Learning to Solve Inverse
Problems in Imaging
Rebecca Willett, University of Chicago
Greg Ongie,

UChicago
Davis Gilton, 

UW-Madison
β
Inverse problems in imaging
• Inpainting
• Deblurring
• Superresolution
• Compressed
Sensing
• MRI
• Radar
Observe: y = Xβ + ε
Goal: Recover β from y
y
Classical approach: Tikhonov regularization (1943)
• Example: deblurring
• Least squares solution:
Wildly	Different	Solutions
Blurred	image	with	noiseBlurred	image
Deblurred the	“naïve”	way Deblurred the	“naïve”	way
ˆβ = (X X) 1
X y<latexit 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Wildly different solutions
Classical approach: Tikhonov regularization (1943)
• Example: deblurring
• Least squares solution:
Wildly	Different	Solutions
Blurred	image	with	noiseBlurred	image
Deblurred the	“naïve”	way Deblurred the	“naïve”	way
• Tikhonov regularization

(aka “ridge regression”)
better conditioned; suppresses noise
ˆβ = arg min
β
y Xβ 2
2 + λ β 2
2
=(X X + λI) 1
X y<latexit 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ˆβ = (X X) 1
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Wildly different solutions
Classical approach: Tikhonov regularization (1943)
• Example: deblurring
• Least squares solution:
Wildly	Different	Solutions
Blurred	image	with	noiseBlurred	image
Deblurred the	“naïve”	way Deblurred the	“naïve”	way
• Tikhonov regularization

(aka “ridge regression”)
better conditioned; suppresses noise
ˆβ = arg min
β
y Xβ 2
2 + λ β 2
2
=(X X + λI) 1
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ˆβ = (X X) 1
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Wildly different solutions
Tikhonov regularization
Geometric models of images
Total variation
Noisy
Patches
Denoised
Patches
Patch
Denoising
Combine to
estimate
denoised
pixel
Patch subspaces and manifolds
(Wavelet) sparsity
Regularization in inverse problems
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sha1_base64="3qRIWc7pIXZR0YKMqzYlnbgFQBU=">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</latexit>
ˆβ = arg min
β
y Xβ 2
2 + r(β)
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Regularization in inverse problems
y ˆβ<latexit 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ˆβ = arg min
β
y Xβ 2
2 + r(β)
<latexit 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Classical: r(β) is a pre-defined
smoothness-promoting regularizer 

(e.g. Tikhinov or ridge estimation)
Geometric: r(β) reflects image geometry 

(e.g. sparsity, patch redundancy, total variation)
Learned: use training data to learn r(β)
Examples in recent literature
• Deep CNN’s for signal recovery
• Compressed sensing with GANs
• Unrolled algorithms for solving inverse problems
• Deep proximal gradient descent nets
• Deep ADMM nets
• Deep half-quadratic splitting
• Deep primal-dual nets
Chen, Yu, Pock, 2015
Sun, Li, Xu, 2016
Chang, Li, Poczos, Kumar, Sankaranarayanan, 2017
Adler and Öktem, 2018
Bora, Jalal, Price, Dimakis, 2017
Mousavi and Baraniuk, 2017
Dong, Loy, He, Tang, 2014
Mardani et al, 2018
Jin, McCann, Froustey, Unser, 2017
Zhang, Zuo, Gu, Zhang, 2017
Ye, Han, Cha, 2018
Classes of methods
Model Agnostic 

(Ignore X)
Decoupled 

(First learn, then reconstruct)
Unrolled Optimization
Neumann Networks 

(this talk!)
X-1~
raw data
(low resolution image)
approximate
high-resolution
image
Train deep CNN
to remove artifacts
reconstruction
blurry/blocky
artifacts due to
re-scaling
bicubic
interpolation
Pictures from: http://webdav.tuebingen.mpg.de/pixel/enhancenet/
Super-resolution with CNNs
Model Agnostic 

(Ignore X)
Classes of methods
Model Agnostic 

(Ignore X)
Decoupled 

(First learn, then reconstruct)
Unrolled Optimization
Neumann Networks 

(this talk!)
GANs for inverse problems
y ˆβ<latexit 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ˆβ = arg min
β
y Xβ 2
2 + r(β)
<latexit 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sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">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</latexit><latexit sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">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</latexit><latexit 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r(β) =
0, β on image manifold
, otherwise<latexit 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“Good” image on manifold
“Bad” image off manifold
Decoupled 

(First learn, then
reconstruct)
GANs for inverse problems
y ˆβ<latexit 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sha1_base64="3qRIWc7pIXZR0YKMqzYlnbgFQBU=">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</latexit>
ˆβ = arg min
β
y Xβ 2
2 + r(β)
<latexit 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sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">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</latexit><latexit sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">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</latexit><latexit 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r(β) =
0, β on image manifold
, otherwise<latexit 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GANs for inverse problems
y ˆβ<latexit 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sha1_base64="3qRIWc7pIXZR0YKMqzYlnbgFQBU=">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</latexit>
ˆβ = arg min
β
y Xβ 2
2 + r(β)
<latexit 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sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">AAAQWXiczVfdr9M2FA/si3UblPG4F7NeJNBC1YSWWx4qMYYGkxhjbBeQ6nLlJG7rXecDx+ltZfLX7a+Y9rqXvW7/wI6d9CNJhwCJaa0S2ef8fOzz6RMv4SyVvd5vZ86+9/4HH3507uPWJ59+dv5C++LnT9I4Ez498mMei2ceSSlnET2STHL6LBGUhB6nT72TbzT/6YKKlMXRz3KV0ElIZhGbMp9IIB23JwobIePH9+5MlDvo2+WTK/zw/gM2m8vWwQGeE4k9KgkaIUzELGTRcTHHL1fXn5khfnnsPnfRV0hcNfNrBwf5cbvT6/bMDzUHTjnoWOXv0fHF83/gIPazkEbS5yRNx46TyIlKOQtomrdwltKE+CdkRscwjEhI04kyGuToClACNI0FPJFEhrq7QpEwTVehB8iQyHla52niPt44k9PhRLEoySSN/GKjacaRjJG2KQqYoL7kKxgQXzDJfOTPiSC+BMtXd1mQSIIaKQVPRTM5x5Iu5SkL5Fzd7A5YVFFReSHMAzrFcqowoKhkPKBqmpdkT87B0gp7ocJmuGZACPBUYa2K56nHa3LhuoIeJ0rPEQbKRt7SyFqu58tELZ/jRLCQ5q0rhhTQFwqnEg4oKIcR6JvI8i1XnGIpGIlmnOZqtBZj1qzpL4AY0VM/DkMSBQp/vaD52IE45HQqMTcY1HEQFjr4sCiEVdeQzZodeAlsVSy4zCLmx4E5/i6Zy6UUpGpt4/iE+lWqjqWCekW7LSQs0iR1j3GOfiJRumHAas25epfNmEztB5BikX1PUHpy7Q3R9ylfgK998pBm9LrJwiJoCOfrA6RvtnK9bNxCCH0Hgpj/rc4S/Ruh6rKCbWvknZgHG1wTqdkb3I7UfbiCPakesXCsPtiUhIyv1kXndY5pgK9/1iMOHjdrXufEW/Tec7+taf8PZ97NJVN/RMinOp02Jb/j5HkDBZdLFYUh+uYkoQWaU2lg3pRPS6kNEVldgjdNqWA03bth1txxja9sXUv5jNPw1TeFqUpwYZoZpP0MbhuSI6TEzMtVd2j37O6whkp4Fm4hNzXkZg0ygxSMNhgNsHtVa0N1hbONXahcWsFiGeigVMfN8wbWi5cG3MLS9yDQ1FSAGsjU2ZGScVLccaOOYyMvljION4TL/cFlj4P+uW1uQ9MajDb2swFXkApQC/tgAyp0G6H0bvFyTKM5iXwa2ExzGFyrcxYENLKBCbYdXXfCcN0+qNM5k1DyCzXyqkMKPbUyRpFMdyueIGKl0hOmK2JV7dP/TG27UBplSUJ1VHh0Brfj9rhvbQJMQZEdtUsVabRgIo50c6PSFxlL57rDy1sKUrvc3BDUAfYyDgl1kOusRwptGwYFoU9D2BwiLJF5UTZ22QkRhusCt9Vkg/lqi+urdxF7BOhrOoTGAboG5XQHEM9VNvEoNw0NsHUi7pGgIcUWxXr4lzBlLGeMoG12l0ITKOj3cFn+AC4iEsyMtYGwnWbeL9B0YRu6MGzn/4otwF3ZfQWIzUAX865nIFzTRaNReEfPBVvmHceccz3Vy3YLAQcXeRwut7ISdG/ZyLycvAkUNChxTomCVw0XEHFSEdjTKPM63APdiux1hwONMqLdGnQOJZWbK2F3f4MfvKsaGes2ba0HSOn2dY2sQLan7x6CiDq7qLLVGlsB7BpKH6FuIT/2CIfgV/DJA0r3bMft2YO6FBYFbBbnG9ANwB3WtYFisgtzb7i2efp1+0kSrTd09Y5DePqNLRexWG1gg4FtngYMki+Ot+J6gz1GKs28K8sZNFARWWz206Zy3LpHIUuZ8brB3OrbN4Zw+LpBV2THIQP73UVP44Zt6rTWHG0SYF+MQaOko6ywTn9gl55roMpg23q3wL5ZxNYDcqi/i536V3Bz8MTtOpC6P7qd23fKL+Rz1hfWl9ZVy7EOrdvWfeuRdWT51q/Wn9Zf1t8Xfm+faZ9rtwro2TPlmktW5de+9A+mU82X</latexit><latexit sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">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</latexit><latexit 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r(β) =
0, β on image manifold
, otherwise<latexit 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sha1_base64="TZlHvKtOKIKwvBb/Te4jyN8zrRk=">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</latexit>
z
G(z)
r(β) =
0, β range(G)
, otherwise<latexit 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Learn generator G that outputs β ∈ ℝd given z ∈ ℝd’ for d’ < d
GANs for inverse problems
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sha1_base64="3qRIWc7pIXZR0YKMqzYlnbgFQBU=">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</latexit>
ˆβ = arg min
β
y Xβ 2
2 + r(β)
<latexit 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sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">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</latexit><latexit sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">AAAQWXiczVfdr9M2FA/si3UblPG4F7NeJNBC1YSWWx4qMYYGkxhjbBeQ6nLlJG7rXecDx+ltZfLX7a+Y9rqXvW7/wI6d9CNJhwCJaa0S2ef8fOzz6RMv4SyVvd5vZ86+9/4HH3507uPWJ59+dv5C++LnT9I4Ez498mMei2ceSSlnET2STHL6LBGUhB6nT72TbzT/6YKKlMXRz3KV0ElIZhGbMp9IIB23JwobIePH9+5MlDvo2+WTK/zw/gM2m8vWwQGeE4k9KgkaIUzELGTRcTHHL1fXn5khfnnsPnfRV0hcNfNrBwf5cbvT6/bMDzUHTjnoWOXv0fHF83/gIPazkEbS5yRNx46TyIlKOQtomrdwltKE+CdkRscwjEhI04kyGuToClACNI0FPJFEhrq7QpEwTVehB8iQyHla52niPt44k9PhRLEoySSN/GKjacaRjJG2KQqYoL7kKxgQXzDJfOTPiSC+BMtXd1mQSIIaKQVPRTM5x5Iu5SkL5Fzd7A5YVFFReSHMAzrFcqowoKhkPKBqmpdkT87B0gp7ocJmuGZACPBUYa2K56nHa3LhuoIeJ0rPEQbKRt7SyFqu58tELZ/jRLCQ5q0rhhTQFwqnEg4oKIcR6JvI8i1XnGIpGIlmnOZqtBZj1qzpL4AY0VM/DkMSBQp/vaD52IE45HQqMTcY1HEQFjr4sCiEVdeQzZodeAlsVSy4zCLmx4E5/i6Zy6UUpGpt4/iE+lWqjqWCekW7LSQs0iR1j3GOfiJRumHAas25epfNmEztB5BikX1PUHpy7Q3R9ylfgK998pBm9LrJwiJoCOfrA6RvtnK9bNxCCH0Hgpj/rc4S/Ruh6rKCbWvknZgHG1wTqdkb3I7UfbiCPakesXCsPtiUhIyv1kXndY5pgK9/1iMOHjdrXufEW/Tec7+taf8PZ97NJVN/RMinOp02Jb/j5HkDBZdLFYUh+uYkoQWaU2lg3pRPS6kNEVldgjdNqWA03bth1txxja9sXUv5jNPw1TeFqUpwYZoZpP0MbhuSI6TEzMtVd2j37O6whkp4Fm4hNzXkZg0ygxSMNhgNsHtVa0N1hbONXahcWsFiGeigVMfN8wbWi5cG3MLS9yDQ1FSAGsjU2ZGScVLccaOOYyMvljION4TL/cFlj4P+uW1uQ9MajDb2swFXkApQC/tgAyp0G6H0bvFyTKM5iXwa2ExzGFyrcxYENLKBCbYdXXfCcN0+qNM5k1DyCzXyqkMKPbUyRpFMdyueIGKl0hOmK2JV7dP/TG27UBplSUJ1VHh0Brfj9rhvbQJMQZEdtUsVabRgIo50c6PSFxlL57rDy1sKUrvc3BDUAfYyDgl1kOusRwptGwYFoU9D2BwiLJF5UTZ22QkRhusCt9Vkg/lqi+urdxF7BOhrOoTGAboG5XQHEM9VNvEoNw0NsHUi7pGgIcUWxXr4lzBlLGeMoG12l0ITKOj3cFn+AC4iEsyMtYGwnWbeL9B0YRu6MGzn/4otwF3ZfQWIzUAX865nIFzTRaNReEfPBVvmHceccz3Vy3YLAQcXeRwut7ISdG/ZyLycvAkUNChxTomCVw0XEHFSEdjTKPM63APdiux1hwONMqLdGnQOJZWbK2F3f4MfvKsaGes2ba0HSOn2dY2sQLan7x6CiDq7qLLVGlsB7BpKH6FuIT/2CIfgV/DJA0r3bMft2YO6FBYFbBbnG9ANwB3WtYFisgtzb7i2efp1+0kSrTd09Y5DePqNLRexWG1gg4FtngYMki+Ot+J6gz1GKs28K8sZNFARWWz206Zy3LpHIUuZ8brB3OrbN4Zw+LpBV2THIQP73UVP44Zt6rTWHG0SYF+MQaOko6ywTn9gl55roMpg23q3wL5ZxNYDcqi/i536V3Bz8MTtOpC6P7qd23fKL+Rz1hfWl9ZVy7EOrdvWfeuRdWT51q/Wn9Zf1t8Xfm+faZ9rtwro2TPlmktW5de+9A+mU82X</latexit><latexit 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r(β) =
0, β on image manifold
, otherwise<latexit 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sha1_base64="TZlHvKtOKIKwvBb/Te4jyN8zrRk=">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</latexit><latexit sha1_base64="TZlHvKtOKIKwvBb/Te4jyN8zrRk=">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</latexit><latexit sha1_base64="TZlHvKtOKIKwvBb/Te4jyN8zrRk=">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</latexit>
Bora, Jalal, Price, Dimakis, 2017
Choose β ∈ range(G) that best fits data:
ˆβ = arg min
β range(G)
y Xβ 2
2
=G(ˆz)
ˆz = arg min
z
y XG(z) 2
2
<latexit 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z
G(z)
r(β) =
0, β range(G)
, otherwise<latexit 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Learn generator G that outputs β ∈ ℝd given z ∈ ℝd’ for d’ < d
How much training data?
Original
β
Observed
y
Reconstruction with
convolutional neural
network (CNN) trained
with 80k samples
Reconstruction with
convolutional neural
network (CNN) trained
with 2k samples
How much training data?
Original
β
Observed
y
Reconstruction with
convolutional neural
network (CNN) trained
with 2k samples
How much training data?
Original
β
Observed
y
What people think he’s
referring to:
What he’s actually
referring to:
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Deep proximal gradient
y ˆβ<latexit 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ˆβ = arg min
β
y Xβ 2
2 + r(β)
<latexit 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<latexit sha1_base64="THBQUnH4haDFXD6GJWNbmMCy54I=">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</latexit>
Deep proximal gradient
Replace with learned neural network
z(k) β(k+1)
Chang, Li, Poczos, Kumar, & Sankaranarayanan 2017
y ˆβ<latexit 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ˆβ = arg min
β
y Xβ 2
2 + r(β)
<latexit 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Learning a proximal operator or learning a generative
model both implicitly require estimating p(β)
p(β)
{β: ‖y-Xβ‖=0}
probability density
over space of all
images
If β ∈ ℝd and p(β) ∈ Bα (Besov-α smooth functions), then the minimax
rate for learning p(β)
No neural network can beat this rate!
min
ˆp
max
p Bα
E ˆp(β) p(β) 2 = O n
α
2α+d
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Learning a proximal operator or learning a generative
model both implicitly require estimating p(β)
Donoho, Johnstone, Kerkyacharian, and Picard 1996
p(β)
{β: ‖y-Xβ‖=0}
probability density
over space of all
images
Prior vs. conditional density estimation
p(β)
Prior vs. conditional density estimation
pixels irrelevant

to inpainting
p(β)
Prior vs. conditional density estimation
We need conditional density p(X⟂β | Xβ)
pixels irrelevant

to inpainting
p(β)
p(X⟂β | Xβ)
If X⟂β only depends on d’ elements in Xβ, then the minimax rate is
Conditional density estimation
Conditional density [p(X⟂β | Xβ)] estimation can be
much easier than density [p(β)] estimation
Efromovich 2007
Bertin, Lacour, Rivoirard 2016

Nguyen 2018
Pixels irrelevant
for inpainting
min
ˆp
max
p Bα
E ˆp(X β|Xβ) p(X β|Xβ) 2 = O n
α
2α+d
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To reach a target squared pointwise error
of ½:
• estimating p(β) requires n ≈ 1060
• estimating p(X⟂β | Xβ) requires n ≈ 106d = 400

d’ = 40
βobs βmissing
p(β)
βobs
βmissing
p(βmissing|βobs)
Conditional density estimation
Conditional density [p(X⟂β | Xβ)] estimation can be
much easier than density [p(β)] estimation
p(βmissing|βobs) =
p(βmissing, βobs)
p(βobs)
=
p(β)
p(βobs)<latexit 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Can be smooth
Either may not be smooth
Smoother
densities are
easier to
estimate
Implications for learning to regularize
Estimating conditional density p(X⟂β | Xβ) can require far
fewer samples than estimating full density p(β)


X should be fully utilized in learning process
Classes of methods
Model Agnostic 

(Ignore X)
Decoupled 

(First learn, then reconstruct)
Unrolled Optimization
Neumann Networks 

(this talk!)
Assume r(β) differentiable.
ˆβ = arg min
β
y Xβ 2
2 + r(β)
set ˆβ
(1)
and stepsize η > 0
for k = 1, 2, . . .
ˆβ
(k+1)
= ˆβ
(k)
+ ηX (y Xˆβ
(k)
) + η r(ˆβ
(k)
)<latexit 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“Unrolled” gradient descent
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+…ˆβ
(0)
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ˆβ
(1)
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ˆβ
(2)
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ˆβ = ˆβ
(B)
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Unrolled
Optimization
Assume r(β) differentiable.
ˆβ = arg min
β
y Xβ 2
2 + r(β)
set ˆβ
(1)
and stepsize η > 0
for k = 1, 2, . . .
ˆβ
(k+1)
= ˆβ
(k)
+ ηX (y Xˆβ
(k)
) + η r(ˆβ
(k)
)<latexit 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sha1_base64="Pu5/Uf2JjbVre55GjOb1FeuMfWY=">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</latexit>
“Unrolled” gradient descent
Replace with learned neural network
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+…ˆβ
(0)
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ˆβ
(1)
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ˆβ
(2)
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ˆβ = ˆβ
(B)
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Unrolled
Optimization
Assume r(β) differentiable.
ˆβ = arg min
β
y Xβ 2
2 + r(β)
set ˆβ
(1)
and stepsize η > 0
for k = 1, 2, . . .
ˆβ
(k+1)
= ˆβ
(k)
+ ηX (y Xˆβ
(k)
) + η r(ˆβ
(k)
)<latexit 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sha1_base64="Pu5/Uf2JjbVre55GjOb1FeuMfWY=">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</latexit>
“Unrolled” gradient descent
Replace with learned neural network
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+…ˆβ
(0)
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ˆβ
(1)
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ˆβ
(2)
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ˆβ = ˆβ
(B)
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Unrolled
Optimization
Assume r(β) differentiable.
ˆβ = arg min
β
y Xβ 2
2 + r(β)
set ˆβ
(1)
and stepsize η > 0
for k = 1, 2, . . .
ˆβ
(k+1)
= ˆβ
(k)
+ ηX (y Xˆβ
(k)
) + η r(ˆβ
(k)
)<latexit 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sha1_base64="Pu5/Uf2JjbVre55GjOb1FeuMfWY=">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</latexit>
“Unrolled” gradient descent
Replace with learned neural network
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+…ˆβ
(0)
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ˆβ
(1)
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ˆβ
(2)
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ˆβ = ˆβ
(B)
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“Unrolled” optimization framework trained end-to-end
Unrolled
Optimization
Neumann series
Assume r(β) differentiable.

Let A be a linear operator. Then the Neumann series is
If A is contractive, we know higher-order terms are smaller.
Assume r(β) differentiable.
ˆβ = arg min
β
y Xβ 2
2 + r(β)
=(X X + r) 1
X y
=
k=1
( 1)k
(X X I + r)k
X y
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(1)
Can we estimate β by approximating (1) using (2)?

(e.g. A = I - X⊤
X + ∇r if ∇r is linear)
(2)(I A) 1
=
k=0
Ak
= I + A + A2
+ A3
+ · · ·
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Neumann
Networks
Neumann networks
Neumann network:
Assume r(β) differentiable.
ˆβ = arg min
β
y Xβ 2
2 + r(β)
=(X X + r) 1
X y
B
k=1
(I ηX X η r)k
ηX y
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[I - ηX⊤
X](⋅)
-η∇r(⋅)
+
[I - ηX⊤
X](⋅)
-η∇r(⋅)
+
[I - ηX⊤
X](⋅)
-η∇r(⋅)
+…
+ + +… +
β
(0)
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β
(1)
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β
(2)
<latexit 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β
(B)
<latexit 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β =
B
k=0
β
(k)
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Neumann networks
Replace with learned
neural network
Neumann network:
Assume r(β) differentiable.
ˆβ = arg min
β
y Xβ 2
2 + r(β)
=(X X + r) 1
X y
B
k=1
(I ηX X η r)k
ηX y
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[I - ηX⊤
X](⋅)
-η∇r(⋅)
+
[I - ηX⊤
X](⋅)
-η∇r(⋅)
+
[I - ηX⊤
X](⋅)
-η∇r(⋅)
+…
+ + +… +
β
(0)
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β
(1)
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β
(2)
<latexit 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β
(B)
<latexit 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β =
B
k=0
β
(k)
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sha1_base64="WIS1/KfonBYRAz7uBT3Rg6betzg=">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</latexit><latexit sha1_base64="WIS1/KfonBYRAz7uBT3Rg6betzg=">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</latexit>
Comparison
Neumann network
Gradient descent network
[I - ηX⊤
X](⋅)
-η∇r(⋅)
+
[I - ηX⊤
X](⋅)
-η∇r(⋅)
+
[I - ηX⊤
X](⋅)
-η∇r(⋅)
+…
+ + +… +
β
(0)
<latexit 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β
(1)
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β
(2)
<latexit 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β
(B)
<latexit 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β =
B
k=0
β
(k)
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[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+…ˆβ
(0)
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ˆβ
(1)
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sha1_base64="EBFh1kwZWtfk4Cfvlgksr9WqfMk=">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</latexit><latexit sha1_base64="EBFh1kwZWtfk4Cfvlgksr9WqfMk=">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</latexit>
ˆβ
(2)
<latexit sha1_base64="RmMIqqV/5Cr1t/Dk4lgLodpBaKk=">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</latexit><latexit sha1_base64="RmMIqqV/5Cr1t/Dk4lgLodpBaKk=">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</latexit><latexit sha1_base64="RmMIqqV/5Cr1t/Dk4lg
ˆβ = ˆβ
(B)
Preconditioning
[I - ηX⊤
X](⋅)
-η∇r(⋅)
+
[I - ηX⊤
X](⋅)
-η∇r(⋅)
+
[I - ηX⊤
X](⋅)
-η∇r(⋅)
+…
+ + +… +
β
(0)
<latexit 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β
(1)
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β
(2)
<latexit 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β
(B)
<latexit 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β =
B
k=0
β
(k)
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sha1_base64="WIS1/KfonBYRAz7uBT3Rg6betzg=">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</latexit><latexit sha1_base64="WIS1/KfonBYRAz7uBT3Rg6betzg=">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</latexit>
Neumann network: [I-ηX⊤
X] is linear and ∇r is nonlinear
ηλ[X
⊤
X+λI]
-1
(⋅)
-η∇r(⋅)
+
ηλ[X
⊤
X+λI]
-1
(⋅)
-η∇r(⋅)
+
ηλ[X
⊤
X+λI]
-1
(⋅)
-η∇r(⋅)
+…
+ + +… +
β
(0)
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β
(1)
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β
(2)
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β
(B)
β =
B
k=0
β
(k)
Preconditioned Neumann net: ηλ[I+λX⊤
X]-1
is linear and ∇r nonlinear
=ηX
⊤
y
η[X
⊤
X+λI]-1
X
⊤
y
Experiments
Comparison Methods
z
G(z)
Design-agnostic GAN
ˆβ = arg min
β range(G)
y Xβ 2
2
=G(ˆz)
1. Train
2. Reconstruct
Bora, Jalal, Price, Dimakis, 2017
Unrolled Gradient Descent
Neumann Network
ˆβ<latexit 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sha1_base64="3qRIWc7pIXZR0YKMqzYlnbgFQBU=">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</latexit><latexit 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X⊤
y
“skip connection”
Residual Autoencoder
Mao, Shen, Yang, 2016
[I - ηX⊤
X](⋅)
-η∇r(⋅)
+
[I - ηX⊤
X](⋅)
-η∇r(⋅)
+
[I - ηX⊤
X](⋅)
-η∇r(⋅)
+…
+ + +… +
β
(0)
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β
(1)
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β
(2)
<latexit 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β
(B)
<latexit 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β =
B
k=0
β
(k)
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[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+
[I-ηX⊤
X](⋅) 

+ ηX⊤
y
-η∇r(⋅)
+…ˆβ
(0)
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sha1_base64="BCWAouX/c2U2orEOKh9L3AF6NBU=">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</latexit>
ˆβ
(1)
<latexit sha1_base64="EBFh1kwZWtfk4Cfvlgksr9WqfMk=">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</latexit><latexit sha1_base64="EBFh1kwZWtfk4Cfvlgksr9WqfMk=">AAAL8niczVbNbhs3EN6kf6nUNk577GVT2YADbAStUbu+CEjToHGBNE1/nAQQFYO7Oyux5nLXJNeWQOwT9A16K3rtpYde2ifp23TIlWStpAZODkXXsEDOfEPON+RwJio4U7rX+/va9TfefOvtd26822q/9/4HN7duffhU5aWM4TjOeS6fR1QBZwKONdMcnhcSaBZxeBadfmH1z85BKpaLH/S0gGFGR4KlLKYaRSdbO+Tx0SM2GmufqIxy7re
ˆβ
(2)
ˆβ = ˆβ
(B)
[ - ηX⊤
X](⋅)
-η∇r(⋅)
+
[ - ηX⊤
X](⋅)
-η∇r(⋅)
+
[ - ηX⊤
X](⋅)
-η∇r(⋅)
+…
+ + +… +
β
(0)
β
(1)
β
(2)
β
(B)
β =
B
k=0
β
(k)
[ -ηX⊤
X](⋅)
+ ηX⊤
y
-η∇r(⋅)
+
[ -ηX⊤
X](⋅)
+ ηX⊤
y
-η∇r(⋅)
+
[ -ηX⊤
X](⋅)
+ ηX⊤
y
-η∇r(⋅)
+…ˆβ
(0)
ˆβ
(1)
ˆβ
(2)
ˆβ = ˆβ
(B)
Summary of Results
Sample Complexity
Sample Complexity
Application: MRI
reconstruction
Patch-based regularization
Extract

Patches
Subtract

Means
Add Back 

Means
Pass Through

Deep Patchwise

Regularizer
+
Recombine

Patches
Patch-based regularization
Extract

Patches
Subtract

Means
Add Back 

Means
Pass Through

Deep Patchwise

Regularizer
+
Recombine

Patches
Patch-based regularization
For the Neumann network, shrinking the patch size from
64x64 to 8x8 has comparable effect on PSNR as 10x
increase in training set size
The architecture of the neural network determines the
function space of regularizers and hence is a form of prior
Theory
Neumann series for nonlinear operators?
Can we justify Neumann net as an estimator
beyond the linear setting?
If A is a nonlinear operator, Neumann series identity does not hold:
In our case, A = I - ηX
⊤
X -η∇r, where ∇r may be nonlinear
(I A) 1
=
k=0
Ak
<latexit 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Case Study: Union of Subspaces Models
Model images as belonging to a union of low-dimensional subspaces
Case Study: Union of Subspaces Models
Images from: Extended Yale B dataset
& http://dhpark22.github.io/greedysc.html
Model images as belonging to a union of low-dimensional subspaces
Neumann nets and union of subspaces
For simplicity, assume:
• X has orthonormal rows
• measurements are noise-free: y = Xβ ∈ ℝm
• maximum subspace dimension < m/2
• the union of subspaces is “generic”
• β
Lemma:
• Optimal “oracle” regularizer ∇r is piecewise linear in β 







• Neumann network with ReLU activations can closely approximate this
• Outputs of all Neumann net blocks are in the same Sk for some k

for a fixed input, ∇r behaves linearly 

Neumann series foundation is justifiable and accurate
Sk = set of points closer to
subspace k than any other
subspace
r (β) =
R1β if β S1
... ...
RKβ if β SK<latexit 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I-ηXTX
“learned component”
Neumann nets and union of subspaces
For simplicity, assume:
• X has orthonormal rows
• measurements are noise-free: y = Xβ ∈ ℝm
• maximum subspace dimension < m/2
• the union of subspaces is “generic”
Theorem (informal):
For a given step size 0 < η < 1 and number of blocks B

there exists a Neumann network estimator

with a piecewise linear learned component such that 





for all β in the union of subspaces.
ˆβ(Xβ) β (1 η)B+1
Xβ<latexit 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ˆβ(Xβ) β (1 η)B+1
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• β
I-ηXTX
“learned component”
Neumann nets and union of subspaces
For simplicity, assume:
• X has orthonormal rows
• measurements are noise-free: y = Xβ ∈ ℝm
• maximum subspace dimension < m/2
• the union of subspaces is “generic”
Theorem (informal):
For a given step size 0 < η < 1 and number of blocks B

there exists a Neumann network estimator

with a piecewise linear learned component such that 





for all β in the union of subspaces.
ˆβ(Xβ) β (1 η)B+1
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ˆβ(Xβ) β (1 η)B+1
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arbitrarily small reconstruction error
• β
Empirical support for theory
2
Experiments on synthetic data show that when ∇r is a deep ReLU
network, the trained ∇r behaves as the predicted ∇r*
Test of Piecewise Linearity of ∇r
Outputs of all blocks
in same subspace
R = ∇r only
affects β in X’s
null space
R = ∇r reflects union of
subspaces structure
Why do Neumann nets give a performance boost?
Hypothesis: friendlier optimization landscape
Training curves for Neumann Network (NN) 

and Unrolled Gradient Descent (GDN)

on CIFAR10 Deblurring
In our experience, gradient descent networks tended to be
more sensitive to initialization and step size tuning.
• Explicitly accounting for design (X)
during training can dramatically reduce
sample complexity.
• Networks that include X in training, such
as unrolling approaches and Neumann
networks, perform well in the low-
sample regime.
• Neumann networks are mathematically
justified for union of subspaces.
• Further benefits from Neumann
networks, likely due to friendlier
optimization landscape.
Conclusions
{β: ‖y-Xβ‖=0}
probability density
over space of all
images p(β)
Learned: use training data to learn r(β)
Next: using theory to guide network
architecture design
Classical: r(β) is a pre-defined
smoothness-promoting regularizer 

(e.g. Tikhinov or ridge estimation)
Bayesian: r(β) = -log p(β) 

Uses a prior distribution over space of β’s 

(e.g. sparsity, patch redundancy, total variation)
References
• Gregor, K, and LeCun, Y. (2010). ”Learning fast approximations of sparse coding."  ICML 2010.
• Radford, A., Metz, L., & Chintala, S. (2015). Unsupervised representation learning with deep convolutional generative
adversarial networks. arXiv:1511.06434.
• Mousavi, A., & Baraniuk, R. G. (2017). Learning to invert: Signal recovery via deep convolutional networks. ICASSP 2017.
• Dong, C., Loy, C. C., He, K., & Tang, X. (2014). Learning a deep convolutional network for image super-resolution. ECCV
2014.
• Y. Chen, W. Yu, & T. Pock. (2015). On learning optimized reaction diffusion processes for effective image restoration.
CVPR 2015.
• Jin, K. H., McCann, M. T., Froustey, E., & Unser, M. (2017). Deep convolutional neural network for inverse problems in
imaging. IEEE TIP, 26(9), 4509-4522.
• Ye, J. C., Han, Y., & Cha, E. (2018). Deep convolutional framelets: A general deep learning framework for inverse
problems. SIAM Journal on Imaging Sciences, 11(2), 991-1048.
• Mardani, M., Sun, Q., Vasawanala, S., Papyan, V., Monajemi, H., Pauly, J., & Donoho, D. (2018). Neural Proximal Gradient
Descent for Compressive Imaging. arXiv:1806.03963.
• Sun, J., Li, H., & Xu, Z. (2016). Deep ADMM-Net for compressive sensing MRI. NIPS 2016.
• Chang, J. H. R., Li, C. L., Poczos, B., Kumar, B. V., & Sankaranarayanan, A. C. (2016). One Network to Solve Them All-
Solving Linear Inverse Problems using Deep Projection Models. ICCV 2016.
• Adler, J., & Öktem, O. (2018). Learned primal-dual reconstruction. IEEE TMI, 37(6), 1322-1332.
• Zhang, K., Zuo, W., Gu, S., & Zhang, L. (2017). Learning deep CNN denoiser prior for image restoration. CVPR 2017.
• Bora, A., Jalal, A., Price, E., & Dimakis, A. G. (2017). Compressed Sensing using Generative Models. ICML 2017.
• Donoho, D. L., & Low, M. G. (1992). Renormalization exponents and optimal pointwise rates of convergence. The Annals
of Statistics, 944-970.
• Huang, G., Liu, Z., Van Der Maaten, L., & Weinberger, K. Q. (2017). Densely Connected Convolutional Networks.
CVPR 2017.

Deep Learning Opening Workshop - Learning to Solve Inverse Problems in Imaging- Rebecca Willett, August 13, 2019

  • 1.
    Learning to SolveInverse Problems in Imaging Rebecca Willett, University of Chicago Greg Ongie,
 UChicago Davis Gilton, 
 UW-Madison
  • 2.
    β Inverse problems inimaging • Inpainting • Deblurring • Superresolution • Compressed Sensing • MRI • Radar Observe: y = Xβ + ε Goal: Recover β from y y
  • 3.
    Classical approach: Tikhonovregularization (1943) • Example: deblurring • Least squares solution: Wildly Different Solutions Blurred image with noiseBlurred image Deblurred the “naïve” way Deblurred the “naïve” way ˆβ = (X X) 1 X y<latexit 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Wildly different solutions
  • 4.
    Classical approach: Tikhonovregularization (1943) • Example: deblurring • Least squares solution: Wildly Different Solutions Blurred image with noiseBlurred image Deblurred the “naïve” way Deblurred the “naïve” way • Tikhonov regularization
 (aka “ridge regression”) better conditioned; suppresses noise ˆβ = arg min β y Xβ 2 2 + λ β 2 2 =(X X + λI) 1 X y<latexit 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Wildly different solutions
  • 5.
    Classical approach: Tikhonovregularization (1943) • Example: deblurring • Least squares solution: Wildly Different Solutions Blurred image with noiseBlurred image Deblurred the “naïve” way Deblurred the “naïve” way • Tikhonov regularization
 (aka “ridge regression”) better conditioned; suppresses noise ˆβ = arg min β y Xβ 2 2 + λ β 2 2 =(X X + λI) 1 X y<latexit 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Wildly different solutions Tikhonov regularization
  • 6.
    Geometric models ofimages Total variation Noisy Patches Denoised Patches Patch Denoising Combine to estimate denoised pixel Patch subspaces and manifolds (Wavelet) sparsity
  • 7.
    Regularization in inverseproblems y ˆβ<latexit 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ˆβ = arg min β y Xβ 2 2 + r(β) <latexit 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sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">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</latexit><latexit sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">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</latexit><latexit sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">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</latexit>
  • 8.
    Regularization in inverseproblems y ˆβ<latexit 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ˆβ = arg min β y Xβ 2 2 + r(β) <latexit 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Classical: r(β) is a pre-defined smoothness-promoting regularizer 
 (e.g. Tikhinov or ridge estimation) Geometric: r(β) reflects image geometry 
 (e.g. sparsity, patch redundancy, total variation) Learned: use training data to learn r(β)
  • 9.
    Examples in recentliterature • Deep CNN’s for signal recovery • Compressed sensing with GANs • Unrolled algorithms for solving inverse problems • Deep proximal gradient descent nets • Deep ADMM nets • Deep half-quadratic splitting • Deep primal-dual nets Chen, Yu, Pock, 2015 Sun, Li, Xu, 2016 Chang, Li, Poczos, Kumar, Sankaranarayanan, 2017 Adler and Öktem, 2018 Bora, Jalal, Price, Dimakis, 2017 Mousavi and Baraniuk, 2017 Dong, Loy, He, Tang, 2014 Mardani et al, 2018 Jin, McCann, Froustey, Unser, 2017 Zhang, Zuo, Gu, Zhang, 2017 Ye, Han, Cha, 2018
  • 10.
    Classes of methods ModelAgnostic 
 (Ignore X) Decoupled 
 (First learn, then reconstruct) Unrolled Optimization Neumann Networks 
 (this talk!)
  • 11.
    X-1~ raw data (low resolutionimage) approximate high-resolution image Train deep CNN to remove artifacts reconstruction blurry/blocky artifacts due to re-scaling bicubic interpolation Pictures from: http://webdav.tuebingen.mpg.de/pixel/enhancenet/ Super-resolution with CNNs Model Agnostic 
 (Ignore X)
  • 12.
    Classes of methods ModelAgnostic 
 (Ignore X) Decoupled 
 (First learn, then reconstruct) Unrolled Optimization Neumann Networks 
 (this talk!)
  • 13.
    GANs for inverseproblems y ˆβ<latexit 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ˆβ = arg min β y Xβ 2 2 + r(β) <latexit 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r(β) = 0, β on image manifold , otherwise<latexit 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“Good” image on manifold “Bad” image off manifold Decoupled 
 (First learn, then reconstruct)
  • 14.
    GANs for inverseproblems y ˆβ<latexit 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ˆβ = arg min β y Xβ 2 2 + r(β) <latexit 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r(β) = 0, β on image manifold , otherwise<latexit 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  • 15.
    GANs for inverseproblems y ˆβ<latexit 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ˆβ = arg min β y Xβ 2 2 + r(β) <latexit 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r(β) = 0, β on image manifold , otherwise<latexit 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z G(z) r(β) = 0, β range(G) , otherwise<latexit 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Learn generator G that outputs β ∈ ℝd given z ∈ ℝd’ for d’ < d
  • 16.
    GANs for inverseproblems y ˆβ<latexit 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ˆβ = arg min β y Xβ 2 2 + r(β) <latexit 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r(β) = 0, β on image manifold , otherwise<latexit 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Bora, Jalal, Price, Dimakis, 2017 Choose β ∈ range(G) that best fits data: ˆβ = arg min β range(G) y Xβ 2 2 =G(ˆz) ˆz = arg min z y XG(z) 2 2 <latexit 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z G(z) r(β) = 0, β range(G) , otherwise<latexit 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Learn generator G that outputs β ∈ ℝd given z ∈ ℝd’ for d’ < d
  • 17.
    How much trainingdata? Original β Observed y Reconstruction with convolutional neural network (CNN) trained with 80k samples
  • 18.
    Reconstruction with convolutional neural network(CNN) trained with 2k samples How much training data? Original β Observed y
  • 19.
    Reconstruction with convolutional neural network(CNN) trained with 2k samples How much training data? Original β Observed y What people think he’s referring to: What he’s actually referring to:
  • 20.
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Deep proximalgradient y ˆβ<latexit 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ˆβ = arg min β y Xβ 2 2 + r(β) <latexit 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sha1_base64="/+r4aBrVfxqpWL/MVPFAYn0tVFs=">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</latexit><latexit 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  • 21.
    <latexit 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Deep proximalgradient Replace with learned neural network z(k) β(k+1) Chang, Li, Poczos, Kumar, & Sankaranarayanan 2017 y ˆβ<latexit 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ˆβ = arg min β y Xβ 2 2 + r(β) <latexit 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  • 22.
    Learning a proximaloperator or learning a generative model both implicitly require estimating p(β) p(β) {β: ‖y-Xβ‖=0} probability density over space of all images
  • 23.
    If β ∈ℝd and p(β) ∈ Bα (Besov-α smooth functions), then the minimax rate for learning p(β) No neural network can beat this rate! min ˆp max p Bα E ˆp(β) p(β) 2 = O n α 2α+d <latexit 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Learning a proximal operator or learning a generative model both implicitly require estimating p(β) Donoho, Johnstone, Kerkyacharian, and Picard 1996 p(β) {β: ‖y-Xβ‖=0} probability density over space of all images
  • 24.
    Prior vs. conditionaldensity estimation p(β)
  • 25.
    Prior vs. conditionaldensity estimation pixels irrelevant
 to inpainting p(β)
  • 26.
    Prior vs. conditionaldensity estimation We need conditional density p(X⟂β | Xβ) pixels irrelevant
 to inpainting p(β) p(X⟂β | Xβ)
  • 27.
    If X⟂β onlydepends on d’ elements in Xβ, then the minimax rate is Conditional density estimation Conditional density [p(X⟂β | Xβ)] estimation can be much easier than density [p(β)] estimation Efromovich 2007 Bertin, Lacour, Rivoirard 2016
 Nguyen 2018 Pixels irrelevant for inpainting min ˆp max p Bα E ˆp(X β|Xβ) p(X β|Xβ) 2 = O n α 2α+d <latexit 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To reach a target squared pointwise error of ½: • estimating p(β) requires n ≈ 1060 • estimating p(X⟂β | Xβ) requires n ≈ 106d = 400
 d’ = 40
  • 28.
    βobs βmissing p(β) βobs βmissing p(βmissing|βobs) Conditional densityestimation Conditional density [p(X⟂β | Xβ)] estimation can be much easier than density [p(β)] estimation p(βmissing|βobs) = p(βmissing, βobs) p(βobs) = p(β) p(βobs)<latexit 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Can be smooth Either may not be smooth Smoother densities are easier to estimate
  • 29.
    Implications for learningto regularize Estimating conditional density p(X⟂β | Xβ) can require far fewer samples than estimating full density p(β) 
 X should be fully utilized in learning process
  • 30.
    Classes of methods ModelAgnostic 
 (Ignore X) Decoupled 
 (First learn, then reconstruct) Unrolled Optimization Neumann Networks 
 (this talk!)
  • 31.
    Assume r(β) differentiable. ˆβ= arg min β y Xβ 2 2 + r(β) set ˆβ (1) and stepsize η > 0 for k = 1, 2, . . . ˆβ (k+1) = ˆβ (k) + ηX (y Xˆβ (k) ) + η r(ˆβ (k) )<latexit 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sha1_base64="Pu5/Uf2JjbVre55GjOb1FeuMfWY=">AAANM3iczVZfjxs1EM+Vf2U5oIVHXlxyRXd0G2UjWvoS1JaKFqmU8ufak+K7k3fXm5jzevds710Odz9WvwVfAPGGeOWFV3hgbGdz2eSo2j5UbKTInvnNeGY8M5645Ezpfv/XtXOvvf7Gm2+dfzt4Z/3d996/cPGDR6qoZEK3k4IXcicminIm6LZmmtOdUlKSx5w+jg++tPzHR1QqVogf9UlJd3MyFixjCdFA2r+4RvGDe/fZeKIRVjnhHAW3lKpyijbkJo6pJlsbKGVZRiUVmhHQ2wuAPmbCEM7G4tM6+ARPCIhbMBoiTOQ4Z2Lf7/GTk6s7bomf7A/2BugKavRiDJJ5XEwNljlSVKManWraM5vRFhDmACJSpDQtFfuZWqTV/gXqt7VkhQTeAZgRhYMQ87TQCjnI4WFF0rb+gyv2hOESEUhXvPadPayLchMcWIFsNRgsICTE+rSMCDAVaROj/Qvdfq/vPrS6iGaLbmf2Pdy/GDzFaZHARQidcKLUKIpKvWsUZylVdYArRUuSHJAxHcFSkJyqXePyoUaXgZK6WGSF0MhRFyUMyZU6yWNA5kRP1DLPEs/ijSqd3dg1TJSVpiLxB2UVR7pANrkgUSRNND+BBUkk0yxByYRIkmhIwfYpR0RocAOunVMx1hOs6VQfs1RPzPXeNSZaLpo4h31KM6wzgwFFNeMpNVk9I8d6ApE3OM4NdsuGAbXAlcHWlTg23zdkn6OeXpTG7iHV4NRG39Tpmjb7aWmme7iULKd1cNmRUnposNJgoKQcVuBvqWf/+oRTrCUjYsxpbYaNGifT0A+BKOhxUuQ55LbBt45oPYp2DeY005g7DOpGCEtbnlh6ZW0ZMpdZgM+AQSuC00qwpEid+Ytkrqdakna03cWXNGlTbS556mV7bTlhwpLMXQZd4wci1JwB0pazeYeNmVbhfeg1IrwrKT3YekH0PcqP4K4T8oBW9KrrUz5poFM1BqgXk2zERgFC6GtQxJKvbJXYb4jaYp4dWuTtgqdz3CrSsue4Ba1n4Tx7t22iv1hrWEZyxk+atvw8Zjrg89u6zeHGnczzWHyKPtPulw3t/8HmxVpy/UfmPLPlZJrod6O6XkHBK9tGYci+CSmpR3OqHSzOeDbTuqKiWtYQZ4pKRtWZB1arJzb41tFLJV9xmj/7pXBdCSYHt4OyH8NrQ2qEjBzHtendCPth78YSquRVfgq5biHXlyBjKEExx1hA2G9HG7or2DYaQOeyDnox8MGY7qCuV7DwvDtwgHUSQ6KZTIIbyPXZoYFH2r9xw24UorjQusjnhEufXbsEL3RyUIfuNXQz0nAevxBwnuRBAU4gBlTaecrY04rpiIoJEQlNQ2Y5DJ7VCUtTKkJgQmyHV6M8B0e8E8cTpqHlezfq9oV4P60zzpHKjm2xJPLEqANmO2Lb7eNX5nbonUZVWVKbFX6+OzX3pUPgZqAFt2cuUnHEZCHscGPUYcXUxI66dWCgtGeHO4LZwHHFoaA2alv1yKDTgcFA6tMcDocMK3Xt28YiuyTScQfADVbZEL4l4WXpRcQZCuwzncPgAFODiXrXIJ/bbBJT7gYaYNtCPEODhfgjvDz8ZjDjIueCYGN2h8IQKOk38Fh+C1dENIQZ2wDhUFXxTzB04RCmMBzW/4n14J7uPQMEs2pt3P9yBcIz7QcNfzt2L9m07kbOzmZbBzDoRstj7eri0aAX9XvRd4Puzduzkfd856POx53NTtT5vHOzc6/zsLPdSdZ+Wftr7e+1f9afrv+2/vv6Hx56bm0m82Gn9a3/+S+y+7J+</latexit> “Unrolled” gradient descent [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) + [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) + [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) +…ˆβ (0) <latexit 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Unrolled Optimization
  • 32.
    Assume r(β) differentiable. ˆβ= arg min β y Xβ 2 2 + r(β) set ˆβ (1) and stepsize η > 0 for k = 1, 2, . . . ˆβ (k+1) = ˆβ (k) + ηX (y Xˆβ (k) ) + η r(ˆβ (k) )<latexit 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“Unrolled” gradient descent Replace with learned neural network [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) + [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) + [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) +…ˆβ (0) <latexit 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Unrolled Optimization
  • 33.
    Assume r(β) differentiable. ˆβ= arg min β y Xβ 2 2 + r(β) set ˆβ (1) and stepsize η > 0 for k = 1, 2, . . . ˆβ (k+1) = ˆβ (k) + ηX (y Xˆβ (k) ) + η r(ˆβ (k) )<latexit 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“Unrolled” gradient descent Replace with learned neural network [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) + [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) + [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) +…ˆβ (0) <latexit 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Unrolled Optimization
  • 34.
    Assume r(β) differentiable. ˆβ= arg min β y Xβ 2 2 + r(β) set ˆβ (1) and stepsize η > 0 for k = 1, 2, . . . ˆβ (k+1) = ˆβ (k) + ηX (y Xˆβ (k) ) + η r(ˆβ (k) )<latexit 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“Unrolled” gradient descent Replace with learned neural network [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) + [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) + [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) +…ˆβ (0) <latexit 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“Unrolled” optimization framework trained end-to-end Unrolled Optimization
  • 35.
    Neumann series Assume r(β)differentiable.
 Let A be a linear operator. Then the Neumann series is If A is contractive, we know higher-order terms are smaller. Assume r(β) differentiable. ˆβ = arg min β y Xβ 2 2 + r(β) =(X X + r) 1 X y = k=1 ( 1)k (X X I + r)k X y <latexit 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(1) Can we estimate β by approximating (1) using (2)?
 (e.g. A = I - X⊤ X + ∇r if ∇r is linear) (2)(I A) 1 = k=0 Ak = I + A + A2 + A3 + · · · <latexit 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Neumann Networks
  • 36.
    Neumann networks Neumann network: Assumer(β) differentiable. ˆβ = arg min β y Xβ 2 2 + r(β) =(X X + r) 1 X y B k=1 (I ηX X η r)k ηX y <latexit 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[I - ηX⊤ X](⋅) -η∇r(⋅) + [I - ηX⊤ X](⋅) -η∇r(⋅) + [I - ηX⊤ X](⋅) -η∇r(⋅) +… + + +… + β (0) <latexit 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β = B k=0 β (k) <latexit 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  • 37.
    Neumann networks Replace withlearned neural network Neumann network: Assume r(β) differentiable. ˆβ = arg min β y Xβ 2 2 + r(β) =(X X + r) 1 X y B k=1 (I ηX X η r)k ηX y <latexit 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[I - ηX⊤ X](⋅) -η∇r(⋅) + [I - ηX⊤ X](⋅) -η∇r(⋅) + [I - ηX⊤ X](⋅) -η∇r(⋅) +… + + +… + β (0) <latexit 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β = B k=0 β (k) <latexit 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  • 38.
    Comparison Neumann network Gradient descentnetwork [I - ηX⊤ X](⋅) -η∇r(⋅) + [I - ηX⊤ X](⋅) -η∇r(⋅) + [I - ηX⊤ X](⋅) -η∇r(⋅) +… + + +… + β (0) <latexit 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β = B k=0 β (k) <latexit 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[I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) + [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) + [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) +…ˆβ (0) <latexit 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sha1_base64="EBFh1kwZWtfk4Cfvlgksr9WqfMk=">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</latexit> ˆβ (2) <latexit sha1_base64="RmMIqqV/5Cr1t/Dk4lgLodpBaKk=">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</latexit><latexit sha1_base64="RmMIqqV/5Cr1t/Dk4lgLodpBaKk=">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</latexit><latexit sha1_base64="RmMIqqV/5Cr1t/Dk4lg ˆβ = ˆβ (B)
  • 39.
    Preconditioning [I - ηX⊤ X](⋅) -η∇r(⋅) + [I- ηX⊤ X](⋅) -η∇r(⋅) + [I - ηX⊤ X](⋅) -η∇r(⋅) +… + + +… + β (0) <latexit 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sha1_base64="VCuA1ZZhGJXUK4p3YvU2iT80Nn4=">AAAL+XiczVZfbxw1EN+Wf+UCNIVHXlwukVJpe7qNSMhLpFIqGqRSyp+0lc7XyLs7e2fi9W5sb3KRtV+Cb8Ab4hUh8Qrfgm/D2Ht3ub0LVdoHxEY52TO/sec39ngmLgXXpt//+9r1N9586+13brzbWXvv/Q9urt/68KkuKpXAYVKIQj2PmQbBJRwabgQ8LxWwPBbwLD7+wumfnYLSvJA/mPMShjkbSZ7xhBkUHa2H9PHBIz4aG0J1zoQgnY0NesZTMFykQGgMhr2wW/079cZG52i92+/1/UdWB9F00A2m35OjW53faVokVQ7SJIJpPYii0gytFriDrju00lCy5JiNYIBDyXLQQ+tZ1WQTJSnJCoX/0hAvXbSwLNf6PI8RmTMz1ss6J7xMN6hMtje0XJaVAZk0G2WVIKYgLkQk5QoSI85xwBLFDU9IMmaKJQYD2d7llEmDNDRg4OXIjKmBicHwmbHd7e1w2aJo4xznKWTUZPYiyDarp+LYjDHclsa5pX44U+CJCm2poxLH9ruZmKlRzmUjL0rr5oSiZL7exK81mc0npZ28oKXiOdSdTS9K4cRSbdBBBQJHyLc0019zLoAaxZkcCajt/mwZbzOTn6BQwllS5DmTqaWfn0I9iIaWCsgMFR5DuhGhyt0yqprF2jZsbrMAnwI7rQhOKsmTIvXuL4qFmRjF2tH2B19C0pa6u9RIN92x5YxLJ7IPOV7+75nUcwVaO83WAz7iRoePMGNk+FABHN95RfQBiFM864Q9hgru+nRrLg0m3MwB/WqWM7NBhxDyFS7Eky9dlrhvn7TNGnXokPcLkc5xq0innuMWVr0M16iHbRebg3WOZSzn4nz2ulzFTQ+8uq+HAk/c21zF4wv0pX6/bmj/Dz4v5pJ/f1QuMpdOdhb9blTXKyisFW0Uxds3ZiU0aAHGw+JMZNNVV5aolleIMw2Kg750w2p1xxm+tfVSylcC8pdXCv8qYf3zM0z7EVYbVhNi1SiubW8v7Ie9vSVUKar8ArLrILtLkBGmoJxjHCDst6ONryv6NtjGl8sRbMyQg7Xd7bpewcbFxIM71CQxXjSbKaRB/Du7b01RNjVuvxuFJC6MKfK54PanO7djgfzr0FdDX+n35/ELEdeIGlCHJhgDUK4rsG63YjIAOWYygTTkTsOxrI55moIMUYmx3b8b5TkSaUicjbnBJ7+hUbcPpOHpyHgilWs+YsXUudXH3L2Ibdpn/xntsCFNqrIEdytiGGF1vHD3tUNAAYks0J5SBHnKVSFdc2P1ScX12DVsdcdiak839wK7QeNKYEJt1C7riSUXDYPFqw85bo43rDR182wsqkumvHYbtZ1VNYZvyXjZehFxyQKuTOfYOGDXYKPeDt7ntprFIHxDg2qXiJes4CDNFo09/k1h1kfOB8HF7AFgE6jgayyW3+ARMYNhpi5ANNRV/CM2XTTELoyG9b9iG3DP9F4C4iPk4n+XMxDLdNNoNKfj5opP6m7k/ZxNa9foRstt7erg6XYv6veib7e79+5PW94bwcfBJ8FWEAWfBfeCg+BJcBgkwU/BH8GfwV9rdu3ntV/Wfm2g169NbT4KWt/ab/8ARkhGOQ==</latexit><latexit sha1_base64="VCuA1ZZhGJXUK4p3YvU2iT80Nn4=">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</latexit><latexit sha1_base64="VCuA1ZZhGJXUK4p3YvU2iT80Nn4=">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</latexit> β (1) <latexit 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β = B k=0 β (k) <latexit 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Neumann network: [I-ηX⊤ X] is linear and ∇r is nonlinear ηλ[X ⊤ X+λI] -1 (⋅) -η∇r(⋅) + ηλ[X ⊤ X+λI] -1 (⋅) -η∇r(⋅) + ηλ[X ⊤ X+λI] -1 (⋅) -η∇r(⋅) +… + + +… + β (0) <latexit 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β (2) <latexit 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sha1_base64="z9gUNd2znDnQNKstdKCm3u β (B) β = B k=0 β (k) Preconditioned Neumann net: ηλ[I+λX⊤ X]-1 is linear and ∇r nonlinear =ηX ⊤ y η[X ⊤ X+λI]-1 X ⊤ y
  • 40.
  • 41.
    Comparison Methods z G(z) Design-agnostic GAN ˆβ= arg min β range(G) y Xβ 2 2 =G(ˆz) 1. Train 2. Reconstruct Bora, Jalal, Price, Dimakis, 2017 Unrolled Gradient Descent Neumann Network ˆβ<latexit 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X⊤ y “skip connection” Residual Autoencoder Mao, Shen, Yang, 2016 [I - ηX⊤ X](⋅) -η∇r(⋅) + [I - ηX⊤ X](⋅) -η∇r(⋅) + [I - ηX⊤ X](⋅) -η∇r(⋅) +… + + +… + β (0) <latexit 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β (1) <latexit 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β = B k=0 β (k) <latexit 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[I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) + [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) + [I-ηX⊤ X](⋅) 
 + ηX⊤ y -η∇r(⋅) +…ˆβ (0) <latexit 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ˆβ (1) <latexit 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sha1_base64="EBFh1kwZWtfk4Cfvlgksr9WqfMk=">AAAL8niczVbNbhs3EN6kf6nUNk577GVT2YADbAStUbu+CEjToHGBNE1/nAQQFYO7Oyux5nLXJNeWQOwT9A16K3rtpYde2ifp23TIlWStpAZODkXXsEDOfEPON+RwJio4U7rX+/va9TfefOvtd26822q/9/4HN7duffhU5aWM4TjOeS6fR1QBZwKONdMcnhcSaBZxeBadfmH1z85BKpaLH/S0gGFGR4KlLKYaRSdbO+Tx0SM2GmufqIxy7re ˆβ (2) ˆβ = ˆβ (B) [ - ηX⊤ X](⋅) -η∇r(⋅) + [ - ηX⊤ X](⋅) -η∇r(⋅) + [ - ηX⊤ X](⋅) -η∇r(⋅) +… + + +… + β (0) β (1) β (2) β (B) β = B k=0 β (k) [ -ηX⊤ X](⋅) + ηX⊤ y -η∇r(⋅) + [ -ηX⊤ X](⋅) + ηX⊤ y -η∇r(⋅) + [ -ηX⊤ X](⋅) + ηX⊤ y -η∇r(⋅) +…ˆβ (0) ˆβ (1) ˆβ (2) ˆβ = ˆβ (B)
  • 42.
  • 43.
  • 44.
  • 45.
  • 46.
    Patch-based regularization Extract
 Patches Subtract
 Means Add Back
 Means Pass Through
 Deep Patchwise
 Regularizer + Recombine
 Patches
  • 47.
    Patch-based regularization Extract
 Patches Subtract
 Means Add Back
 Means Pass Through
 Deep Patchwise
 Regularizer + Recombine
 Patches
  • 48.
    Patch-based regularization For theNeumann network, shrinking the patch size from 64x64 to 8x8 has comparable effect on PSNR as 10x increase in training set size The architecture of the neural network determines the function space of regularizers and hence is a form of prior
  • 49.
  • 50.
    Neumann series fornonlinear operators? Can we justify Neumann net as an estimator beyond the linear setting? If A is a nonlinear operator, Neumann series identity does not hold: In our case, A = I - ηX ⊤ X -η∇r, where ∇r may be nonlinear (I A) 1 = k=0 Ak <latexit 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  • 51.
    Case Study: Unionof Subspaces Models Model images as belonging to a union of low-dimensional subspaces
  • 52.
    Case Study: Unionof Subspaces Models Images from: Extended Yale B dataset & http://dhpark22.github.io/greedysc.html Model images as belonging to a union of low-dimensional subspaces
  • 53.
    Neumann nets andunion of subspaces For simplicity, assume: • X has orthonormal rows • measurements are noise-free: y = Xβ ∈ ℝm • maximum subspace dimension < m/2 • the union of subspaces is “generic” • β Lemma: • Optimal “oracle” regularizer ∇r is piecewise linear in β 
 
 
 
 • Neumann network with ReLU activations can closely approximate this • Outputs of all Neumann net blocks are in the same Sk for some k
 for a fixed input, ∇r behaves linearly 
 Neumann series foundation is justifiable and accurate Sk = set of points closer to subspace k than any other subspace r (β) = R1β if β S1 ... ... RKβ if β SK<latexit 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  • 54.
    I-ηXTX “learned component” Neumann netsand union of subspaces For simplicity, assume: • X has orthonormal rows • measurements are noise-free: y = Xβ ∈ ℝm • maximum subspace dimension < m/2 • the union of subspaces is “generic” Theorem (informal): For a given step size 0 < η < 1 and number of blocks B
 there exists a Neumann network estimator
 with a piecewise linear learned component such that 
 
 
 for all β in the union of subspaces. ˆβ(Xβ) β (1 η)B+1 Xβ<latexit 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ˆβ(Xβ) β (1 η)B+1 <latexit 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• β
  • 55.
    I-ηXTX “learned component” Neumann netsand union of subspaces For simplicity, assume: • X has orthonormal rows • measurements are noise-free: y = Xβ ∈ ℝm • maximum subspace dimension < m/2 • the union of subspaces is “generic” Theorem (informal): For a given step size 0 < η < 1 and number of blocks B
 there exists a Neumann network estimator
 with a piecewise linear learned component such that 
 
 
 for all β in the union of subspaces. ˆβ(Xβ) β (1 η)B+1 Xβ<latexit 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ˆβ(Xβ) β (1 η)B+1 <latexit 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arbitrarily small reconstruction error • β
  • 56.
    Empirical support fortheory 2 Experiments on synthetic data show that when ∇r is a deep ReLU network, the trained ∇r behaves as the predicted ∇r* Test of Piecewise Linearity of ∇r Outputs of all blocks in same subspace R = ∇r only affects β in X’s null space R = ∇r reflects union of subspaces structure
  • 57.
    Why do Neumannnets give a performance boost? Hypothesis: friendlier optimization landscape Training curves for Neumann Network (NN) and Unrolled Gradient Descent (GDN) on CIFAR10 Deblurring In our experience, gradient descent networks tended to be more sensitive to initialization and step size tuning.
  • 58.
    • Explicitly accountingfor design (X) during training can dramatically reduce sample complexity. • Networks that include X in training, such as unrolling approaches and Neumann networks, perform well in the low- sample regime. • Neumann networks are mathematically justified for union of subspaces. • Further benefits from Neumann networks, likely due to friendlier optimization landscape. Conclusions {β: ‖y-Xβ‖=0} probability density over space of all images p(β)
  • 59.
    Learned: use trainingdata to learn r(β) Next: using theory to guide network architecture design Classical: r(β) is a pre-defined smoothness-promoting regularizer 
 (e.g. Tikhinov or ridge estimation) Bayesian: r(β) = -log p(β) 
 Uses a prior distribution over space of β’s 
 (e.g. sparsity, patch redundancy, total variation)
  • 60.
    References • Gregor, K,and LeCun, Y. (2010). ”Learning fast approximations of sparse coding."  ICML 2010. • Radford, A., Metz, L., & Chintala, S. (2015). Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv:1511.06434. • Mousavi, A., & Baraniuk, R. G. (2017). Learning to invert: Signal recovery via deep convolutional networks. ICASSP 2017. • Dong, C., Loy, C. C., He, K., & Tang, X. (2014). Learning a deep convolutional network for image super-resolution. ECCV 2014. • Y. Chen, W. Yu, & T. Pock. (2015). On learning optimized reaction diffusion processes for effective image restoration. CVPR 2015. • Jin, K. H., McCann, M. T., Froustey, E., & Unser, M. (2017). Deep convolutional neural network for inverse problems in imaging. IEEE TIP, 26(9), 4509-4522. • Ye, J. C., Han, Y., & Cha, E. (2018). Deep convolutional framelets: A general deep learning framework for inverse problems. SIAM Journal on Imaging Sciences, 11(2), 991-1048. • Mardani, M., Sun, Q., Vasawanala, S., Papyan, V., Monajemi, H., Pauly, J., & Donoho, D. (2018). Neural Proximal Gradient Descent for Compressive Imaging. arXiv:1806.03963. • Sun, J., Li, H., & Xu, Z. (2016). Deep ADMM-Net for compressive sensing MRI. NIPS 2016. • Chang, J. H. R., Li, C. L., Poczos, B., Kumar, B. V., & Sankaranarayanan, A. C. (2016). One Network to Solve Them All- Solving Linear Inverse Problems using Deep Projection Models. ICCV 2016. • Adler, J., & Öktem, O. (2018). Learned primal-dual reconstruction. IEEE TMI, 37(6), 1322-1332. • Zhang, K., Zuo, W., Gu, S., & Zhang, L. (2017). Learning deep CNN denoiser prior for image restoration. CVPR 2017. • Bora, A., Jalal, A., Price, E., & Dimakis, A. G. (2017). Compressed Sensing using Generative Models. ICML 2017. • Donoho, D. L., & Low, M. G. (1992). Renormalization exponents and optimal pointwise rates of convergence. The Annals of Statistics, 944-970. • Huang, G., Liu, Z., Van Der Maaten, L., & Weinberger, K. Q. (2017). Densely Connected Convolutional Networks. CVPR 2017.