Tensor Network Representation for Machine Learning - Recent Advances - - PowerPoint PPT Presentation

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Tensor Network Representation for Machine Learning - Recent Advances - - PowerPoint PPT Presentation

Tensor Network Representation for Machine Learning - Recent Advances and Perspectives Qibin ZHAO Tensor Learning Unit RIKEN AIP AIP Symposium (Mar. 19, 2019) Tensor Learning Unit - Members Postdoctoral Researchers (2) Ming Hou, Chao Li


slide-1
SLIDE 1

Tensor Network Representation for Machine Learning - Recent Advances and Perspectives

Qibin ZHAO Tensor Learning Unit RIKEN AIP AIP Symposium (Mar. 19, 2019)

slide-2
SLIDE 2

2

Tensor Learning Unit - Members

Postdoctoral Researchers (2)

  • Ming Hou, Chao Li

Part-timer (2)

  • Longhao Yuan (PhD student), Xuyang Zhao (PhD student)

Interns (4)

  • Canada, Japan, China

Visitors (9)

  • Andrzej Cichocki, Toshihisa Tanaka, Jianting Cao
  • Guillaume Rabusseau, Justin Dauwels, Danilo Mandic, Brahim Chib-

draa, Cesar F. Caiafa, Jordi Sole Casals

slide-3
SLIDE 3

3

Background and Problems

Kernel learning

  • Problems become easier when mapping to higher

dimensional space.

  • Curse of dimensionality, grows exponentially
  • Weights can be exponentially big
  • “kernelization” scales quadratically with training set
  • size. In the era of big data, this issue is cited as one

reason why neural nets have overtaken kernel methods.

  • Low generalization due to representer theorem

Kernel Learning

Perfect Problem for Tensor Networks to solve

f(x)

W · Φ(x)

= Rank-1 tensor

em says exact W = X

j

αjΦ(xj)

slide-4
SLIDE 4

4

Background and Problems

Neural Networks

  • Weight matrix is huge but highly redundant.
  • Low-rank compression: limited compression rate
  • Computational inefficient due to huge parameters
  • Not applicable for small devices

Multi-modal deep learning, multi-task deep learning

Neural Nets

Tensor Networks is a natural tool to solve these problems

f(x) = Φ2 ⇣ M2Φ1

  • M1x

slide-5
SLIDE 5

5

Neural Network (NN) vs. Tensor Network (TN)

Similarity

  • Assembling simple units (neurons or tensors) into complicated

functions

Difference

  • Decision functions in ML vs. wavefunctions in quantum mechanics
  • Nonlinear in NN vs. linear in TN
  • NN do non-linear things to low-dimensional space vs. TN do linear

things in high-dimensional space

slide-6
SLIDE 6

6

What Are Tensor Networks (TNs) ?

  • A powerful tool to describe strongly entangled quantum

many-body systems in physics

  • Decompose a high-order tensor into a collection of low-
  • rder tensors connected according to a network pattern
  • Tensor network diagram

a Scalar a Vector I I A Matrix A I J I J 3rd-order tensor A I1 I2 I3 I1 I2 I3 Λ R R R R R R 3rd-order diagonal tensor

  • Λ

x I J A

=

I b=Ax B I J A

=

I C=AB K K B J K A M I P L

=

C

  • K

k =1

ai,j,k bk,l,m,p = ci,j,l,m,p

J I M P L

slide-7
SLIDE 7

7

TT/MPS Representation and Properties

  • Efficient to represent data values by

parameters

  • Efficient to compute or optimize TT/MPS by DMRG

algorithm

I1 . . . R1 R2 R1 I2

(2)

G

(1)

Gi1

( ) n

G

( ) N

G R1 R2 I2 IN I1 In G(2) G( )

n

G(1) G( )

N

Rn-1 Rn RN-1

... ...

Rn-1 In Rn RN-1 IN . . . ... . . . . . . . . . ... . . . i2 in

N

i

O(NIR2)

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IN

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TT: tensor train decomposition; MPS: matrix product state

[V. Oseledets, SIAM J. Sci. Comput., 2011]

slide-8
SLIDE 8
  • Input:
  • Nonlinear mapping by tensor product (Hilbert space)
  • Decision function - is an Nth-order tensor
  • TT representation of weight parameter

x = [x1, x2, x3, . . . , xN]

8

TNs for Weight Compression & Kernel Learning

s1 s2 s3 s4 s5 s6

= φs1 φs2 φs3 φs4 φs5 φs6 · · · sN φsN Φ(x)

[E. Stoudenmire, NIPS 2016]

=

sj

[ [

1 xj

Φ(x) W

=

f(x)

W · Φ(x)

=

[A. Novikov, NIPS 2015]

W

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=

Φ(x) W f(x)

W =

Space

2N

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slide-9
SLIDE 9
  • Optimization algorithm scaling:

m: TT rank, : Sample size

  • Without “kernel trick”, avoiding scaling problem
  • Without deep layers transformation
  • Feature sharing for multi-class function
  • Adaptive learning

=

f `(x) `

9

TNs for Weight Compression & Kernel Learning

[E. Stoudenmire, NIPS 2016]

O(NNT m3)

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NT

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N 2

T

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Decision function f `(x) = W ` · Φ(x)

slide-10
SLIDE 10

Fundamental theoretical questions:

  • Are deep networks efficient w.r.t. shallow one for

ConvNets?

  • What kind of func can different network arch

represent?

  • What is the inductive bias of conv/pool window

geometry?

  • Do overlapping operations introduce efficiency?
  • Can connectivity scheme be justified in terms of

efficiency?

10

Theoretical Analysis of ConvNets

DenseNet ResNet

ResNet

slide-11
SLIDE 11

11

Relations Between TNs and DNNs

  • Equivalence of Restricted Boltzmann Machines and Tensor Networks
  • Equivalence of Deep Convolutional Network and Hierarchical Tucker

Decomposition

  • Recurrent Neural Networks and Tensor Train
  • Powerful tools to study theory behind DNN

TNS RBM

Sec II Sec III

  • ,
d i

rep i d f

  • x

input representation 1x1 conv pooling 1x1 conv pooling dense (output)

hidden layer 0 hidden layer L-1 (L=log2N)

i

x M r r

1 L

r

1 L

r Y

  • 0, ,

, , ,:

j

conv j rep j

  • a
  • '
2 1,2

, ',

j j j

pool j conv j

  • 1
1 ' 1,2

',

L L j

pool conv j

  • ,1
1 , ,

:

L y L
  • ut y

pool a X

corresponds to Hierarchical Tucker (HT) decomposition:

Tensor Decompositions Deep Learning CP-decomposition shallow network TT-decomposition RNN HT-decomposition CNN rank of the decomposition width of the network G2 G1 G3 Gd

fθ(x1)

f✓(x2) f✓(x3) f✓(xd) z1 z2 zd−1

ly(X)

[Chen et al, Physical Review B, 2018] [N. Cohen & A. Shashua, ICML 2016] [Khrulkov, ICLR 2018]

network structure ← → decomposition type (depth, width, pooling etc) (dim tree, internal ranks etc) network weights ← → decomposition parameters

[Carleo et al, Science, 2017]

slide-12
SLIDE 12

12

Tensor Networks for Large-Scale Optimization Problems

  • Fast ALS/DMRG algorithm
  • Applicable to large-scale

SVD/PCA/CCA and etc

R1 P1 R1 I1 I1 In-1 In-1 Rn -1 Pn-1 Rn-1 Rn Pn Rn Rn+1 Pn+1 Rn+1 In In In+1 In+1 IN IN In+2 In+2 Rn+2 Pn+2 Rn+2

}

x A x

T

A(1) A( -1)

n

A( )

n

A( +1)

n

A( +2)

n

A( )

N

X( )

n

} }

X(1) X( -1)

n

X( +1)

n

X( +2)

n

X( )

N

X(1) X( -1)

n

X( +1)

n

X( +2)

n

X( )

N

X( )

n

Eigenvalue problem:

max xT Ax

X( )

n

X( )

n

x ( )

n

A( )

n

_

x ( )

n

Rn -1 In Rn Rn -1 In Rn In In

a

I I1 = I2…IN … … … I1 I2 I3 IN A … I I1 = I2…IN J J1 = J2…JN … I1 I2 I3 IN J1 J2 J3 JN

  • TT format of a large matrix
  • TT format of a large vector
slide-13
SLIDE 13

13

Research Scheme

  • Study the fundamental principle of tensor networks
  • Investigate tensor networks for data representation
  • Investigate tensor networks for model representation
  • Explore the potential applications of tensor methods
slide-14
SLIDE 14

14

Fundamental Tensor Network Model

TT representation

  • Powerful but still some limitations
  • TT-ranks of middle cores are large

Tensor ring representation

  • Generalized TT without constraints on boundary cores
  • Sum of TT with shared core tensors
  • Efficient computation for multilinear operations
  • Highly expressive model

T n1 nd · · · nk · · · n2

=

Z1 Zd · · · Zk · · · Z2 n1 nd · · · nk · · · n2 r1 r2 rd rk+1 rk r3

Tensor Ring

…IN … … … I1 I2 I3 IN

as xi1,i2,...,iN tr G 1

i1

G 2

i2

G N

iN

Tensor Train [Zhao et al, ICLR workshop 2018, ICASSP 2019]

slide-15
SLIDE 15

15

Tensor Networks for Data Representation

Real data is often high-dimensional

  • Recommender system (user x item x time)
  • Gene expression, remote sensing, fMRI

Real data is often incomplete

  • Low-rank approximation via convex optimization (high computation cost)
  • Decomposition based approach (model selection problem)
  • How much structure information can be used?
slide-16
SLIDE 16

16

Tensor Networks for Data Imputation

Tensor completion based on TT/TR decomposition

Recovered data Approximated entries P r e d i c t i

  • n

Observed entries Observed data High-order 
 tensorization

i2 = 1 = 1 i2 = 2 = 2 i2 = 3 = 3 i2 = 4

Higher-order tensor Low-rank TT/TR approximation TT/TR decomposition

= 1 , · · · , N

  • r

= 1 , · · · , N

slide-17
SLIDE 17

17

From Tensorization to Linear Transformation

In the simplest case, the completion problem can be solved by the following optimization problem:

min

X2Rm1×m2 kQ(X)k⇤

s.t. kPΩ(X) PΩ(Y)kF  δ, (5)

Linear transformation

With mild conditions, the solution of the above problem obeys

k ˆ M M0kF  2δ · cond(Q) 1 kRΛk2 s min{n1, n2}(p + k [Q]h2i k2

2)

p .

ˆ M

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Estimation

M0

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Ground truth

cond(·)

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Condition number

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A matrix related to dual certificate

[Chao et al, CVPR’19]

slide-18
SLIDE 18

18

Beyond Unfolding: Reshuffling Operation

(a) a 3rd-order tensor with the size 3x3x3

Mode-1 Unfolding Reshuffling

(b) The unfolded matrix X along the frist mode (c) The matrix Y by the propsoed tensor reshuffling mode-1 mode-3 mode-2

The correspondence mode-1

mode-2 mode-2 mode-2 mode-3 high rank low rank

“Restore the Rubix Cube” Mode-3 Unfolding

  • Fig. Difference between tensor unfolding and reshuffling.
slide-19
SLIDE 19

19

Reshuffled Tensor Decomposition

Image steganography is to hide a secret image into cover image

strength para. container cover secret

  • ur method

reconstruction Random permutation

  • Fig. 14 Block diagram of the system scheme used in the experiment.

min

Ai, i∈[N] N

ÿ

i=1

ÎAiÎ∗, s.t., X =

N

ÿ

i=1

Ri(Ai),

slide-20
SLIDE 20

20

Tensor Networks for Model Representation

Deep Multi-task Learning

  • Cannot handle data from

multiple sources/modalities

  • Cannot deal with

heterogeneous network for individual task

  • Lack flexibility in knowledge-

sharing mechanism

TN Prior TN Prior

Task A Task B

DMTRL-T MRN

[Long et al. NIPS 2017]

Task A Task B

ucker DMTRL-TT

[Yang et al, ICLR 2017]

slide-21
SLIDE 21

21

Tensor Ring Multi-task Learning

  • Heterogeneous DNN for each task
  • Subset of TR-cores are shared among

tasks

  • Flexibility in knowledge-sharing pattern
  • High efficiency by sharing information in

latent space

  • Disadvantages: choosing the best cores

for sharing is difficult.

Task A Task B

TRMTL

slide-22
SLIDE 22

22

AI Support for Epileptic Diagnosis

Brain signal (EEG) examination (surface) Drug therapy to control seizures Brain signal (iEEG) examination (inside) Surgical removal Need to record data for one week Not effective for all patients Need Craniotomy and long time record Difficult to determine the location of remove

Challenging problems

  • Need special doctors, only about 600 eligible doctors in Japan.
  • Need several weeks high-quality iEEG data.
  • Time-consuming by several doctors’ visual judgment.
  • Focal detection is not reliable.

順天堂

slide-23
SLIDE 23

23

AI Support for Epileptic Diagnosis

  • Mission: Automatic localization of epileptic focal from iEEG

signals as a support technology for doctors 順天堂

  • High accuracy

Entropies of different frequency bands for feature extraction and CNN for classification

  • End to end model

Discovery of iEEG focal without handcraft feature extraction

  • Less labels

Only need a few labelled data by PU learning [Prof. Sugiyama’s PU algorithms]

Positive Unlabeled PU learning

Health zone

Epileptic zone

Focal

slide-24
SLIDE 24

24

Future Work: Tensor Network for Graphical Model

  • CP decomposition
  • Markov random field models as tensor train
  • Undirected graphical model represented as a TT model

H X1 X2 Xn

· · ·

H1 H2 H3 Hn X1 X2 X3 Xn

· · · HMM

(a) (b) (c) (d)

[Glasser et al, 2018]

P(x1, x2, x3, x4) =

  • h

P(x1|h)P(x2|h)P(x3|h)P(x4|h)P(h)

PEPS

[Novikov et al., ICML 2014]

10 variables, 10 states each −→ 1010 entries

MERA

slide-25
SLIDE 25

25

Future Work - Acceleration of Tensor Convolution

  • High-order convolution (computation and storage)
  • Fast convolution via tensor network representation

N

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N

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N

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N

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· · ·

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N

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N

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N

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N

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=

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2N + 1

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2N + 1

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2N + 1

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2N + 1

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O(N 2d)

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N

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N

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· · ·

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N

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N

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N

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N

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N

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N

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r

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r

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r

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2N + 1

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2N + 1

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2N + 1

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2N + 1

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=

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2N + 1

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2N + 1

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2N + 1

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2N + 1

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O(N dr4)

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core-based conv.

O(dNr4)

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slide-26
SLIDE 26

26

Collaborations within AIP

  • A novel schema for hyper-spectral image restoration
  • Dementia detection via tensorizing neural networks
  • Gene data completion via tensor network

[He et al., CVPR2019] [Ruikowski et al., NeurlPS 2018 workshop] [Iwata et al., ISMB/ECCB 2019]

Naoto Yokoya Mihoko Otake Yasuo Tabei

slide-27
SLIDE 27

27

Summary

  • Tensor networks are intriguing alternative to

traditional machine learning models

  • Better scaling, efficient algorithms, opportunities for

theoretical insights

  • Promising as a framework for machine learning with

quantum computing

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SLIDE 28

28

Achievements in FY2018

Publications (32)

  • Conference (19) including AAAI, IJCAI, CVPR, ICASSP, NeurIPS

Workshop, ICLR workshop and etc

  • Journal (13) including IEEE TNNLS, Signal Processing and etc

Awards

  • The 3rd IEEE SPS Japan Best Paper Award
  • 2018 SPS Signal Processing Magazine Best Paper