Deconvolution for the Wasserstein metric and topological inference

28/08/2013
Auteurs : Bertrand Michel
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Deconvolution for the Wasserstein metric and topological inference

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Deconvolution for the Wasserstein metric and topological inference C. Caillerie, F. Chazal, J. Dedecker et B. Michel Geometric Science of Information 2013 B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 1 / 37 Outline 1 Introduction : distance function to a measure 2 Measure deconvolution 3 Rates of convergence 4 Simulations B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 2 / 37 Outline 1 Introduction : distance function to a measure 2 Measure deconvolution 3 Rates of convergence 4 Simulations B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 3 / 37 Topological and geometric inference Aim : Infer the topology of an unknown object G using data sampled on G or close to G : Method : find an approximating object ˆG « close to » G such that the topological properties of ˆG et G are similar. B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 4 / 37 Homotopy Two continuous functions from one topological space to another are called homotopic if one can be “continuously deformed” into the other. Two spaces X and Y are homotopy equivalent (homotopy type) if there exist continuous maps f : X → Y et g : Y → X such that g ◦ f is homotopic to idX ; f ◦ g is homotopic to idY ; Two curves with the same homotopy type : Many concepts are homotopy invariant. B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 5 / 37 First step : topo. inference with a “deterministic approach" 1 a distance function : G compact, dG (x) = infy∈G d(x, y). 2 approximating sets : offsets Gr of G : d−1 G ([0, r]). 3 a distance between objects : Hausdorff distance : dH. Theorem : Chazal Leutier [CL07] G G compact in Rn s.t. dH(G, G ) < ε, wfz(G) > 2ε and wfz(G ) > 2ε. Then, for 0 < α ≤ 2ε, Gα and G α have the same homotopy type. In practice : find the topology of G using the offsets G r of a point cloud G sampled on G. B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 6 / 37 2d step : replacing compact sets by measures [CCSM11] What if data come with outliers ? Hausdorff distance is no longer relevant to formalize the notion of approximation. Replacing compact sets by measures : For a point x → Dirac distribution δx ; For point cloud → empirical measure a manifold M of dimension k → µM(B) = vol(B∩M) vol(M) B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 7 / 37 (preliminary) distance function to a probability measure [CCSM11] Definition Let µ be a probability measure on Rd and let 0 ≤ m ≤ 1, then δµ,m(x) := inf{r > 0 : µ(B(x, r) > m} µ G x δmu,m(x) m δµ,m is the minimum radius r such that B(x, r) supports at least a fraction m of the measure µ. B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 8 / 37 Distance function to a probability measure [CCSM11] In order to gain regularity : Definition Let µ be a positive probability measure on Rd and let 0 ≤ m0 ≤ 1, the distance function to µ with parameter m0 is defined by : d2 µ,m0 (x) = 1 m0 m0 0 δ2 µ,m(x)dm Example : let C = {p1, . . . , pn} and µ = 1 n n i=1 δpi . Take m0 = k0/n : d2 µ,m0 (x) = 1 k0 k0 k=1 ||x − pk C (x)||2 where pk C (x) is the k nearest closest neighbor of x in C. p1 p2 Pk x B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 9 / 37 Wasserstein distances µ and ν two probability measures on Rd , a transport plan between µ and ν is a probability measure π on Rd × Rd with margins µ et ν. 2-cost of π : C2(π) = Rd ×Rd x − y 2 π(x, y) 1/2 Wasserstein distance W2 between µ and ν : optimal transport plan : W 2 2 (µ, ν) = inf π∈M(µ,ν) x − y 2 π(dx, dy) . Discrete probabilities µ = ci δxi and ν = dj δyj : πi,j ≥ 0 s.t. i πi,j = dj and j πi,j = ci . C2(π) = πij ||xi − yj ||2 W2(µ, ν) = infπ C(π). 2d step : topological inference : measure approach [CCSM11] A “measure approach” of topological inference : 1 distance function to a measure µ : dµ,m0 . 2 approximating objects : levels of dµ,m0 . 3 a distance between measures : Wasserstein distance [CCSM11] : under “some general assumptions”, if ν is a probability distribu- tion close to µ for the Wasserstein metric, then the sublevel sets of dν,m0 provide a topologically correct approximation of the support of µ. If ones knows a measure ν that is close to µ for the Wasserstein metric, the level sets of dν,m0 can be used to infer the topology of the sublevel sets of the distance function to the support of µ. In practice we take for ν the empirical measure. B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 11 / 37 Example : square with outliers Data Sublevel sets of dµ,m Analyze the topology of the sublevels of distance function to the empirical measure µn. B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 12 / 37 But if there is to much noise ... qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q −6 −4 −2 0 2 4 6 −6−4−20246 ... the procedure fails because µn and µ are not close enough. Aim of this work : to propose a better candidate then µn, for the Wasserstein metric, to estimate µ. B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 13 / 37 Outline 1 Introduction : distance function to a measure 2 Measure deconvolution 3 Rates of convergence 4 Simulations B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 14 / 37 Convolution model Observations in Rd : Zi = Xi + εi (Xi )1≤i≤n : i.i.d. distribution µ supported on a compact G ; (εi )1≤i≤n are i.i.d., independent of (Xi )1≤i≤n, with a known law µε ; Note that ( µn is the empirical measure of the observations) W2(µn, µ) −→ W2(µ µε, µ) > 0 ( if µε = δ0), n → ∞. We need a deconvolution estimator of µ. B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 15 / 37 Univariate density deconvolution Assume µ has a density f on R. We have µZ = f µε ←→ µ∗ Z = f ∗ µ∗ ε where ∗ denotes the Fourier transform : f ∗ (t) := exp(itx)f (x)dx = E(itX1). First idea : estimating µ∗ Z using the empirical characteristic function : ˆΨZ (t) = 1 n n i=1 exp(itZi ), and we take ˆf ∗ (t) = ˆΨZ (t)/µ∗ ε(t) and finally by taking the inverse F.t : ˆfnaive(x) = 1 2π exp(−itx)ˆf ∗ (t)dt. Nevertheless : ˆfnaive is not well defined since ˆΨZ (t) is not integrable. B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 16 / 37 Univariate density deconvolution [CH88, SC90] Zi = Xi + εi We estimate the density m of the distribution of the Zi ’s with a kernel estimator with K ∈ L1(R) ∩ L2(R) : ˆm(x) := 1 nh n i=1 K x − Zi h Then the F.t of ˆm exists : ( ˆm)∗ (t) := 1 nh n i=1 exp(itx)K x − Zi h dx = ... = ˆΨZ (t)K∗ (th). and we take ˆΨZ (t)K∗(th)/µ∗ ε(t) as an estimator of f ∗. Using the kernel K(x) = sin(x)/(πx), if µ∗ ε does not vanish we can take ˆf (x) = 1 2π exp(−itx)K∗ (th) 1 n n i=1 exp(itZi )/µ∗ ε(t) B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 17 / 37 Measure deconvolution in Rd Observations in Rd : Zi = Xi + εi (Xi )1≤i≤n : i.i.d. distribution µ supported on a compact G ; (εi )1≤i≤n are i.i.d., symmetric, indep. of (Xi )1≤i≤n, known law µε with moments of order 6 and s.t. µ∗ ε does not vanish on Rd . B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 18 / 37 Measure deconvolution in Rd Observations in Rd : Zi = Xi + εi . ˆfn(x) := (|H|n)−1 n i=1 VH(H−1 (x − Zi )) where VH(x) = 1 (2π)d Rd ei K∗(u) µ∗ ε((H−1)tu) du, and H = Hn is d × d inversible matrix s.t. tr(HtH) → 0, n → ∞. K is a symmetric kernel on Rd with a moment of order 2, its Fourier transform is three times differentiable and has a compact support. Problem : we need a true probability measure to consider the distance function to this probability but ˆfn is not necessary positive. We then consider the probability measure ˜µn of density ˆgn = ˆαn ˆf + n , where ˆf + n = max{ˆfn, 0} and ˆαn = 1/ ˆf + n dx. B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 19 / 37 Outline 1 Introduction : distance function to a measure 2 Measure deconvolution 3 Rates of convergence 4 Simulations B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 20 / 37 Bias - variance decomposition Note that E(ˆfn(x)) = µ KH(x) where KH(x) = |H|−1K(H−1x). By the triangular inequality : E(W 2 2 (˜µn, µ)) ≤ 2W 2 2 (µ KH, µ) biais term + 2E(W 2 2 (˜µn, µ KH)) variance term . Proposition We have W 2 2 (µ KH, µ) ≤ C1tr(Ht H) E(W 2 2 (˜µn, µ KH)) ≤ C2 (1 + x 2 ) Var(ˆfn(x))dx and finally E(W 2 2 (˜µn, µ)) ≤ C tr(Ht H) + (1 + x 2 ) Var(ˆfn(x))dx where C1 depends of K and C2 and C depend on K and µ. B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 21 / 37 Bias term Proof : we have W 2 2 (µ KH, µ) = min{ U − V 2 2 : U ∼ µ, V ∼ µ KH} . Let YH be a random variable of distribution KH, independent of X1 ∼ µ. Since X1 + YH ∼ µ KH, we propose the transport plan π := (X1, X1 + YH) and we get W 2 2 (µ KH, µ) ≤ X1 + YH − X1 2 = YH 2 2 (equal if µ = δ0), where YH 2 2 = Rd (ut Ht Hu)K(u)du ≤ C1tr(Ht H) . B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 22 / 37 Variance term (1) Lemma [Vil08] Theorem 6.15 Let µ and ν be two probability measures on Rd with densities f and g with respect to the Lebesgue measure, then W 2 2 (µ, ν) ≤ 2 Rd x 2 |f (x) − g(x)| dx Using the result, we get W 2 2 (˜µn, µ KH) ≤ 2 x 2 ˆgn(x) − E(ˆfn(x)) dx Then, x 2 ˆgn(x) − E(ˆfn(x)) dx ≤ ˆαn x 2 ˆf + n (x) − E(ˆfn(x)) dx + (1 − ˆαn) x 2 E(ˆfn(x))dx ≤ x 2 ˆfn(x) − E(ˆfn(x)) dx + (1 − ˆαn) x 2 E(ˆfn(x))dx B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 23 / 37 Variance term (2) Since (1 − ˆαn) ≤ (ˆf + n − ˆfn)(x)dx, we find that E(1 − ˆαn) ≤ E ˆf + n (x) − E(ˆfn(x)) dx ≤ E ˆfn(x) − E(ˆfn(x)) dx Note that CH := x 2 E(ˆfn(x))dx = x 2 µ KH(x)dx ≤ C2 because CH → x 2µ(x)dx as n → ∞. We finally get that E(W 2 2 (˜µn, µ KH)) ≤ C2 (1 + x 2 )E ˆfn(x) − E(ˆfn(x)) dx ≤ C2 (1 + x 2 ) Var(ˆfn(x))dx. B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 24 / 37 bidimensional case µε = µ1 ⊗ µ2 Assume that d = 2 and that µε = µ1 ⊗ µ2. ˆfn is defined using the kernel K(x1, x2) = k1(x1)k2(x2) and a diagonal matrix H : ˆfn(x) = 1 n n i=1 1 h1 V1,h1 x1 − Zi,1 h1 1 h2 V2,h2 x2 − Zi,2 h2 , with V1,h1 (x) = 1 2π R eiux k∗ 1 (u) µ∗ 1(u/h1) du , V2,h2 (x) = 1 2π R eiux k∗ 2 (u) µ∗ 2(u/h2) du . B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 25 / 37 Variance term Using Hölder’s inequality : (1 + x2 1 + x2 2 ) Var(ˆfn(x1, x2))dx1dx2 ≤ C √ n E (1 + x6 1 )(1 + x6 2 ) h1h2 V1,h1 x1 − Z1,1 h1 2 V2,h2 x2 − Z1,2 h2 2 dx1dx2 Standard Fourier calculations give that this term is bounded by Cn−1/2 [I(h1, µ1)I(h2, µ2)] where I(h1, µ1) = 1/h1 −1/h1 1 |µ∗ 1(x)|2 + |µ∗ 1 (x)|6 |µ∗ 1(x)|8 + |µ∗ 1 (x)|2 |µ∗ 1(x)|4 + |µ∗ 1 (x)|2 |µ∗ 1(x)|4 dx . B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 26 / 37 Gaussian noise Isotropic Gaussian noise : µ∗ 1(x) = µ∗ 2(x) = e−x2/2. Taking h1 = h2 = h, we find that E(W 2 2 (˜µn, µ)) ≤ C h2 + 1 h5 √ n eh−2 . With h = σ/ ln(n) we find that E(W 2 2 (˜µn, µ)) = O 1 ln(n) . This is the correct rate of convergence for super smooth noise distributions as shown in [DM13]. Same rate of convergence for a Gaussian noise in only one direction. B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 27 / 37 No noise In this case : µ∗ 1 = 1 et µ∗ 2 = 1. Taking h1 = h2 = h, we find that E(W 2 2 (˜µn, µ)) ≤ C h2 + 1 h √ n . With the bandwidth h = 1 n1/6 . we find the upper bound E(W 2 2 (˜µn, µ)) = O 1 n1/3 . In this context, Horowitz and Karandikar [HK94] have found this rate for ˜µn. B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 28 / 37 Outline 1 Introduction : distance function to a measure 2 Measure deconvolution 3 Rates of convergence 4 Simulations B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 29 / 37 Complete procedure 1 Some points Xi are simulated on geometric shape (in R2). 2 Distribution of the noise : Gaussian, Laplace. Zi = Xi + εi with µε = µ1 ⊗ µ2. 3 We compute the values of ˆfn on a grid pi,j . 4 We consider the discrete probability measure : ˜µd = i,j αi,j δpi,j where αi,j = ˆf + n (pi,j ) sumk,l ˆf + n (pi,j ) 5 We plot the sublevels of the distance function dˆµ,m0 and we compare these last with the sublevels of dµn,m0 . B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 30 / 37 Two nested circles B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 31 / 37 Two nested circles B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 32 / 37 3 Disks B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 33 / 37 3 Disks B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 34 / 37 Unidirectionnal noise : empirical measure qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q −6 −4 −2 0 2 4 6 −6−4−20246 B. Michel (UPMC) Wass. deconvolution & Topo Inference GSI 2013 35 / 37 Unidirectionnal noise : deconvolution q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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