Extrinsic Geometrical Methods for Neural Blind Deconvolution Simone Fiori [email protected]

Dipartimento di Elettronica, Intelligenza Artificiale e Telecomunicazioni Universit`a Politecnica delle Marche Ancona (Italy, EU)

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.1/34

Talk overview Blind deconvolution (BD): Signal model, basic assumptions, Bayesian (‘Bussgang’-type) BD.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.2/34

Talk overview Blind deconvolution (BD): Signal model, basic assumptions, Bayesian (‘Bussgang’-type) BD. Automatic gain control (AGC): Source of geometrical structure of the parameter space.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.2/34

Talk overview Blind deconvolution (BD): Signal model, basic assumptions, Bayesian (‘Bussgang’-type) BD. Automatic gain control (AGC): Source of geometrical structure of the parameter space. Algorithms: Geodesic-based and projection-based.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.2/34

Talk overview Blind deconvolution (BD): Signal model, basic assumptions, Bayesian (‘Bussgang’-type) BD. Automatic gain control (AGC): Source of geometrical structure of the parameter space. Algorithms: Geodesic-based and projection-based. Numerical experiments and comparison.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.2/34

BD: Channel Model Channel output signal model: T

xn = h sn + νn , def

sn = [sn sn−1 sn−2 . . . sn−Lh +1 ]T is the system’s input vector-stream at time n = 1, . . . , N,

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.3/34

BD: Channel Model Channel output signal model: T

xn = h sn + νn , def

sn = [sn sn−1 sn−2 . . . sn−Lh +1 ]T is the system’s input vector-stream at time n = 1, . . . , N, sn denotes the sampled source signal,

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.3/34

BD: Channel Model Channel output signal model: T

xn = h sn + νn , def

sn = [sn sn−1 sn−2 . . . sn−Lh +1 ]T is the system’s input vector-stream at time n = 1, . . . , N, sn denotes the sampled source signal, νn represents a zero-mean white measurement disturbance independent of the source signal,

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.3/34

BD: Channel Model Channel output signal model: T

xn = h sn + νn , def

sn = [sn sn−1 sn−2 . . . sn−Lh +1 ]T is the system’s input vector-stream at time n = 1, . . . , N, sn denotes the sampled source signal, νn represents a zero-mean white measurement disturbance independent of the source signal, Lh denotes the length of system impulse response h.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.3/34

BD: Filter model FIR filter output signal mode: zm,n = wTm xn , wm = [w0 w2 w3 . . . wLw −1 ]T denotes the filter’s impulse response at learning-iteration m = 1, . . . , M,

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.4/34

BD: Filter model FIR filter output signal mode: zm,n = wTm xn , wm = [w0 w2 w3 . . . wLw −1 ]T denotes the filter’s impulse response at learning-iteration m = 1, . . . , M, def

xn = [xn xn−1 xn−2 . . . xn−Lw +1 ]T denotes the filter input samples at time n = 1, . . . , N,

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.4/34

BD: Filter model FIR filter output signal mode: zm,n = wTm xn , wm = [w0 w2 w3 . . . wLw −1 ]T denotes the filter’s impulse response at learning-iteration m = 1, . . . , M, def

xn = [xn xn−1 xn−2 . . . xn−Lw +1 ]T denotes the filter input samples at time n = 1, . . . , N, Lw denotes the length of the filter impulse response.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.4/34

BD: Channel-Filter Model Channel-filter cascade output model: zm,n = cm sn−δm + Nm,n , where: cm denotes instantaneous amplitude distortion,

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.5/34

BD: Channel-Filter Model Channel-filter cascade output model: zm,n = cm sn−δm + Nm,n , where: cm denotes instantaneous amplitude distortion, δm instantaneous group delay,

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.5/34

BD: Channel-Filter Model Channel-filter cascade output model: zm,n = cm sn−δm + Nm,n , where: cm denotes instantaneous amplitude distortion, δm instantaneous group delay, N m,n denotes so-termed deconvolution noise:

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.5/34

BD: Channel-Filter Model Channel-filter cascade output model: zm,n = cm sn−δm + Nm,n , where: cm denotes instantaneous amplitude distortion, δm instantaneous group delay, N m,n denotes so-termed deconvolution noise: zero-mean, white, Gaussian random process,

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.5/34

BD: Channel-Filter Model Channel-filter cascade output model: zm,n = cm sn−δm + Nm,n , where: cm denotes instantaneous amplitude distortion, δm instantaneous group delay, N m,n denotes so-termed deconvolution noise: zero-mean, white, Gaussian random process, incorrelated with the source signal. Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.5/34

BD: Basic Hypotheses Channel’s impulse response satisfies hT h = 1 and its inverse has finite energy.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.6/34

BD: Basic Hypotheses Channel’s impulse response satisfies hT h = 1 and its inverse has finite energy. Channel is time-invariant or slowly time-varying.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.6/34

BD: Basic Hypotheses Channel’s impulse response satisfies hT h = 1 and its inverse has finite energy. Channel is time-invariant or slowly time-varying. Source stream sn is a stationary, ergodic, independent identically distributed (IID) random process with mean IE s [sn ] = 0 and variance IE s [s2n ] = 1.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.6/34

BD: Basic Hypotheses Channel’s impulse response satisfies hT h = 1 and its inverse has finite energy. Channel is time-invariant or slowly time-varying. Source stream sn is a stationary, ergodic, independent identically distributed (IID) random process with mean IE s [sn ] = 0 and variance IE s [s2n ] = 1. The probability density function p s(s) of the source signal is symmetric around zero and non-Gaussian.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.6/34

BD: Applications Equalization of communication channels.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.7/34

BD: Applications Equalization of communication channels. Optomagnetic memory-support storage and retrieval enhancement.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.7/34

BD: Applications Equalization of communication channels. Optomagnetic memory-support storage and retrieval enhancement. Image deblurring.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.7/34

BD: Applications Equalization of communication channels. Optomagnetic memory-support storage and retrieval enhancement. Image deblurring. Geophysical measurements analysis.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.7/34

BD: Bussgang filtering The model reveals that the relationship between zm,n and cm sn−δm is deterministic but for the deconvolution noise.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.8/34

BD: Bussgang filtering The model reveals that the relationship between zm,n and cm sn−δm is deterministic but for the deconvolution noise.

⇓

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.8/34

BD: Bussgang filtering The model reveals that the relationship between zm,n and cm sn−δm is deterministic but for the deconvolution noise.

⇓

An estimator of the source sequence having form B(zm,n ) can be designed according to Bayesian estimation theory.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.8/34

BD: Filter Structure -

Delay

-

Delay

···

-

-

Delay

xn ··· w0,m

j z

? P  9

wLw −1,m

zm,n ?

ˆ B(z) cm sˆn−δm ?

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.9/34

A ‘Virtuous Cycle’ -

Inverse filter updating

Works better

Works ner

Source stream estimation



Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.10/34

BD as Optimization Problem On the basis of the available Bayesian estimator, the error criterion may be minimized: def 1

h i
1 2 2 C(wm ) = IENm,n [Nm,n ] = IEzm,n zm,n − B(zm,n ) . 2 2

Thanks to ergodicity, the ensemble average IE[·] is estimated by: N X 1 Φ(zm,n ) , IEzm,n [Φ(zm,n )] ≈ N n=1

for any vector-valued function Φ : R → R p . Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.11/34

BD: Automatic Gain Control For practical reasons, it is customary to set the energy-contraint: w20 + w22 + w23 + · · · + w2Lw −1 = 1 . Namely, the filter’s impulse response should belong – at any time – to the unit hyper-sphere: S

p−1 def

= {v ∈ R p |vT v = 1} .

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.12/34

Geometry of S

p−1

At every point v ∈ S p−1 , the tangent space has structure: TvS

p−1 def

= {u ∈ R p |uT v = 0} .

If S p−1 ֒→ R p , which is equipped with the standard Euclidean metric, then the normal space has structure: p−1 def Nv S = {λv|λ ∈ R} .

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.13/34

Riemannian Gradient on S

p−1

Riemannian gradient of a smooth function f : S p−1 → R S p−1 is a vector ∇v f that satisfies: Tangency:

S p−1 ∇v

f ∈ T v S p−1 ,

Compatibility: h∇Sv With the above setting: S p−1 ∇v

p−1

f, uiv =

 ∂ f T ∂v

u, ∀u ∈ T v S p−1 .

∂f f = (I p − vv ) . ∂v T

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.14/34

Geodesics on S

p−1

A geodesic v(t) = G(t, v0 , g) is a curve on which a particle, amanating from v0 with velocity g, slides with constant scalar speed k gk.  p−1  ¨ v ∈ N S ,  v    p−1 v(0) = v ∈ S ,  0     v˙ (0) = g ∈ T v S p−1 . 0 The solution is:

g G(t, v0 , g) = cos(k gkt)v0 + sin(k gkt) . k gk Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.15/34

S p−1 ∇v -based

Optimization

As an optimization law for searching for the minimum (or local minima) of a regular function f : S p−1 → R over S p−1 , we may use the Riemannian-gradient based rule: ( dv S p−1 f, dt = −∇v v(0) = v0 ∈ S p−1 .

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.16/34

BD: Geodesic-based Rule The general-purpose differential equation may be customized as: dw T ∂C(w) = −(I p − ww ) , dt ∂w with p = Lw and:  ∂C(w)    ∂w = IE x [γ(z)x] ,    γ(z)def = (B(z) − z)(B′ (z) − 1) .

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.17/34

BD: Geodesic-based Algorithm It is suggested to approximate the exact flow of the differential equation on a manifold via piece-wise geodesic arcs:   p−1 wm = G tm , wm−1 , −∇Swm−1 C(w) , m ∈ {1, . . . , M} . where: tm denotes an appropriate sequence of adaptation stepsizes,

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.18/34

BD: Geodesic-based Algorithm It is suggested to approximate the exact flow of the differential equation on a manifold via piece-wise geodesic arcs:   p−1 wm = G tm , wm−1 , −∇Swm−1 C(w) , m ∈ {1, . . . , M} . where: tm denotes an appropriate sequence of adaptation stepsizes, w0 ∈ S p−1 is arbitrarily selected.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.18/34

BD: Geodesic-based Algorithm It is suggested to approximate the exact flow of the differential equation on a manifold via piece-wise geodesic arcs:   p−1 wm = G tm , wm−1 , −∇Swm−1 C(w) , m ∈ {1, . . . , M} . where: tm denotes an appropriate sequence of adaptation stepsizes, w0 ∈ S p−1 is arbitrarily selected.

Up to numerical error, wm ∈ S p−1 at every iteration. Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.18/34

BD: Projection-based algorithm By the embedding S p−1 ֒→ R p , updates along the Euclidean gradient direction: ! ∂C(w) def v . wm = Π wm−1 − tm , Π (v) = √ ∂w w=wm−1 vT v where: tm denotes an appropriate sequence of adaptation stepsizes,

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.19/34

BD: Projection-based algorithm By the embedding S p−1 ֒→ R p , updates along the Euclidean gradient direction: ! ∂C(w) def v . wm = Π wm−1 − tm , Π (v) = √ ∂w w=wm−1 vT v where: tm denotes an appropriate sequence of adaptation stepsizes, w0 ∈ S p−1 is arbitrarily selected,

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.19/34

BD: Projection-based algorithm By the embedding S p−1 ֒→ R p , updates along the Euclidean gradient direction: ! ∂C(w) def v . wm = Π wm−1 − tm , Π (v) = √ ∂w w=wm−1 vT v where: tm denotes an appropriate sequence of adaptation stepsizes, w0 ∈ S p−1 is arbitrarily selected,

Π : R p → S p−1 is the selected back-projector. Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.19/34

Short Geodesic Arcs If the time to within the geodesic is extended is short enough, the geodesic-based algorithm traces the Riemannian-gradient flow. In fact, for t small enough, the S p−1 -geodesic may be approximated as: ! 2 2 k gk t G(t, v0 , g) ≈ 1 − v0 + gt , 2 which gives rise to the expression: wm − wm−1 ≈− t

S p−1 k∇wm−1 C(w)k2 t

2

wm−1 −

S p−1 ∇wm−1 C(w)

.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.20/34

Pre-whitening The source stream sn is IID. After passing through the channel, the samples gain second-order statistical correlation. Second-order correlation is easy to remove by data pre-whitening. Let us define: def

R xx = IE xn [xn xTn ] . Whitened filter-input vector-stream: − 12 xˆ n = R xx xn def

.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.21/34

BD Algorithms at a Glance Collect the filter-input stream and build-up the multivariate stream xn .

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.22/34

BD Algorithms at a Glance Collect the filter-input stream and build-up the multivariate stream xn . Whiten the multivariate signal xn .

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.22/34

BD Algorithms at a Glance Collect the filter-input stream and build-up the multivariate stream xn . Whiten the multivariate signal xn . Choose a starting point for the inverse filter impulse response w0 and learning parameters.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.22/34

BD Algorithms at a Glance Collect the filter-input stream and build-up the multivariate stream xn . Whiten the multivariate signal xn . Choose a starting point for the inverse filter impulse response w0 and learning parameters. Compute the final inverse filter impulse response w M by the geodesic-based algorithm or the projection-based algorithm applied to the whitened input stream.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.22/34

Figures of Performance Residual inter-symbol interference (ISI): def

ISIm =

2 TmT Tm − T m,max 2 T m,max

,

def

where Tm = h ⊗ wm and T m,max denotes the component of Tm having maximal absolute value.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.23/34

Figures of Performance Residual inter-symbol interference (ISI): def

ISIm =

2 TmT Tm − T m,max 2 T m,max

,

def

where Tm = h ⊗ wm and T m,max denotes the component of Tm having maximal absolute value. Elapsed run-time on a 1.86GHz – 512MB platform.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.23/34

Figures of Performance Residual inter-symbol interference (ISI): def

ISIm =

2 TmT Tm − T m,max 2 T m,max

,

def

where Tm = h ⊗ wm and T m,max denotes the component of Tm having maximal absolute value. Elapsed run-time on a 1.86GHz – 512MB platform. c Flops (counted by Matlab 5.3).

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.23/34

Source and Bayesian Estimator It is assumed that sn is a white random signal, √ √ uniformly distributed within [− 3, + 3], counting N = 5, 000 samples. In this case, a suitable Bayesian estimator is ˆ = κ tanh(λz). B(z) Parameters κ and λ may be pre-learnt on the basis, e.g., of the procedure introduced in S. F IORI, Analysis of modified ‘Bussgang’ algorithms (MBA) for channel equalization, IEEE Trans. on Circuits and Systems - Part I, Vol. 51, No. 8, pp. 1552 – 1560, August 2004.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.24/34

Experiments on a Toy Channel The channel’s impulse response is h = [1] (Lh = 1) and the base manifold is S 2 (Lw = 3). In this experiment, the channel-filter-cascade impulse response Tm = h ⊗ wm = wm . If we let the learning trajectories depart from randomly generated w0 ∈ S 2 , they should eventually converge to one of the six attractors [±1 0 0]T , [0 ± 1 0]T or [0 0 ± 1]T .

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.25/34

Toy Channel – Geodesic Numerical results on 100 independent trials, M = 100 learning iterations per trial, learning stepsize 0.5.

Neuron weight w

3,m

1

0.5

0

−0.5

−1 1 0.5

1 0.5

0 0

−0.5 Neuron weight w

2,m

−0.5 −1

−1 Neuron weight w

1,m

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.26/34

Toy Channel – Projection Numerical results on 100 independent trials, M = 100 learning iterations per trial, learning stepsize 0.9.

Neuron weight w

3,m

1

0.5

0

−0.5

−1 1 0.5

1 0.5

0 0

−0.5 Neuron weight w

2,m

−0.5 −1

−1 Neuron weight w

1,m

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.27/34

Experiments on Telephonic Channel Sampled telephonic channel having duration Lh = 14. Zeros of BGR channel

Channel amplitude frequency response 2

1.5 1

|H(e )| (dB)

0.5

0

jω

0 −0.5 −1

−1 −2

−1.5 −2

−1

0 Real Part

1

−3 −4

2

Approximate inverse BGR channel

−2

0 ω

2

4

Channel phase frequency response

1

10 0

arg[H(e )] (rad)

0.5

−10

jω

Imaginary Part

1

0

−20 −30

−0.5

0

10

20

30

−40 −4

−2

0 ω

2

4

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.28/34

Experiments on BGR: Data Filter of length Lw = 14.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.29/34

Experiments on BGR: Data Filter of length Lw = 14. w0 = [0 0 0 0 0 0 1 0 0 0 0 0 0 0]T .

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.29/34

Experiments on BGR: Data Filter of length Lw = 14. w0 = [0 0 0 0 0 0 1 0 0 0 0 0 0 0]T . Noiseless channel (i.e., with νn ≡ 0 identically).

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.29/34

Experiments on BGR: Data Filter of length Lw = 14. w0 = [0 0 0 0 0 0 1 0 0 0 0 0 0 0]T . Noiseless channel (i.e., with νn ≡ 0 identically). Learning stepsize: 1 for the geodesic-based algorithm and 0.9 for the projection-based algorithm.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.29/34

Experiments on BGR: Data Filter of length Lw = 14. w0 = [0 0 0 0 0 0 1 0 0 0 0 0 0 0]T . Noiseless channel (i.e., with νn ≡ 0 identically). Learning stepsize: 1 for the geodesic-based algorithm and 0.9 for the projection-based algorithm. Learning iterations: M = 80 for both algorithms.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.29/34

Experiments on BGR: Results 0

−12

Cost function C(wm)

m

ISI (dB)

−5 −10 −15 −20

−12.5

−13

−13.5

−25 0

20

40 Iterations

60

−14

80

1

1.2

0.8

1

Convolution TM=h*wM

Filter impulse response wM

−30

0.6 0.4 0.2 0 −0.2 −0.4

0

20

40 Iterations

60

80

0.8 0.6 0.4 0.2 0

1

2

3

4 5 6 7 8 9 10 11 12 13 Discrete−time index n

−0.2

0

5

10 15 20 Discrete−time index n

25

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.30/34

Numerical Complexity Comparison Algorithms were run on the same batch of 5, 000 channel output samples.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.31/34

Numerical Complexity Comparison Algorithms were run on the same batch of 5, 000 channel output samples. learning iterations: M = 50.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.31/34

Numerical Complexity Comparison Algorithms were run on the same batch of 5, 000 channel output samples. learning iterations: M = 50. The flops count refers to the number of floating point operations required by the implemented code to run, averaged over the total number of samples passing by (5, 000 × 50).

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.31/34

Numerical Complexity Comparison Algorithms were run on the same batch of 5, 000 channel output samples. learning iterations: M = 50. The flops count refers to the number of floating point operations required by the implemented code to run, averaged over the total number of samples passing by (5, 000 × 50). The time count refers to the total time required by each algorithm to run on the specified platform.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.31/34

Complexity Comparison: Results Results of computational-complexity comparison of the geodesic-based algorithm and the projection-based algorithm.

A LGORITHM ISI (dB) Flops Time (sec.s) Geodesic-based −25.057 80.594 0.328 Projection-based −25.056 81.582 0.313

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.32/34

Summary Both algorithms are well-behaving.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.33/34

Summary Both algorithms are well-behaving. The deconvolution performances are comparable for the two algorithms.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.33/34

Summary Both algorithms are well-behaving. The deconvolution performances are comparable for the two algorithms. The geodesic-based algorithm may exhibit steadier convergence.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.33/34

Summary Both algorithms are well-behaving. The deconvolution performances are comparable for the two algorithms. The geodesic-based algorithm may exhibit steadier convergence. The projection-based algorithm may be slightly lighter from a computational point of view.

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.33/34

Many thanks to... The organizers and E.T. Jaynes Foundation!

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.34/34

Many thanks to... The organizers and E.T. Jaynes Foundation! Everybody for the kind attention!

Extrinsic Geometrical Methods for Neural Blind Deconvolution – p.34/34

Many thanks to... The organizers and E.T. Jaynes Foundation! Everybody for the kind attention! The Italian team for winning the World Cup!!!!

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