Tensor Networks for Generative Modeling From Boltzmann machines to - - PowerPoint PPT Presentation

Institute of Physics, Beijing Chinese Academy of Sciences

Tensor Networks for Generative Modeling

Lei W ang ()

https://wangleiphy.github.io

From Boltzmann machines to Born machines, and back

Han, Wang, Fan, LW, Zhang, PRX18’ Chen, Cheng, Xie, LW, Xiang, PRB 18’ Cheng, Chen, LW, Entropy 18’+unpublished

Ψ

Quantum circuits architecture and parametrization

TNSAA: Live long and prosper

Neural networks and Probabilistic graphical models

TNSAA in the era of quantum computing

Kim and Swingle, 1711.07500

ψi+1 = ψi |0i |0i |0i |0i · · ·

Noise-resilient quantum circuits with TNS architecture

+ quantum tomography, quantum annealing, quantum error correction, holographic duality, and more

Simulation and validation of quantum circuits

Villalonga et al, 1811.09599

Tensor Networks Graphical Models

• Widely used in speech recognition, bioinformatics, cryptography…
• Learning, inference, and sampling algorithms (Forward-backward message

passing, Viterbi, Baum-Welch) => Think of positive MPS

Chen, Cheng, Xie, LW, Xiang, 1701.04831 Critch, Morton, 1210.2812 Robeva, Seigal, 1710.01437

TNSAA in the era of deep learning

Temme, Verstraete,1003.2545

Hidden Markov Model

p(ht|ht−1) p(ot|ht)

… Hidden Observation

Zhao, Jaeger, Neural Comput. 2010 Bailly, J. Mach. Learn. Res. 2011

Unleash the power of tensor network states and algorithms for generative modeling

Discriminative and generative learning

• r

p(x, y)

y = f(x)

p(y|x)

Discriminative and generative learning

“What I can not create, I do not understand”

Generated Arts

https://www.christies.com/Features/A-collaboration-between-two-artists-one-human-one-a-machine-9332-1.aspx

$432,500 25 October 2018 Christie’s New York

Generated Arts

https://www.christies.com/Features/A-collaboration-between-two-artists-one-human-one-a-machine-9332-1.aspx

Gaussian Noise Generative Network

$432,500 25 October 2018 Christie’s New York

Generate Molecules

Physical configurations Latent attributes

Sanchez-Lengeling & Aspuru-Guzik, Science 2018

Simple Distributions Complex Distribution Generate Inference

p(x)

Probabilistic Generative Modeling

How to express, learn, and sample from a high-dimensional probability distribution ?

• “random” images

“natural” images

p(x)

Probabilistic Generative Modeling

How to express, learn, and sample from a high-dimensional probability distribution ?

• “random” images

“natural” images

Page 159 “… the images encountered in AI applications occupy a negligible proportion of the volume of image space.”

p(x)

Probabilistic Generative Modeling

How to express, learn, and sample from a high-dimensional probability distribution ?

https://blog.openai.com/generative-models/

latent space

Boltzmann Machines

statistical physics

p(x) = e−E(x) Z

Generative Modeling using Boltzmann Machines

x1 x2 . . . xN h1 h2 h3 . . . hM

Learn

Negative log-likelihood loss ℒ = − 1 |풟| ∑

x∈풟

ln p(x)

Generative Modeling using Boltzmann Machines

x1 x2 . . . xN h1 h2 h3 . . . hM

Learn

Negative log-likelihood loss ℒ = − 1 |풟| ∑

x∈풟

ln p(x)

Generate

Generative Modeling using Boltzmann Machines

x1 x2 . . . xN h1 h2 h3 . . . hM

Learn

Negative log-likelihood loss ℒ = − 1 |풟| ∑

x∈풟

ln p(x)

Generate

Generative Modeling using Boltzmann Machines

x1 x2 . . . xN h1 h2 h3 . . . hM

Learn

Negative log-likelihood loss

= ⟨E(x)⟩x∼풟 + ln Z

ℒ = − 1 |풟| ∑

x∈풟

ln p(x)

Generate

Generative Modeling using Boltzmann Machines

x1 x2 . . . xN h1 h2 h3 . . . hM

Learn

Negative log-likelihood loss

= ⟨E(x)⟩x∼풟 + ln Z

ℒ = − 1 |풟| ∑

x∈풟

ln p(x) ∇ℒ = ⟨∇E⟩x∼풟 − ⟨∇E⟩x∼p(x)

Generate

Generative Modeling using Boltzmann Machines

x1 x2 . . . xN h1 h2 h3 . . . hM

Learn

Negative log-likelihood loss

= ⟨E(x)⟩x∼풟 + ln Z

ℒ = − 1 |풟| ∑

x∈풟

ln p(x) ∇ℒ = ⟨∇E⟩x∼풟 − ⟨∇E⟩x∼p(x)

Reducing the Dimensionality of Data with Neural Networks

• G. E. Hinton* and R. R. Salakhutdinov

D

• data. A simple and widely used method is

Renaissance of deep learning

Wavefunctions ansatz Quantum state tomography Quantum error correction Monte Carlo updates Renormalization group …… see next talk !

Feedback to physics

State-of-the-Art: Autoregressive Models

p(x) = ∏

i

p(xi|x<i)

1601.06759, 1606.05328

x1 xi xn

xn2 xn2

WaveNet 1609.03499,1711.10433 PixelCNN Speech data Image data

State-of-the-Art: Autoregressive Models

p(x) = ∏

i

p(xi|x<i)

1601.06759, 1606.05328

x1 xi xn

xn2 xn2

WaveNet 1609.03499,1711.10433 PixelCNN Speech data Image data

= p(x1)p(x2|x1)p(x3|x1, x2)⋯

WaveNet in the Real World

https://deepmind.com/blog/wavenet-generative-model-raw-audio/ https://deepmind.com/blog/high-fidelity-speech-synthesis-wavenet/ https://deepmind.com/blog/wavenet-launches-google-assistant/

2018 Google I/O

Born Machines

quantum physics p(x) = |Ψ(x)|2 Z

Born Machines

quantum physics Hilbert Space Area Law Entanglement p(x) = |Ψ(x)|2 Z

Quantum inspired generative modeling

Ψ

Han, Wang, Fan, LW, Zhang, 1709.01662, PRX 18’

Learn

Quantum inspired generative modeling

Ψ

Han, Wang, Fan, LW, Zhang, 1709.01662, PRX 18’

Learn Generate

Quantum inspired generative modeling

Ψ

Han, Wang, Fan, LW, Zhang, 1709.01662, PRX 18’

“Teach a quantum state to write digits”

Learn Generate

Quantum inspired generative modeling

Ψ

Han, Wang, Fan, LW, Zhang, 1709.01662, PRX 18’

Generative modeling using Tensor Network States

Stoudenmire, Schwab NIPS 2016 Stoudenmire Q. Sci. Tech. 2018 Liu et al 1710.04833 Liu et al 1803.09111 Glasser et al 1806.05964 Hallam et al 1711.03357 Gallego, Orus 1708.01525 Pestun et al 1711.01416…

Tensor Network Machine Learning

Cichocki et al 1604.05271,1609.00893,1708.09165… Novikov et al 1509.06569 Kossaifi et al 1707.08308

50 100 150 200 250 300 350 10 20 30 40 50 60 70 Dmax

L

The ultimate goal is “learn to forget”

ℒ = − 1 |풟| ∑

x∈풟

ln p(x)

Lower bound ln(|풟|)

Name of the game

Bond dimension

| |2 /Z

50 100 150 200 250 300 350 10 20 30 40 50 60 70 Dmax

L

The ultimate goal is “learn to forget”

ℒ = − 1 |풟| ∑

x∈풟

ln p(x)

Lower bound ln(|풟|)

Name of the game

Test Train

Bond dimension

| |2 /Z

50 100 150 200 250 300 350 10 20 30 40 50 60 70 Dmax

L

The ultimate goal is “learn to forget”

ℒ = − 1 |풟| ∑

x∈풟

ln p(x)

Lower bound ln(|풟|)

Name of the game

Test Train

Bond dimension

| |2 /Z

Nice features of an MPS Born Machine

Expressibility Learning Inference Sampling

Glasser, Clark, Deng, Gao, Chen, Huang… 17’

p(x) = | |2 /Z

Tractable Likelihood

Efficient & unbiased learning compared to models with intractable partition functions

Z =

tractable via efficient tensor contraction

*applies to TTN and MERA as well

∂Z/∂ ( ) = 2 ×

Adaptive Learning

Adaptively grows the bond dimensions, thus dynamically tuning the expressibility Bond dimensions Training images

*applies to 2-site

• ptimization

Direct Generation

p(x<i) =

No thermalization issue compared to slow mixing Gibbs sampling of Boltzmann Machines

Ferris & Vidal 2012

p(x) = ∏

i

p(xi|x<i) = ∏

i

p(x<i+1) p(x<i)

*applies to TTN and MERA as well

Direct Generation

p(x<i+1) =

No thermalization issue compared to slow mixing Gibbs sampling of Boltzmann Machines

Ferris & Vidal 2012

p(x) = ∏

i

p(xi|x<i) = ∏

i

p(x<i+1) p(x<i)

*applies to TTN and MERA as well

Direct Generation

p(x<i+2) =

No thermalization issue compared to slow mixing Gibbs sampling of Boltzmann Machines

Ferris & Vidal 2012

p(x) = ∏

i

p(xi|x<i) = ∏

i

p(x<i+1) p(x<i)

*applies to TTN and MERA as well

Direct Generation

p(x<i+2) =

“Zipper Sampling” No thermalization issue compared to slow mixing Gibbs sampling of Boltzmann Machines

Ferris & Vidal 2012

p(x) = ∏

i

p(xi|x<i) = ∏

i

p(x<i+1) p(x<i)

*applies to TTN and MERA as well

Image Restoration

Han, Wang, Fan, LW, Zhang, 1709.01662, PRX 18’

Arbitrary order, in contrast to autoregressive models PixelCNN

Image Restoration

Han, Wang, Fan, LW, Zhang, 1709.01662, PRX 18’

?

Quantum Perspective on Deep Learning

(a)

Quantum Perspective on Deep Learning

Q: How to quantify our inductive biases ? A: Information pattern of probability distributions

(a)

Quantum Perspective on Deep Learning

Q: How to quantify our inductive biases ? A: Information pattern of probability distributions

(a) (a)

Quantum Perspective on Deep Learning

Q: How to quantify our inductive biases ? A: Information pattern of probability distributions

p(

) p( )

p(

) p( )

(b)

×

(a)

×

Quantum Perspective on Deep Learning

Classical mutual information Quantum Renyi entanglement entropy

Striking similarity implies common inductive bias

Cheng, Chen, LW, 1712.04144, Entropy 18’

+Quantitative & interpretable approaches +Principled structure design & learning

Quantum Perspective on Deep Learning

I = − ⟨ln ⟨ p(x, y′)p(x′, y) p(x′, y′)p(x, y)⟩x′,y′⟩

x,y

S = − ln ⟨⟨ Ψ(x, y′)Ψ(x′, y) Ψ(x′, y′)Ψ(x, y)⟩x′,y′⟩

x,y

DEEP LEARNING AND QUANTUM ENTANGLEMENT: FUNDAMENTAL CONNECTIONS WITH IMPLICATIONS

TO NETWORK DESIGN

ABSTRACT

• networks. Specifically, we show a structural equivalence between the function re-

1 INTRODUCTION

DEEP LEARNING AND QUANTUM ENTANGLEMENT: FUNDAMENTAL CONNECTIONS WITH IMPLICATIONS

TO NETWORK DESIGN

ABSTRACT

• networks. Specifically, we show a structural equivalence between the function re-

1 INTRODUCTION

Mean Field Theory

Physicists’ gifts to Machine Learning

U

Quantum Computing Tensor Networks Monte Carlo Methods

Ψ

Quantum Circuit Born Machine

Train the quantum circuit as a probabilistic generative model Quantum sampling complexity underlines the “quantum supremacy”

With Liu, Zeng, Wu, Hu 1804.04168, 1808.03425

Qubit efficient scheme

Product state Measured qubits

see also Cramer et al, Nat. Comm. 2010

Tensor network inspired quantum circuit architecture

Qubit efficient scheme

Product state Measured qubits

see also Cramer et al, Nat. Comm. 2010

Qubit efficient scheme

Product state Measured qubits

see also Cramer et al, Nat. Comm. 2010

Reuse!

Qubit efficient scheme

Product state Measured qubits

(a)

see also Cramer et al, Nat. Comm. 2010

Reuse!

Qubit efficient scheme

Product state Measured qubits

(a)

D = 23

see also Cramer et al, Nat. Comm. 2010

Reuse!

MPS with exponentially large bond dimensions

W e!, back to Boltzmann Machines

Quantitative Performance

PixelCNN RBM ~84.6 DBM 81.3 Binarized MNIST test NLL, lower is better* ~86.3

*Lower test NLL may not imply better sample quality, Theis et al, 1511.01844

Quantitative Performance

PixelCNN RBM ~84.6 DBM 81.3 Binarized MNIST test NLL, lower is better* ~86.3

*Lower test NLL may not imply better sample quality, Theis et al, 1511.01844

MPS (D=100) 101.5

Han et al, PRX 18’

Quantitative Performance

PixelCNN RBM ~84.6 DBM 81.3 Binarized MNIST test NLL, lower is better* ~86.3

*Lower test NLL may not imply better sample quality, Theis et al, 1511.01844

TTN (D=50) 94.3

With Song Cheng & Pan Zhang, unpublished

(a)

MPS (D=100) 101.5

Han et al, PRX 18’

“Positive” PEPS

p(x) =

/Z

= ∑

σ

σ Probability model

• Smaller truncation error
• No need for double layer tensor

Why positive? aka hidden Markov random field model

p(x) = ∑

h ∏ i

f i(xi, h)/Z

“Positive” PEPS

p(x) =

/Z

= ∑

σ

σ

ℒ = − ⟨ln ( )⟩x∼풟 + ln ( )

Contract for each given sample Contract once Loss function Probability model

• Smaller truncation error
• No need for double layer tensor

Why positive? aka hidden Markov random field model

p(x) = ∑

h ∏ i

f i(xi, h)/Z

“Positive” PEPS

p(x) =

/Z

= ∑

σ

σ

ℒ = − ⟨ln ( )⟩x∼풟 + ln ( )

Contract for each given sample Contract once Loss function Probability model Contract 28x28 PEPS 60000+1 times per epoch!

• Smaller truncation error
• No need for double layer tensor

Why positive? aka hidden Markov random field model

p(x) = ∑

h ∏ i

f i(xi, h)/Z

Quantitative Performance

PixelCNN RBM ~84.6 DBM 81.3 ~86.3 MPS (D=100) 101.5 TTN (D=50) 94.3 Binarized MNIST test NLL, lower is better

ℒ = − ⟨ln ( )⟩x∼풟 + ln ( )

Contract for each given sample Contract once Loss function Contract 28x28 PEPS 60000+1 times per epoch!

Quantitative Performance

PixelCNN RBM ~84.6 DBM 81.3 ~86.3 MPS (D=100) 101.5 TTN (D=50) 94.3 PEPS+ (D=5) 92 Binarized MNIST test NLL, lower is better

ℒ = − ⟨ln ( )⟩x∼풟 + ln ( )

Contract for each given sample Contract once Loss function Contract 28x28 PEPS 60000+1 times per epoch!

With Song Cheng & Pan Zhang, unpublished

Quantitative Performance

PixelCNN RBM ~84.6 DBM 81.3 ~86.3 MPS (D=100) 101.5 TTN (D=50) 94.3 PEPS+ (D=5) 92 Binarized MNIST test NLL, lower is better

Quantitative Performance

PixelCNN RBM ~84.6 DBM 81.3 ~86.3 MPS (D=100) 101.5 TTN (D=50) 94.3 PEPS+ (D=5) 92 Binarized MNIST test NLL, lower is better

Bonus: What does deep learning offer to tensor network states & algorithms?

Differentiable tensor libraries with GPU/TPU/FPGA/… support

ℒ Loss

∂ℒ ∂xℓ = ∂ℒ ∂xℓ+1 ( ∂xℓ+1 ∂xℓ )

… …

The engine of deep learning: Back-Propagation algorithm

Jacobian-Vector Product

Computes gradients efficiently & accurately Via reverse mode automatic differentiation

f` f`+1 x` x`+1 x`+2

computation graph

• Computationally homogeneous

Benefits

• Simple to bake into silicon
• Constant running time
• Constant memory usage
• Highly portable & agile
• Modules can meld into an optimal whole
• Better than humans

Andrej Karpathy

Differentiable Programming

Writing software 2.0 by gradient search in the program space

Differentiable Scientific Programming

• Most linear algebra operations (Eigen, SVD!) are differentiable
• Loop/Condition/Sort/Permutations are also differentiable
• Differentiable ray tracer
• Differentiable Monte Carlo/Tensor Network/Functional RG/

Dynamical Mean Field Theory/Density Functional Theory…

• ODE integrators are differentiable with O(1) memory

Differentiable fluid simulations and

Differentiable programming is more than training neural networks

Differentiable Eigensolver

HΨ = ΨΛ

Forward mode: What happen if H = H + dH Perturbation theory Reverse mode: How should I change ? ∂ℒ/∂Ψ ∂ℒ/∂Λ and ? Inverse perturbation theory

H given

Hamiltonian engineering via differentiable programming

https://github.com/wangleiphy/DL4CSRC/tree/master/2-ising

Differentiable Levin-Nave TRG

computation graph ln Z β

https://github.com/ wangleiphy/TRG

SVD w/ truncation contraction

Automatic differentiation for high order gradients

• Exact gradient
• Not finite-difference
• No additional codes
• Efforts comparable

to the forward pass

Z = ∑

{σ}

exp ( 1 2 Kijσiσj)

Contraction ln Z

Differentiable spin glass solver

random couplings Kij

Wang, Qin, Zhou, PRB 2014 Rams et al, 1811.06518 TNS studies of Ising spin glasses:

Z = ∑

{σ}

exp ( 1 2 Kijσiσj)

Contraction ln Z

Differentiable spin glass solver

random couplings Kij

Wang, Qin, Zhou, PRB 2014

d ln Z dKij = ⟨σiσj⟩

Differentiation correlations

Rams et al, 1811.06518 TNS studies of Ising spin glasses:

Z = ∑

{σ}

exp ( 1 2 Kijσiσj)

Contraction ln Z

Differentiable spin glass solver

random couplings Kij

Wang, Qin, Zhou, PRB 2014

d ln Z dKij = ⟨σiσj⟩

Differentiation correlations

Rams et al, 1811.06518 TNS studies of Ising spin glasses:

Tensor network solver for inverse Ising problems Gradient descend

Gradient based variational optimization

Human only cares about tensor contraction Differentiable programing takes care of the gradients

Vanderstraeten et al, PRB 16’

ℒ = ln⟨Ψ| ̂ H|Ψ⟩ − ln ⟨Ψ|Ψ⟩

ℒ = ⟨E(x)⟩x∼풟 + ln Z

Gradient optimization works fine Generative Modeling Variational ground state, why not ?

Pan Zhang Song Cheng Jing Chen Tao Xiang Zhiyuan Xie Jin-Guo Liu

Thank You!