quantization: We “force” Markov property to write a Quantized Backward Dynamic Programming Principle b h ( b Y n = X n ) `b ´ b b E k ( b X k , b E k ( b Y k +1 ) , b Y k = Y k +1 ) + ∆ n f k ζ k 1 b b E k ( b ζ k = Y k +1 ( W t n k +1 − W t n k )) ∆ n where b E k = E ( · | b X k ) . ⊲ By induction b v k ( b Y k = ˆ X k ) , k = 0 , . . . , n . so that `b ´ ` ´ b = b v k +1 ( b k +1 − W t n k +1 − W t n E k Y k +1 ( W t n k ) E k ˆ X k +1 )( W t n k ) . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 35 / 81
Application to BSDE Quanrization tree ⊲ A Quantization tree for ( b X k ) k =0 ,..., n : N = N 0 + · · · + N n , N k = size of layer t n k . Figure: A typical (small!) 1-dimensional quantization tree ⊲ At time k (i.e. t k ) ` ´ b with Γ k = { x k 1 , . . . , x k X t k = Proj Γ k X t k Nk } is a grid of size N k . ⊲ What kind of tree a quantization tree is ? A quantization tree is not re-combining. But its size can designed a priori (and subject to possible optimization). Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 36 / 81
Application to BSDE Calibrating the quantization tree ⊲ To implement the above Quantized Backward Dynamic Programming Principled we need to compute repeatedly conditional expectations of the form ` ´ ` ´ ϕ ( b X k +1 ) | b ϕ ( b X k +1 )∆ W t k +1 | b E X k and E X k ⊲ First, one has N k +1 X ` ´ ϕ ( b π k ij ϕ ( x k +1 X k +1 ) 1 { b = b ) E X k = x k i } j j =1 where ` ´ π k b ij = P X k +1 ∈ C j (Γ k +1 ) & X k ∈ C i (Γ k ) π k so we need to estimate the hyper-matrix [ˆ ij ] i , j , k . ⊲ Weights for the Z term N k +1 X ` ´ ϕ ( b π W , k ϕ ( x k +1 X k +1 )∆ W t k +1 1 { b = e ) E X k = x k j i } ij j =1 where “ ” π W , k e = E 1 { X k +1 ∈ C j (Γ k +1 ) }∩{ , b i } ∆ W t k +1 X k = x k ij Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 37 / 81
Application to BSDE Quantized forward Kolmogorov equations (on weights) ⊲ Note that by elementary Bayes formula N k − 1 X ` ´ p k X ∈ C j (Γ k ) π k − 1 j := P = ˆ ij i =1 so that we may compute ` ´ ϕ ( b ` ´ X k +1 ) 1 { X k ∈ C i (Γ k ) } = E ϕ ( b X k +1 ) | b E X k P ( X ∈ C i (Γ k )) ⊲ Initialization: Quantize X 0 (often X 0 = x 0 ). Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 38 / 81
Application to BSDE Grid optimization and calibration (offline) ⊲ Simulability Exact X k = X t k when possible. A discretization scheme X k = ¯ X k . Let ( X m k , ∆ W m t k +1 ) 0 ≤ k ≤ n , m = 1 : M be i.i.d. copies of ( X k , ∆ W m t k +1 ) 0 ≤ k ≤ n . ⊲ Grid Optimization: Let the sample “pass” through the quantization tree using either Randomized Lloyd procedure. or CLVQ . to optimize the grids Γ k at each time level. π k π k ⊲ Calibrate b ij and e ij : n o X M 1 π k m : X m k ∈ C i (Γ k ) & X m b ij = lim Card k +1 ∈ C j (Γ k +1 ) , 1 ≤ m ≤ M M M → + ∞ m =1 and h i X M 1 π k ∆ W m e ij = lim E t k +1 1 { X m . k ∈ C i (Γ k ) }∩{ X m k +1 ∈ C j (Γ k +1 ) } M M → + ∞ m =1 ⊲ Embedded optimal quantization: Perform optimization and calibration simultaneously. Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 39 / 81
Application to BSDE Error estimates Theorem (A priori error estimates (Sagna-P., SPA 2017)) Suppose that all the “Lipschitz” assumptions on b, σ , f , h are fulfilled. ( a ) “Price”: Then, for every k = 0 , . . . , n, „ n « X n ‚ ‚ ‚ ‚ e (1+[ f ] Lip )( t n i − t n ‚ ¯ k − b ‚ 2 ‚ ¯ i − b ‚ 2 2 ≤ [ f ] 2 k ) K i ( b , σ, T , f , h ) Y t n Y k X t n X t n 2 = O . Lip 2 i N d i = k ( b ) “Hedge”: n − 1 n − 1 X ‚ ‚ X k ‚ ‚ ‚ ‚ e (1+[ f ] Lip ) t n ‚ 2 ‚ 2 ‚ 2 ‚ ¯ k − b ‚ Y t n k +1 − b ‚ X t n k − b ∆ n ζ t n ζ k 2 ≤ Y t n 2 + K k ( b , σ, T , f , h ) X t n 2 k +1 k k =0 k =0 Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 40 / 81
Application to BSDE Error estimates Theorem (A priori error estimates (Sagna-P., SPA 2017)) Suppose that all the “Lipschitz” assumptions on b, σ , f , h are fulfilled. ( a ) “Price”: Then, for every k = 0 , . . . , n, „ n « X n ‚ ‚ ‚ ‚ e (1+[ f ] Lip )( t n i − t n ‚ ¯ k − b ‚ 2 ‚ ¯ i − b ‚ 2 2 ≤ [ f ] 2 k ) K i ( b , σ, T , f , h ) Y t n Y k X t n X t n 2 = O . Lip 2 i N d i = k ( b ) “Hedge”: n − 1 n − 1 X ‚ ‚ X k ‚ ‚ ‚ ‚ e (1+[ f ] Lip ) t n ‚ 2 ‚ 2 ‚ 2 ‚ ¯ k − b ‚ Y t n k +1 − b ‚ X t n k − b ∆ n ζ t n ζ k 2 ≤ Y t n 2 + K k ( b , σ, T , f , h ) X t n 2 k +1 k k =0 k =0 ( c ) “RBSDE”: The same error bounds hold with Reflected BSDE (so far without Z in f ) by replacing h by h k = h ( t n k , . ) where h ( t , X t ) is the obstacle process in the resulting quantized scheme. What is new (compared to Bally-P. 2003 for reflected BSDE )? +: Z inside the driver f for quantization error bounds. +: Squares everywhere Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 40 / 81
Application to BSDE Distortion mismatch A new result : distortion mismatch/ L s -rate optimality, s > p ⊲ Let Γ ( p ) N , N ≥ 1, be a sequence L p -optimal grids. What about e s ( X , Γ p N ) ( L s -mean quantization error) when X ∈ L s R d ( P ) for s > p ? Theorem ( L p - L s -distortion mismatch, Graf-Luschgy-P. 2005, Luschgy-P. 2015) R d ( P ) and let (Γ ( p ) ( a ) Let X ∈ L p N ) N ≥ 1 be an L p -optimal sequence for grids. Let s ∈ ( p , p + d ) . If sd d + p − s + δ ( P ) , δ > 0 , X ∈ L sd sd (note that d + p − s > s and lim s → p + d d + p − s = + ∞ ), then 1 d e s (Γ ( p ) lim N N N , X ) < + ∞ . ( b ) If P X = f ( | x | ) .λ d ( d ξ ) (radial density) then δ = 0 is admissible. sd 1 d + p − s = + ∞ , then lim N N d e s (Γ ( p ) ( c ) If E | X | N , X ) = + ∞ . ⊲ Possible perspectives: error bounds for quantization based numerical schemes for BSDE with a quadratic Z term ? ⊲ So far, an application to quantized non-linear filtering. Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 41 / 81
Application to BSDE Distortion mismatch Application to non-linear filtering Signal process ( X k ) k ≥ 0 is an R d -valued Markov chain. The observation process ( Y k ) k ≥ 0 is a sequence of R q -valued random vectors such that ( X k , Y k ) k ≥ 0 is a Markov chain. The conditional distribution L ( Y k | X k − 1 , Y k − 1 , X k ) = g k ( X k − 1 , Y k − 1 , X k , y ) λ q ( dy ) Aim : compute Π y 0: n , n ( dx ) = P ( X k ∈ dx | Y 1 = y 1 , · · · , Y n = y n ) Kallianpur-Streibel formula: set y = y 0: n = ( y 0 , . . . , y n ) a vector of observations Π y , n ( dx ) = Π y , n f = π y , n f π y , n 1 with the normalized filter π y 0 , n , n defined by Y n π y 0: n , n f = E ( f ( X n ) L y 0: n , n ) with L y 0: n , n = g k ( X k − 1 , y k − 1 , X k , y k ) , k =1 solution to both a forward and a backward inductionsbased on the kernels H y , k h ( x ) = E ( h ( X k ) g k ( x , y k − 1 , X k , y k ) | X k − 1 = x ) , H y , 0 f ( x ) = E ( f ( X 0 )) , Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 42 / 81
Application to BSDE Distortion mismatch Forward: Start from π y , 0 = H y , 0 and define by a forward induction π y , k f = π y , k − 1 H y , k f , k = 1 , . . . , n . Backward: We define by a backward induction u y , n ( f )( x ) = f ( x ) , u y , k − 1 ( f ) = H y , k u y , k ( f ) , k = 0 , . . . , n . so that π y , n f = u y , − 1 ( f ) This formulation is useful in order to establish the quantization error bound. Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 43 / 81
Application to BSDE Distortion mismatch Quantized Kallianpur-Streibel formula (P.-Pham (2005)) Quantization of the kernel: → b H y 0: n , k f ( x ) = E ( f ( b X k ) g k ( x , y k − 1 , b X k , y k ) | b H y 0: n , k f ( x ) − X k − 1 = x ) Forward quantized dynamics (I): π y , k − 1 b π y , k f = b b H y , k f , k = 1 , . . . , n . Forward quantized dynamics (II): b π y , n f Π y ( dx ) = b b Π y , n f = π y 0: n , n 1 (finitely supported unnormalized filter satisfies formally the same recursions) Weight computation: If b X n = b X Γ n n , Γ n = { x 1 1 , . . . , x n N n } then N n X ` ´ b Π i b with b Π i y , n = b Π y , n ( dx ) = y , n δ x n Π y , n . 1 C i (Γ n ) i i =1 Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 44 / 81
Application to BSDE Distortion mismatch From Lip to θ -Liploc assumptions Standard H Lip assumption for the conditional densities g k ( ., y , ., y ′ ): bounded by K g and Lipschitz continuity. ` ´ x | + | x ′ − b | g k ( x , y , x ′ , y ′ ) − g k ( b x ′ , y ′ ) | [ g k ] Lip ( y , y ′ ) x ′ | x , y , b | x − b ≤ . The kernels P k ( x , d ξ ) = P ( X k ∈ d ξ | X k − 1 = x ) propagate Lipschitz continuity with coefficient [ P k ] Lip s such that k =1 ,..., n [ P k ] Lip < + ∞ max Aim: Switch to a θ -local Lipschitz assumption ( θ : R d → R + , ↑ + ∞ as | x | ↑ + ∞ ). ` ´` ´ x | + | x ′ − b | h ( x , x ′ ) − h (ˆ x ′ ) | ≤ [ h ] loc x ′ | 1 + θ ( x ) + θ ( x ′ ) + θ (ˆ x ′ ) | x − b x , ˆ x ) + θ (ˆ New ( H θ Liploc ) assumption: the functions g k are still bounded by K g and θ -local Lipschitz continuous ` ´` ´ | g k ( x , y , x ′ , y ′ ) − g k ( b x ′ , y ′ ) | ≤ [ g k ] loc ( y , y ′ ) x | + | x ′ − b x ′ | 1+ θ ( x )+ θ ( x ′ )+ θ (ˆ x ′ ) x , y , b | x − b x )+ θ (ˆ The kernels P k ( x , d ξ ) = P ( X k ∈ d ξ | X k − 1 = x ) propagate θ -local Lipschitz continuity with coefficient [ P k ] loc < + ∞ . ` ´ The kernels P k ( x , d ξ ) propagate θ -control: max 0 ≤ k ≤ n − 1 P k ( θ )( x ) ≤ C 1 + θ ( x ) . Typical example: X k = ¯ X n k (Euler scheme with step ∆ n = T n ), θ ( ξ ) = | ξ | α , α > 0. t n Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 45 / 81
Application to BSDE Distortion mismatch Theorem (Sagna-P., SPA ’17) Let s ∈ (1 , 1 + d 2 ) and θ ( x ) = | x | α , α ∈ (0 , 1 d ) . s − 1 − 2 1 Assume ( X k ) and ( g k ) satisfy ( H θ Liploc ) (in particular ( X k ) propagates θ -Lipschitz 2 ds d +2 − 2 s , k = 0 , . . . , n. Then continuity) and assume X k ∈ L n 2( K n g ) 2 X Π y , n f | 2 ≤ | Π y , n f − b B n � X k − b X k � 2 k ( f , y ) × (2) 2 s n ( y ) ∨ b | {z } φ 2 φ 2 n ( y ) k =0 − 2 ≍� X k − b X k � 2 d 2 ≤ c k N (Mismatch!!) k with b φ n ( y ) = b φ n ( y ) = π y , n 1 and π y , n 1 , B n k ( f , y ) := 2[ P ] 2( n − k ) [ f ] 2 loc + 2 � f � 2 ∞ R n , k + � f � ∞ R 2 n , k , loc where h “ n − k ” 2 i s X s − 1 M n R n , k = 8 s [ g k +1 ] 2 loc + [ g k ] 2 [ P ] m − 1 loc + loc (1 + [ P ] loc )[ g k + m ] loc , K 2 g m =1 and ` s − 1 ´ ` s − 1 ´ 2 s 2 s θ ( b M n s := 2 max k =0 ,..., n ( E θ ( X k ) + E X k ) . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 46 / 81
Application to BSDE Distortion mismatch Numerical illustrations (3) Risk-neutral price under historical probability (B&S model, Euler scheme) “ ” rY t + µ − r dY t = Z dt + Z t dW t σ with Y T = h ( X T ) = ( X T − K ) + . ⊲ Model parameters: r = 0 . 1; T = 0 . 1; σ = 0 . 25; S 0 = K = 100. ⊲ Quantization tree calibration: 7 . 5 10 5 MC and NbLloyd = 1. ⊲ Reference call BS ( K , T ) = 3 . 66, Z 0 = 14 . 148. If µ ∈ { 0 . 05 , 0 . 1 , 0 . 15 , 0 . 2 } , n = 10 and N k = ¯ N = 20 : Q -price = 3 . 65, b Z 0 = 14 . 06. n = 10 and N k = ¯ N = 40, Q -price = 3 . 66, b Z 0 = 14 . 08. ⊲ Computation time : – 5 seconds for one contract. – Additional contracts for free (more than 10 5 / s ). ⊲ Romberg extrapolation price = 2 ∗ Q -price( N 2 )- Q -price( N 1 ) does improve the price (and the “hedge”). Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 47 / 81
Application to BSDE Distortion mismatch Numerical illustrations Bid-ask spreads on interest rates : „ ”« “ rY t + µ − r Y t − Z t dY t = Z t + ( R − r ) min σ , 0 dt + Z t dW t σ with Y T = h ( X T ) = ( X T − K 1 ) + − 2( X T − K 2 ) + , K 1 = 95 , K 2 = 105 . µ = 0 . 05 , r = 0 . 01 , σ = 0 . 2 , T = 0 . 25 , R = 0 . 06 ⊲ Reference values: price = 2 . 978, b Z 0 = 0 . 553. ⊲ Crude Quantized prices: n = 10 and N k = ¯ N 1 = 20 : Q -price = 2 . 96, b Z 0 = 0 . 515. n = 10 and N k = ¯ N 2 = 40, Q -price = 2 . 97, b Z 0 = 0 . 531. ⊲ Romberg extrapolated price = 2 ∗ Q -price(¯ N 2 )- Q -price(¯ N 1 ) ≃ 2 . 98 and Romberg extrapolated hedge b Z 0 ≈ 0 . 547. Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 48 / 81
Application to BSDE Distortion mismatch Multidimensional example (due to J.-F. Chassagneux) ⊲ Let W be a d -dimensional B.M. and let e t = exp( t + W 1 t + . . . + W d t ) . ⊲ Consider the non-linear BSDE e T dX t = dW t , − dY t = f ( t , Y t , Z t ) dt − Z t · dW t , Y T = 1 + e T ` ´ y − 2+ d with f ( t , y , z ) = ( z 1 + . . . + z d ) . 2 d ⊲ Solution: e t e t Y t = 1 + e t , Z t = (1 + e t ) 2 . We set d = 2 , 3 and T = 0 . 5, so that Z i Y 0 = 0 . 5 and 0 = 0 . 24 , i = 1 , . . . , d . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 49 / 81
Application to BSDE Distortion mismatch Figure: Convergence rate of the quantization error for the multidimensional example). Abscissa axis: the size N = 5 , . . . , 100 of the quantization. Ordinate axis: The error | Y 0 − b Y N a / N + ˆ 0 | and the graph N �→ ˆ b , where a and ˆ ˆ b are the regression coefficients. d = 3. Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 50 / 81
Other results Local behaviour of optimal quantizers (back to Benett’s conjecture) Theorem (Local behaviour: toward Benett’s conjecture, Graf-Luschgy-P. AoP, 2012) ( a ) If P X is absolutely continuous on R d then N +1 , p ( X ) ≍ N − ( 1+ p d ) . e p N , p ( X ) − e p Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 51 / 81
Other results Local behaviour of optimal quantizers (back to Benett’s conjecture) Theorem (Local behaviour: toward Benett’s conjecture, Graf-Luschgy-P. AoP, 2012) ( a ) If P X is absolutely continuous on R d then N +1 , p ( X ) ≍ N − ( 1+ p d ) . e p N , p ( X ) − e p ( b ) Upper-bounds : Suppose P X = ϕ.λ d ϕ is essentially bounded with compact support and its support is peakless ` ´ ∀ s ∈ (0 , s 0 ) , ∀ x ∈ supp ( P X ) , P X B ( x , s ) ≥ c λ d ( B ( x , s )) , c > 0 . “ 8 ´ ≤ c 1 C i (Γ ∗ , N ) > max N , x i ∈ Γ ∗ , N P X < Z ∃ c , ¯ c ∈ [1 , ∞ ) s.t. ∀ N ∈ N , � ξ − x i � p d P X ( d ξ ) ≤ ¯ cN − (1+ p > d ) . : max x i ∈ Γ ∗ , N C i (Γ ∗ , N ) Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 51 / 81
Other results Local behaviour of optimal quantizers (back to Benett’s conjecture) Theorem (Local behaviour: toward Benett’s conjecture, Graf-Luschgy-P. AoP, 2012) ( a ) If P X is absolutely continuous on R d then N +1 , p ( X ) ≍ N − ( 1+ p d ) . e p N , p ( X ) − e p ( b ) Upper-bounds : Suppose P X = ϕ.λ d ϕ is essentially bounded with compact support and its support is peakless ` ´ ∀ s ∈ (0 , s 0 ) , ∀ x ∈ supp ( P X ) , P X B ( x , s ) ≥ c λ d ( B ( x , s )) , c > 0 . “ 8 ´ ≤ c 1 C i (Γ ∗ , N ) > max N , x i ∈ Γ ∗ , N P X < Z ∃ c , ¯ c ∈ [1 , ∞ ) s.t. ∀ N ∈ N , � ξ − x i � p d P X ( d ξ ) ≤ ¯ cN − (1+ p > d ) . : max x i ∈ Γ ∗ , N C i (Γ ∗ , N ) Z � ξ − a � p d P ( ξ ) ≥ c N − ( 1+ p d ) . ( c ) Lower bounds ∀ n ∈ N , min a ∈ Γ ∗ , N C a (Γ ∗ , N ) “ p ´ d + p C a (Γ ∗ , N ) ϕ ( a ) , a ∈ Γ ∗ , N , as N → + ∞ . ∼ c x ⊲ Benett’s conjecture (1955): P N ⊲ Various extensions to unbounded r.v., including uniform results for radial decreasing Gilles PAG` distribution (Junglen, 2012). ES (LPMA-UPMC) Quantization 19.07.2017 51 / 81
Other results Quantification quadratique optimale de taille 50 de N (0; 1) 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -4 -3 -2 -1 0 1 2 3 4 o Le quantifieur optimal de taille 50 : x (50) = ( x (50) , . . . , x (50) 50 ), 1 � � X ∈ C i ( x (50) ) —- Les poids : x i �→ P � ( ξ − x (50) ) 2 P X ( dξ ) —- L’inertie locale : x i �→ i C i ( x (50) ) Figure: a �→ P ( X ∈ C ( b X ∗ , N ), X ∼ N (0; 1), N = 50 a Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 52 / 81
Other results Applications Applications to Numerical Probability What are these applications using optimal quantization grids? Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 53 / 81
Other results Applications Applications to Numerical Probability What are these applications using optimal quantization grids? Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 53 / 81
Other results Applications Applications to Numerical Probability What are these applications using optimal quantization grids? Obstacle Problems: Valuation of Bermuda and American options, Reflected BSDE’s (Bally-P.-Printems ’01, ’03 and ’05, Illand ’11). Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 53 / 81
Other results Applications Applications to Numerical Probability What are these applications using optimal quantization grids? Obstacle Problems: Valuation of Bermuda and American options, Reflected BSDE’s (Bally-P.-Printems ’01, ’03 and ’05, Illand ’11). δ -Hedging for American options (ibid. ’05). Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 53 / 81
Other results Applications Applications to Numerical Probability What are these applications using optimal quantization grids? Obstacle Problems: Valuation of Bermuda and American options, Reflected BSDE’s (Bally-P.-Printems ’01, ’03 and ’05, Illand ’11). δ -Hedging for American options (ibid. ’05). Optimal Stochastic Control problems (P.-Pham-Printems 06’), Pricing of Swing options (Bouthemy-Bardou-P.’09). . . on massively parallel architecture (GPU, Bronstein-P.-Wilbertz, ’10), Control of PDMP (Dufour-de Sapporta ’13). Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 53 / 81
Other results Applications Applications to Numerical Probability What are these applications using optimal quantization grids? Obstacle Problems: Valuation of Bermuda and American options, Reflected BSDE’s (Bally-P.-Printems ’01, ’03 and ’05, Illand ’11). δ -Hedging for American options (ibid. ’05). Optimal Stochastic Control problems (P.-Pham-Printems 06’), Pricing of Swing options (Bouthemy-Bardou-P.’09). . . on massively parallel architecture (GPU, Bronstein-P.-Wilbertz, ’10), Control of PDMP (Dufour-de Sapporta ’13). Non-linear filtering and stochastic, volatility models (P.-Pham-Printems ’05, Pham-Sellami-Runggaldier’06, Sellami ’09 &’10, Callegaro-Sagna ’10). Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 53 / 81
Other results Applications Applications to Numerical Probability What are these applications using optimal quantization grids? Obstacle Problems: Valuation of Bermuda and American options, Reflected BSDE’s (Bally-P.-Printems ’01, ’03 and ’05, Illand ’11). δ -Hedging for American options (ibid. ’05). Optimal Stochastic Control problems (P.-Pham-Printems 06’), Pricing of Swing options (Bouthemy-Bardou-P.’09). . . on massively parallel architecture (GPU, Bronstein-P.-Wilbertz, ’10), Control of PDMP (Dufour-de Sapporta ’13). Non-linear filtering and stochastic, volatility models (P.-Pham-Printems ’05, Pham-Sellami-Runggaldier’06, Sellami ’09 &’10, Callegaro-Sagna ’10). Discretization of SPDE’s (stochastic Zaka¨ ı & McKean-Vlasov equations) [Gobet-P.-Pham-Printems ’07]. Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 53 / 81
Other results Applications Applications to Numerical Probability What are these applications using optimal quantization grids? Obstacle Problems: Valuation of Bermuda and American options, Reflected BSDE’s (Bally-P.-Printems ’01, ’03 and ’05, Illand ’11). δ -Hedging for American options (ibid. ’05). Optimal Stochastic Control problems (P.-Pham-Printems 06’), Pricing of Swing options (Bouthemy-Bardou-P.’09). . . on massively parallel architecture (GPU, Bronstein-P.-Wilbertz, ’10), Control of PDMP (Dufour-de Sapporta ’13). Non-linear filtering and stochastic, volatility models (P.-Pham-Printems ’05, Pham-Sellami-Runggaldier’06, Sellami ’09 &’10, Callegaro-Sagna ’10). Discretization of SPDE’s (stochastic Zaka¨ ı & McKean-Vlasov equations) [Gobet-P.-Pham-Printems ’07]. Quantization based Universal Stratification (variance reduction) [Corlay-P. ’10]. Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 53 / 81
Other results Applications Applications to Numerical Probability What are these applications using optimal quantization grids? Obstacle Problems: Valuation of Bermuda and American options, Reflected BSDE’s (Bally-P.-Printems ’01, ’03 and ’05, Illand ’11). δ -Hedging for American options (ibid. ’05). Optimal Stochastic Control problems (P.-Pham-Printems 06’), Pricing of Swing options (Bouthemy-Bardou-P.’09). . . on massively parallel architecture (GPU, Bronstein-P.-Wilbertz, ’10), Control of PDMP (Dufour-de Sapporta ’13). Non-linear filtering and stochastic, volatility models (P.-Pham-Printems ’05, Pham-Sellami-Runggaldier’06, Sellami ’09 &’10, Callegaro-Sagna ’10). Discretization of SPDE’s (stochastic Zaka¨ ı & McKean-Vlasov equations) [Gobet-P.-Pham-Printems ’07]. Quantization based Universal Stratification (variance reduction) [Corlay-P. ’10]. CVaR-based dynamical risk hedging [Bardou-Frikha-P., ’15). Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 53 / 81
Other results Applications Applications to Numerical Probability What are these applications using optimal quantization grids? Obstacle Problems: Valuation of Bermuda and American options, Reflected BSDE’s (Bally-P.-Printems ’01, ’03 and ’05, Illand ’11). δ -Hedging for American options (ibid. ’05). Optimal Stochastic Control problems (P.-Pham-Printems 06’), Pricing of Swing options (Bouthemy-Bardou-P.’09). . . on massively parallel architecture (GPU, Bronstein-P.-Wilbertz, ’10), Control of PDMP (Dufour-de Sapporta ’13). Non-linear filtering and stochastic, volatility models (P.-Pham-Printems ’05, Pham-Sellami-Runggaldier’06, Sellami ’09 &’10, Callegaro-Sagna ’10). Discretization of SPDE’s (stochastic Zaka¨ ı & McKean-Vlasov equations) [Gobet-P.-Pham-Printems ’07]. Quantization based Universal Stratification (variance reduction) [Corlay-P. ’10]. CVaR-based dynamical risk hedging [Bardou-Frikha-P., ’15). Fast Marginal quantization [Sagna-P., 2015] Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 53 / 81
Other results Applications First conclusions on optimal (Voronoi) vector quantization ⊲ Download free pre-computed grids of N (0; I d ) distributions at the URL www.quantize.maths-fi.com for d = 1 , . . . , 10 and N = 1 , . . . , 10 4 and many others items related to optimal quantization. Voronoi quantization is optimal for “Lipschitz approximation” Paradox: it does not preserve regularity Second order (stationarity) : (almost) only optimal grids ⇒ lack of flexibility As for cubature: quantization vs uniformly distributed sequences? ( ξ N ) N ≥ 1 , [0 , 1] d -valued sequences s.t. N X 1 R d δ ξ i = ⇒ λ | [0 , 1] d N i =1 R d vs [0 , 1] d [1 − 0]. 1 Lipschitz continuity vs Hardy & Krause finite variation on [0 , 1] d , [2 − 0]. 2 Sequences of N -tuples vs sequences [2 − 1] ( QMC !). 3 Companion weights vs no weights [2 − 2]. 4 Rates n − 1 d vs log n × n − 1 d ( Stoikov, 1987, price for uniform weights!) [3 − 2]. 5 How to “fix” (3) without affecting (4): Greedy quantization. Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 54 / 81
Greedy quantization What greedy quantization is the name for? ⊲ Switch from a sequence of N -tuples toward a sequence of points ( a N ) N ≥ 1 such that a ( N ) = { a 1 , . . . , a N } produces “good” quantization grids. ∀ N ≥ 1 , Among others, the first questions are: How to proceed theoretically? How “good”? How to compute them? How flexible can they be? Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 55 / 81
Greedy quantization Level-by-level “greedy” optimization Let X ∈ L p R d (Ω , A , P ) be a random vector with distribution P X = µ . ⊲ Optimal greedy quantization: We define by induction a sequence ( a N ) N ≥ 1 recursively by a (0) = ∅ , a N +1 ∈ argmin ξ ∈ R d e p ( a ( N ) ∪ { ξ } , X ) . ∀ N ≥ 0 , ⊲ It is a natural and constructive way to answer the above first question. ⊲ Is it the best one? No answer so far. . . ⊲ Note that a 1 always exists and a 1 is the L p ( P )-median (always unique if p > 1). Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 56 / 81
Greedy quantization Existence of an L p - optimal greedy quantization sequence ` ´ = + ∞ and X ∈ L p ( P )) Proposition (Assume card supp ( µ ) ( a ) Existence: There exists an L p - optimal greedy quantization sequence ( a N ) N ≥ 1 and ` ´ e p ( a ( n ) , X ) 1 ≤ n ≤ N is (strictly) decreasing to 0 (and a 1 is an L p -median). ( b ) Space filling: Let q > p. If X ∈ L q R d ( P ) . Then, any L p -optimal greedy quantization sequence ( a N ) N ≥ 1 satisfies N e q ( a ( N ) , X ) = 0 . lim Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 57 / 81
Greedy quantization is rate optimal Greedy quantization is rate optimal ⊲ Main rate optimality result. Theorem (Rate optimality, Luschgy-P. ’15) Let p ∈ (0 , + ∞ ) , X ∈ L p (Ω , A , P ) and let µ = P X . Let ( a N ) N ≥ 1 be an L p -optimal greedy quantization sequence. ( a ) Let p ′ > p. There exists C p , p ′ , d ∈ (0 , + ∞ ) such that, for every R d -valued X r.v. e p ( a ( N ) , X ) ≤ C p , p ′ , d .σ p ′ ( X ) . N − 1 d . ∀ N ≥ 1 , ( b ) If µ = ϕ ( ξ ) λ d ( d ξ ) = f ( | ξ | 0 ) λ d ( d ξ ) , | . | 0 (any) norm on R d and f = R + → R + , bounded and non-increasing outside a compact, and X lies in L p and R d d + p d λ d ( ξ ) < + ∞ , then R d f ( | ξ | 0 ) 1 d e p ( a ( N ) , X ) < + ∞ . lim sup N N Condition in ( b ) is optimal since, if µ = ϕ.λ d , „Z « ( d + p ) / d R d ϕ d / ( d + p ) d λ d 1 d e p , N ( X ) ≥ e Q p , | . | × lim inf N . N ⊲ Main tool: Still micro-macro inequalities. Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 58 / 81
Greedy quantization is rate optimal Flavour of proof ⊲ First we not that by definition of the sequence ( a N ) N ≥ 1 , N +1 := e p ( a ( N ) , X ) p − e p ( a ( N +1) ) ≥ e p ( a ( N ) , X ) p − e p ( a ( N ) ∪ { y } , X ) p ∀ y ∈ R d , ∆ ( a ) So, we start from the micro-macro inequality (0 < b < 1 2 , fixed parameter). ` ´ e p ( a ( N ) , X ) p − e p ( a ( N ) ∪ { y } , X ) p ≥ C p , b d ( y , a ( N ) ) p µ ∀ y ∈ R d , B ( y , b d ( y , a ( N ) )) . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 59 / 81
Greedy quantization is rate optimal Flavour of proof ⊲ First we not that by definition of the sequence ( a N ) N ≥ 1 , N +1 := e p ( a ( N ) , X ) p − e p ( a ( N +1) ) ≥ e p ( a ( N ) , X ) p − e p ( a ( N ) ∪ { y } , X ) p ∀ y ∈ R d , ∆ ( a ) So, we start from the micro-macro inequality (0 < b < 1 2 , fixed parameter). ` ´ e p ( a ( N ) , X ) p − e p ( a ( N ) ∪ { y } , X ) p ≥ C p , b d ( y , a ( N ) ) p µ ∀ y ∈ R d , B ( y , b d ( y , a ( N ) )) . ⊲ Let µ = P X . Integrating w.r.t. a distribution ν ( dy ): Z Z ∆ ( a ) 1 {| ξ − y |≤ b d ( y , a ( N ) ) } d ( y , a ( N ) ) p ν ( dy ) µ ( d ξ ) ≥ C p , b N +1 Z Z b +1 d ( ξ, a ( N ) ) } d ( y , a ( N ) ) p ν ( dy ) µ ( d ξ ) ≥ C p , b 1 {| ξ − y |≤ b d ( y , a ( N ) ) , d ( y , a ( N ) ) ≥ 1 Z Z C ′ b +1 d ( ξ, a ( N ) ) } d ( ξ, a ( N ) ) p ν ( dy ) µ ( d ξ ) ≥ 1 {| ξ − y |≤ b d ( y , a ( N ) ) , d ( y , a ( N ) ) ≥ p , b 1 Z Z C ′ b +1 d ( ξ, a ( N ) ) } d ( ξ, a ( N ) ) p ν ( dy ) ≥ 1 {| ξ − y |≤ p , b b Z “ “ ”” b ∆ ( a ) C ′ b + 1 d ( ξ, a ( N ) ) d ( ξ, a ( N ) ) p µ ( d ξ ) = ν B ξ ; p , b N +1 still by Fubini’s theorem. Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 59 / 81
Greedy quantization is rate optimal ⊲ Let b ∈ (0 , 1 b +1 = 1 b 2 ) be such that 4 . κ ν ( dx ) = ( | x − a 1 | + 5 / 4) d + η λ d ( dx ) . Then, if ρ ≤ 1 4 | x − a 1 | , h i ` ´ 1 ≥ ρ d × g ( ξ ) := κ ′ V d ν B ( ξ, ρ ) . ( | ξ − a 1 | + 1) d + η Noting that d ( ξ, a ( N ) ) ≤ d ( x , a 1 ) = | x − a 1 | yields Z e p ( a ( N ) , X ) p − e p ( a ( N +1) ≥ C ′′ d ( ξ, a ( N ) ) p + d g ( ξ ) µ ( ξ ) . p Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 60 / 81
Greedy quantization is rate optimal ⊲ Let b ∈ (0 , 1 b +1 = 1 b 2 ) be such that 4 . κ ν ( dx ) = ( | x − a 1 | + 5 / 4) d + η λ d ( dx ) . Then, if ρ ≤ 1 4 | x − a 1 | , h i ` ´ 1 ≥ ρ d × g ( ξ ) := κ ′ V d ν B ( ξ, ρ ) . ( | ξ − a 1 | + 1) d + η Noting that d ( ξ, a ( N ) ) ≤ d ( x , a 1 ) = | x − a 1 | yields Z e p ( a ( N ) , X ) p − e p ( a ( N +1) ≥ C ′′ d ( ξ, a ( N ) ) p + d g ( ξ ) µ ( ξ ) . p p + d < 1 and − p p ⊲ Inverse Minkowski Inequality implies with d < 0, yields »Z – p + d »Z – − d p p g ( ξ ) − p ∆ ( a ) N +1 ≥ C ′′ d ( ξ, a ( N ) ) p µ ( d ξ ) d µ ( d ξ ) . p | {z } = e p ( a ( N ) , X ) p + d Now Z Z g ( ξ ) − p | ξ − a 1 | p + η d µ ( ξ ) = E | X | p + η d µ ( ξ ) ≍ d < + ∞ so that e p ( a ( N ) , X ) p − e p ( a ( N +1) ) p ≥ C p , X · e p ( a ( N ) , X ) p + d . Gilles PAG` ES (LPMA-UPMC) Quantization 19.07.2017 60 / 81
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