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Machine learning meets super-resolution
H. N. MhaskarClaremont Graduate University, Claremont.
Inverse Problems and Machine LearningFebruary 10, 2018
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Goals
The problem of super-resolution is dual of the problem of machinelearning, viewed as function approximation.
I How to measure the accuracy
I How to ensure lower bounds
I Common tools
Will illustrate on the (hyper-)sphere Sq of Rq+1.
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1. Machine learning
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Machine learning on Sq
Given data (training data) of the form D = {(xj , yj)}Mj=1, wherexj ∈ Sq, yj ∈ R,
find a function x 7→∑N
k=1 akG (x · zk)
I that models the data well;
I in particular,∑N
k=1 akG (xj · zk) ≈ yj .
Tacit assumption: There exists an underlying function f such thatyj = f (xj) + noise.
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ReLU networks
An ReLU network is a function of form
x 7→N∑
k=1
ak |wk · x + bk |.
wk · x + bk (wk , b) · (x, 1)√
(wk |2 + 1)(|x|2 + 1)
Approximation on Euclidean space approximation on sphere
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Notation on the sphere
Sq := {x = (x1, . . . , xq+1) :∑q+1
k=1 x2k = 1},
ωq = Riemannian volume of Sqρ(x, y) = geodesic distance between x and y.Πqn = class of all spherical polynomials of degree at most n.
Hq` = class of all homogeneous harmonic polynomials of degree `,
dq` = the dimension of Hq
` ,{Y`,k} = orthonormal basis for Hq
` .∆ = Negative Laplace-Beltrami operator.∆Y`,k = `(`+ q − 1)Y`,k = λ2
`Y`,k .
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Notation on the sphere
With p` = p(q/2−1,q/2−1)` (Jacobi polynomial),
d q∑k=1
Y`,k(x)Y`,k(y) = ω−1q−1p`(1)p`(x · y).
If G : [−1, 1]→ R,
G (x · y) =∞∑`=0
G (`)
dq∑k=1
Y`,k(x)Y`,k(y).
For a measure µ on Sq,
µ(`, k) =
∫SqY`,k(y)dµ(y).
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Notation on the sphere
Φn(t) = ω−1q−1
n∑`=0
h
(λ`n
)p`(1)p`(t).
σn(µ)(x) =
∫Sq
Φn(x · y)dµ(y) =n∑`=0
h
(λ`n
) dq∑k=1
µ(`, k)Y`,k(x).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
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Notation on the sphere
Localization(Mh. 2004) If S > q and h is sufficiently smooth,
|Φn(x · y)| ≤ c(h, s)nq
max(1, (nρ(x · y))S)
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Polynomial approximation
(Mh. 2004)En(f ) = min
P∈Πqn
‖f − P‖∞.
Wr = {f ∈ C (Sq) : En(f ) = O(n−r )}.
Theorem TFAE
1. f ∈Wr
2. ‖f − σn(f )‖ = O(n−r )
3. ‖σ2n(f )− σ2n−1(f )‖ = O(2−nr ) (Littlewood-Paley typeexpansion)
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Data-based approximation
For C = {xj} ⊂ Sq, D = {(xj , yj)}Mj=1,
1. Find N and wj ∈ R such that
M∑j=1
wjP(xj) =
∫SqP(x)dx, P ∈ Πq
2N
andM∑j=1
|wjP(xj)| ≤ c
∫Sq|P(x)|dx, P ∈ Πq
2N .
Done by least squares or least residual solutions, to ensure agood condition number.
2.
SN(D)(x) =M∑j=1
wjyjΦN(x · xj)
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Data-based approximation
(Le Gia, Mh., 2008)If {xj}Mj=1 are chosen uniformly from µq, and f ∈Wr , then withhigh probability,
‖f − SN(D)‖∞ - M−r/(2r+q).
If f is locally in Wr , then the results holds locally as well; i.e.,accuracy in approximation adapts itself according to localsmoothness.
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Examples
f (x , y , z) = [0.01− (x2 + y2 + (z − 1)2)]+ + exp(x + y + z)
−14 −12 −10 −8 −6 −4 −20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Percentages of error less than 10x Least square, σ63(h1), σ63(h5).
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Examples
f (x , y , z) = (x − 0.9)3/4+ + (z − 0.9)
3/4+
−11 −10 −9 −8 −7 −6 −5 −4 −3 −20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Percentages of error less than 10x Least square, σ63(h1), σ63(h5).
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Examples
East–west component of earth’s magnetic field
Original data on left (Courtesy Dr. Thorsten Maier),reconstruction with σ46(h7) on right
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ZF networks
Let G (`) v `−β, β > q, Cm a nested sequence of points with
δ(Cm) = maxx∈Sq
minz∈Cm
ρ(x, z) ∼ η(Cm) = minz1 6=z2∈Cm
ρ(z1, z2) ≥ 1/m.
G(Cm) = span{G (◦ · z) : z ∈ Cm}.
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ZF networks
(Mh. 2010)Theorem Let 0 < r < β − q, then f ∈Wr if and only if
dist(f ,G(Cm)) = O(m−r ),
Remark. The theorem gives lower limits for individual functions.
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One problem
xj ’s may not be distributed according to µq; their distribution isunknown.
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Drusen classification
I AMD (Age related Macular Degeneration) is the mostcommon cause of blindness among the elderly in the westernworld.
I AMD ! RPE (Retinal Pigment Epithelium) ! Drusenaccumulation of different kinds
Problem: Automated quantitative prediction of diseaseprogression, based on drusen classification.
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Drusen classification
(Ehler, Filbir, Mh., 2012)We used 24 images (400× 400 pixels each) on each patient, atdifferent frequencies. By preprocessing these images at each pixel,we obtained a data set consisting of 160,000 points on a sphere ina 5 dimensional Euclidean space. We used about 1600 of these astraining set, and classified the drusen in 4 classes.While the current practice is based on spatial appearance, ourmethod is based on multi–spectral information.
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Drusen classification
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2. Super-resolution
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Problem statement
Given observations of the form
L∑m=1
am exp(−ijxm) + noise, |j | ≤ N,
determine L, am’s and xm’s.Hidden periodicities (Lanczos)Direction finding (Krim, Pillai, · · · )Singularity detection (Eckhoff, Gelb, Tadmor, Tanner, Mh.,Prestin, Batenkov, · · · )Parameter estimation (Potts, Tasche, Filbir, Mh., Prestin, · · · )Blind source signal separation (Flandrin, Daubeschies, Wu, Chui,Mh., · · · )
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A simple observation
If ΦN is a highly localized kernel (Mh.-Prestin, 1998), then∑Lm=1 amΦN(x − xm) ≈
∑Lm=1 amδxm .
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A simple observation
Original signal:
f (t) = cos(2πt)+cos(2π(0.96)t)+cos(2π(0.92)t)+cos(2π(0.9)t)+noise
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A simple observation
Original signal:
f (t) = cos(2πt)+cos(2π(0.96)t)+cos(2π(0.92)t)+cos(2π(0.9)t)+noise
Frequencies obtained by our method (Chui, Mh., van der Walt,2015): .
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Super-resolution
Question How large should N be?Answer With η = minj 6=k |xj − xk |, N ≥ cη−1.Super-resolution (Donoho, Candes, Fernandez-Granda) How canwe do this problem with N � η−1?
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Spherical variant
Given
L∑m=1
amY`,k(xm) + noise, k = 1, · · · , dq` , 0 ≤ ` ≤ N,
determine L, am, xm.ObservationWith µ∗ =
∑Lm=1 amδxm ,
µ∗(`, k) =L∑
m=1
amY`,k(xm).
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Super-duper-resolution
Givenµ∗(`, k) + noise, k = 1, · · · , dq
` , ` ≤ N,
determine µ∗.Remark The minimal separation is 0. Any solution based on finiteamount of information is beyond super-resolution.
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Duality
dµN(x) = σN(µ∗)(x)dx =
∫Sq
ΦN(x · y)dµ∗(y)dx.
For f ∈ C (Sq),∫Sqf (x)dµN(x) =
∫SqσN(f )(x)dµ∗(x).
So, ∣∣∣∣∫Sqf (x)d(µN − µ∗)(x)
∣∣∣∣ ≤ |µ∗|TVEN/2(f ).
Thus, µN → µ∗ (weak-*). Also,∫SqP(x)dµN(x) =
∫SqP(x)dµ∗(x), P ∈ Πq
N/2.
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Examples
(Courtesy: D. Batenkov)
Original measure (left), Fourier projection (middle), σ64 (belowleft), thresholded |σ64| (below right).
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Examples
(Courtesy: D. Batenkov)
Original measure (left), Fourier projection (middle), σ64 (below).
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Examples
(Courtesy: D. Batenkov)
Original measure (left), Fourier projection (middle), σ64 (below).
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3. Distance between measures
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Erdos-Turan discrepancy
Erdos, Turan, 1940 If ν is a signed measure on T,
(∗) D[ν] = sup[a,b]⊂T
|ν([a, b])|.
Analogues of (*) hard for manifolds, even sphere.Equivalently, if
G (x) =∑
k∈Z\{0}
e ikx
ik
(∗∗) D[ν] = supx∈T
∣∣∣∣∫TG (x − y)dν(y)
∣∣∣∣Generalization to multivariate case: Dick, Pillisheimer, 2010.
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Wasserstein metric
supf
{∣∣∣∣∫Sqfdν
∣∣∣∣ : maxx,y∈Sq
|f (x)− f (y)| ≤ 1
}.
Replace maxx,y∈Sq |f (x)− f (y)| ≤ 1 by ‖∆(f )‖ ≤ 1.Equivalent metric: ∥∥∥∥∫
SqG (◦ · y)dν(y)
∥∥∥∥1
,
where G is Green kernel for ∆.
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Measuring weak-* convergence
Let G : [−1, 1]→ R, G (`) > 0 for all `, G (`) v `−β, β > q.
DG [ν] =
∥∥∥∥∫SqG (◦ · y)dν(y)
∥∥∥∥1
.
TheoremDG [µN − µ∗] ≤ cN−β|µ∗|TV .
Remark The approximating measure is constructed from O(Nq)pieces of information µ∗(`, k). In terms of the amount ofinformation, M, the rate is O(M−β/q).
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Widths
Let M= set of all Borel measures on Sq having bounded variation,
K = {ν ∈M : |ν|TV ≤ 1}.
S = {S : K → RM ,weak-* continuous},
For A : RM →M, S ∈ S,
ErrM(A, S) = supµ∈K
DG [A(S(µ))− µ].
(width)dM(K) = inf
A,SErrM(A,S) ≥ cM−β/q.
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Under the hood
(Mh. 2010)∥∥∥∥G (◦, y)−∫SqG (z, y)ΦN(◦ · z)dz
∥∥∥∥1
≤ cN−β.
For function approximaton:
σN(f ) Estimate on dist(f ,G(Cm)).
For super-duper-resolution: Estimate on DG [µN − µ∗].
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Under the hood
(Mh. 2010) If F (x) =∑L
k=1 akG (x · zk),
η = min1≤k 6=j≤L
ρ(zk , z`),
thenL∑
k=1
|ak | ≤ cη−β‖F‖1.
For function approximation: Converse theorem for ZFapproximation.For super-duper-resolution: Estimate on the widths.
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Thank you.