deep learning, neural networks and kernel machines ... · generalization, deep learning and kernel...

Deep Learning, Neural Networks and Kernel

Machines: towards a unifying framework

Johan Suykens

KU Leuven, ESAT-STADIUSKasteelpark Arenberg 10

B-3001 Leuven (Heverlee), BelgiumEmail: [email protected]://www.esat.kuleuven.be/stadius/

AI Seminar at BeCentral Brussels, Oct 2019

Outline

• Introduction

• Function estimation, model representations, duality

• Neural networks and kernel machines

• Application examples, large scale methods

• Robustness

• Generative models: GAN, RBM, Deep BM

• Restricted kernel machines (RKM), Gen-RKM, and deep learning

• Explainability

• Recent developments

1

Introduction

1

Self-driving cars and neural networks

in the early days of neural networks:

ALVINN (Autonomous Land Vehicle In a Neural Network)

[Pomerleau, Neural Computation 1991]

2

Self-driving cars and deep learning

(27 million connections)

from: [selfdrivingcars.mit.edu (Lex Fridman et al.), 2017]

3

Convolutional neural networks

[LeCun et al., Proc. IEEE 1998]

Further advanced architectures:

Alexnet (2012): 5 convolutional layers, 3 fully connectedVGGnet (2014): 19 layersGoogLeNet (2014): 22 layersResNet (2015): 152 layers

4

Historical context

1942 McCulloch & Pitts: mathematical model for neuron1958 Rosenblatt: perceptron learning1960 Widrow & Hoff: adaline and lms learning rule1969 Minsky & Papert: limitations of perceptron

1986 Rumelhart et al.: error backpropagating neural networks→ booming of neural network universal approximators

1992 Vapnik et al.: support vector machine classifiers→ convex optimization, kernel machines

1998 LeCun et al.: convolutional neural networks2006 Hinton et al.: deep belief networks2010 Bengio et al.: stacked autoencoders

→ booming of deep neural networks

com

putin

g po

wer

4

Different paradigms

Deep

Learning

Neural

Networks

SVM, LS-SVM &

Kernel methods

5

Different paradigms

Deep

Learning

Neural

Networks

SVM, LS-SVM &

Kernel methods

new synergies?

5

Towards a unifying picture

Model

Dual representation

Primal representation

Duality principle

other

Legendre−Fenchel duality

Lagrange duality

Conjugate feature duality

Kernel−based

other

Parametric

linear, polynomial

finite or infinite dictionary

positive definite kernel

tensor kernel

indefinite kernel

symmetric or non−symmetric kernel

(deep) neural network

[Suykens 2017]

6

multi−scale

multiplex

data fusion

ensembledeep

multi−task

[Suykens 2017]

7

Function estimation and model representations

7

Linear function estimation (1)

• Given (xi, yi)Ni=1 with xi ∈ Rd, yi ∈ R, consider y = f(x) where f is

parametrized asy = wTx+ b

with y the estimated output of the linear model.

• Consider estimating w, b by

minw,b

1

2wTw + γ

1

2

N∑

i=1

(yi − wTxi − b)2

→ one can directly solve in w, b

8

Linear function estimation (2)

• ... or write as a constrained optimization problem:

minw,b,e

12w

Tw + γ12

∑

i e2i

subject to ei = yi − wTxi − b, i = 1, ..., N

Lagrangian: L(w, b, ei, αi) = 12w

Tw + γ 12

∑

i e2i −

∑

i αi(ei − yi + wTxi + b)

• From optimality conditions:

y =∑

i

αi xTi x+ b

where α, b follows from solving a linear system[

0 1TN1N Ω + I/γ

] [

b

α

]

=

[

0

y

]

with Ωij = xTi xj for i, j = 1, ..., N and y = [y1; ...; yN ].

9

Linear model: solving in primal or dual?

inputs x ∈ Rd, output y ∈ R

training set (xi, yi)Ni=1

(P ) : y = wTx+ b, w ∈ Rd

րModel

ց(D) : y =

∑

i αi xTi x+ b, α ∈ R

N

10


few inputs, many data points: d≪ N

primal : w ∈ Rd

dual: α ∈ RN (large kernel matrix: N ×N)

11


many inputs, few data points: d≫ N

primal: w ∈ Rd

dual : α ∈ RN (small kernel matrix: N ×N)

11

Feature map and kernel

From linear to nonlinear model:

(P ) : y = wTϕ(x) + bր

Modelց

(D) : y =∑

i αiK(xi, x) + b

Mercer theorem:K(x, z) = ϕ(x)Tϕ(z)

Feature map ϕ, Kernel function K(x, z) (e.g. linear, polynomial, RBF, ...)

• SVMs: feature map and positive definite kernel [Cortes & Vapnik, 1995]

• Explicit or implicit choice of the feature map

• Neural networks: consider hidden layer as feature map [Suykens & Vandewalle, 1999]

• Least squares support vector machines [Suykens et al., 2002]: L2 loss and regularization

12

Least Squares Support Vector Machines: “core models”

• Regression

minw,b,e

wTw + γ∑

i

e2i s.t. yi = wTϕ(xi) + b+ ei, ∀i

• Classification

minw,b,e

wTw + γ∑

i

e2i s.t. yi(wTϕ(xi) + b) = 1− ei, ∀i

• Kernel pca (V = I), Kernel spectral clustering (V = D−1)

minw,b,e

−wTw + γ∑

i

vie2i s.t. ei = wTϕ(xi) + b, ∀i

• Kernel canonical correlation analysis/partial least squares

minw,v,b,d,e,r

wTw + vTv + ν∑

i

(ei − ri)2 s.t.

ei = wTϕ(1)(xi) + bri = vTϕ(2)(yi) + d

[Suykens & Vandewalle, 1999; Suykens et al., 2002; Alzate & Suykens, 2010]

13

Sparsity: through regularization or loss function

• through regularization: model y = wTx+ b

min∑

j

|wj|+ γ∑

i

e2i

⇒ sparse w (e.g. Lasso)

• through loss function: model y =∑

i αiK(x, xi) + b

min wTw + γ∑

i

L(ei)

⇒ sparse α (e.g. SVM)

−ε 0 +ε

14

SVMs and neural networks

x

oooo

x

xx

x

x x

x

x

x

oo

oo Input space

Feature space

ϕ(x)

Parametric

Nonparametric

Primal space

Dual space

y = sign[wTϕ(x) + b]

y = sign[∑#sv

i=1 αiyiK(x, xi) + b]

K(xi, xj) = ϕ(xi)Tϕ(xj) (Mercer)

y

y

w1

wnh

α1

α#sv

ϕ1(x)

ϕnh(x)

K(x, x1)

K(x, x#sv)

x

x

15

SVMs and neural networks

x

oooo

x

xx

x

x x

x

x

x

oo

oo Input space

Feature space

ϕ(x)

Parametric

Nonparametric

Primal space

Dual space

y = sign[wTϕ(x) + b]

y = sign[∑#sv

i=1 αiyiK(x, xi) + b]

K(xi, xj) = ϕ(xi)Tϕ(xj) (“Kernel trick”)

y

y

w1

wnh

α1

α#sv

ϕ1(x)

ϕnh(x)

K(x, x1)

K(x, x#sv)

x

x

Par

amet

ricN

on−p

aram

etric

[Suykens et al., 2002]

15

Wider use of the “kernel trick”

• Angle between vectors: (e.g. correlation analysis)Input space:

cos θxz =xTz

‖x‖2‖z‖2Feature space:

cos θϕ(x),ϕ(z) =ϕ(x)Tϕ(z)

‖ϕ(x)‖2‖ϕ(z)‖2=

K(x, z)√

K(x, x)√

K(z, z)

• Distance between vectors: (e.g. for “kernelized” clustering methods)Input space:

‖x− z‖22 = (x− z)T (x− z) = xTx+ zTz − 2xTz

Feature space:

‖ϕ(x)− ϕ(z)‖22 = K(x, x) +K(z, z)− 2K(x, z)

x

zθ

16

Interpretation of kernel-based models

Decision making: classification problem (e.g. apples versus tomatoes) Inputdata xi ∈ R

d and class labels yi ∈ −1,+1. N training data.

SVM or LS-SVM classifier: given a new x (e.g. ), obtain

y = sign[∑

i

αiyiK(x, xi) + b]

with xi for i = 1, ..., N :

Here K(x, xi) characterizes the similarity between x and xi.The bias term b can be related to prior class probabilities.

17

Function estimation in RKHS

• Find function f such that [Wahba, 1990; Evgeniou et al., 2000]

minf∈HK

1

N

N∑

i=1

L(yi, f(xi)) + λ‖f‖2K

with L(·, ·) the loss function. ‖f‖K is norm in RKHS HK defined by K.

• Representer theorem: for convex loss function, solution of the form

f(x) =N∑

i=1

αiK(x, xi)

Reproducing property f(x) = 〈f,Kx〉K with Kx(·) = K(x, ·)

• Sparse representation by hinge and ǫ-insensitive loss [Vapnik, 1998]

18

Kernels

Wide range of positive definite kernel functions possible:

- linear K(x, z) = xTz- polynomial K(x, z) = (η + xTz)d

- radial basis function K(x, z) = exp(−‖x− z‖22/σ2)- splines- wavelets- string kernel- kernels from graphical models- kernels for dynamical systems- Fisher kernels- graph kernels- data fusion kernels- additive kernels (good for explainability)- other

[Scholkopf & Smola, 2002; Shawe-Taylor & Cristianini, 2004; Jebara et al., 2004; other]

19

Krein spaces: indefinite kernels

• LS-SVM for indefinite kernel case:

minw+,w−,b,e

1

2(wT+w+ − wT−w−) +

γ

2

N∑

i=1

e2i s.t. yi = wT+ϕ+(xi) + wT−ϕ−(xi) + b+ ei, ∀i

and indefinite kernel K(xi, xj) = K+(xi, xj)−K−(xi, xj)with positive definite kernels K+, K−

K+(xi, xj) = ϕ+(xi)Tϕ+(xj) and K−(xi, xj) = ϕ−(xi)

Tϕ−(xj)

• also: KPCA with indefinite kernel [X. Huang et al. 2017], KSC andsemi-supervised learning [Mehrkanoon et al., 2018]

[X. Huang, Maier, Hornegger, Suykens, ACHA 2017]

[Mehrkanoon, X. Huang, Suykens, Pattern Recognition, 2018]

Related work of RKKS: [Ong et al 2004; Haasdonk 2005; Luss 2008; Loosli et al. 2015]

20

Banach spaces: tensor kernels

• Regression problem:

min(w,b,e)∈ℓr(K)×R×RN

ρ(‖w‖r) + γN

∑Ni=1L(ei)

subject to yi = 〈w,ϕ(xi)〉+ b+ ei , ∀i = 1, ..., N

with r = mm−1 for even m ≥ 2, ρ convex and even.

For m large this approaches ℓ1 regularization.

• Tensor-kernel representation

y = 〈w,ϕ(x)〉r,r∗ + b =1

Nm−1

N∑

i1,...,im−1=1

ui1...uim−1K(xi1, ..., xim−1

, x) + b

[Salzo & Suykens, arXiv 1603.05876; Salzo, Suykens, Rosasco, AISTATS 2018]

related: RKBS [Zhang 2013; Fasshauer et al. 2015]

21

Generalization, deep learning and kernel methods

Recently one has observed in deep learning that over-parametrized neuralnetworks, that would ”overfit”, may still perform well on test data.This phenomenon is currently not yet fully understood. A number ofresearchers have stated that understanding kernel methods in thiscontext is important for understanding the generalization performance.

Related references:

• Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, Oriol Vinyals,

Understanding deep learning requires rethinking generalization, 2016, arXiv:1611.03530

• Amit Daniely, SGD Learns the Conjugate Kernel Class of the Network, 2017,

arXiv:1702.08503

• Arthur Jacot, Franck Gabriel, Clement Hongler, Neural Tangent Kernel: Convergence

and Generalization in Neural Networks, 2018, arXiv:1806.07572

• Tengyuan Liang, Alexander Rakhlin, Just Interpolate: Kernel ”Ridgeless” Regression

Can Generalize, 2018, arXiv:1808.00387

• Mikhail Belkin, Siyuan Ma, Soumik Mandal, To understand deep learning we need to

understand kernel learning, 2018, arXiv:1802.01396

22

Generalization and deep learning - Double U curve

Figure: Mikhail Belkin, Daniel Hsu, Siyuan Ma, Soumik Mandal, Reconciling modern

machine learning and the bias-variance trade-off, 2018, arXiv:1812.11118

23

Example: Black-box weather forecasting (1)

Weather data350 stations located in US

Features:Tmax, Tmin, precipitation,wind speed, wind direction ,...

Black-box forecasting multiple weather stations simultaneously

[Signoretto, Frandi, Karevan, Suykens, IEEE-SCCI, 2014]

24

Black-box weather forecasting

• Black-box weather forecasting: prediction temperature in Brussels

• Multi-view learning:- Multi-view LS-SVM regression [Houthuys, Karevan, Suykens, IJCNN 2017]

- Multi-view Deep Neural Networks [Karevan, Houthuys, Suykens, ICANN 2018]

25

Multi-view learning: kernel-based (1)

• Primal problem:

minw[v],e[v]

12

∑Vv=1w

[v]Tw[v] + 12

∑Vv=1 γ

[v]e[v]Te[v] +ρ

∑Vv,u=1;v 6=u e

[v]T e[u]

subject to y = Φ[v]w[v] + b[v]1N + e[v], v = 1, ..., V

• Dual:[

0V×V 1TMΓM1M + ρ IM1M ΓMΩM + INV + ρ IMΩM

] [

bMαM

]

=

[

0VΓMyM + (V − 1)ρ yM

]

• Prediction:

y(x) =V∑

v=1

βv

N∑

k=1

α[v]k K

[v](x[v],x[v]k ) + b[v]

[Houthuys et al., 2017]

26


• Data set:

– Real measurements for weather elements such as temperature, humidity, etc.

– From 2007 until mid 2014

– Two test sets:

- mid-November 2013 until mid-December 2013

- from mid-April 2014 to mid-May 2014

• Goal: Forecasting minimum and maximum temperature for one to six days ahead in

Brussels Belgium

• Views: Brussels together with 9 neighboring cities

• Tuning parameters:

- kernel parameters for each view

- regularization parameters γ[v]

- coupling parameter ρ

27


Apr/May

Nov/Dec

28

Multi-view learning: deep neural network (1)

• Primal formulation of multi-view LS-SVM:

minw[v],e[v],b[v]

12

∑Vv=1w

[v]Tw[v] + 12

∑Vv=1 γ

[v]e[v]Te[v] + ρ

∑Vv,u=1;v 6=ue

[v]Te[u]

subject to y = Φ[v]w[v] + b[v]1N + e[v] for v = 1, ..., V

• Weighted Multi-view approach:

minw[v],e[v]

1

2

V∑

v=1

s[v](

w[v]Tw[v] + γ[v]e[v]Te[v]

)

+V∑

v,u=1;v 6=u

ρ[v,u]√

s[v]√

s[u] e[v]Te[u]

– s[v]: weight of the view v (can be manually determined by an expert,or calculated during a pre-processing step)

– ρ[v,u]: coupling parameter for pairwise combination of views– 0 ≤ ρ[v,u] ≤ minγ[v], γ[u]

[Karevan et al., 2018]

29

Multi-view learning: deep neural network (2)

• Weather forecasting is a time series prediction problem → Consider eachdelay as a view

• Consider 5 views (i.e. the delay is considered to be 5)

• Tuning parameters: regularization parameter γ[v] and number of neuronsfor each view, and ρ[v,u] coupling parameter for each part of views

• The weight of each view is defined based on its error the validation set:

s[v] = exp(−mse[v]val)

• Forecasting minimum and maximum temperature for one to six daysahead in Brussels, Belgium

30

Fixed-size kernel methods for large scale data

30

Nystrom method

• “big” kernel matrix: Ω(N,N) ∈ RN×N

“small” kernel matrix: Ω(M,M) ∈ RM×M (on subset)

• Eigenvalue decompositions: Ω(N,N) U = U Λ and Ω(M,M)U = U Λ

• Relation to eigenvalues and eigenfunctions of the integral equation

∫

K(x, x′)φi(x)p(x)dx = λiφi(x′)

with

λi =1

Mλi, φi(xk) =

√M uki, φi(x

′) =

√M

λi

M∑

k=1

ukiK(xk, x′)

[Williams & Seeger, 2001] (Nystrom method in GP)

31

Fixed-size method: estimation in primal

• For the feature map ϕ(·) : Rd → Rh obtain an approximation

ϕ(·) : Rd → RM

based on the eigenvalue decomposition of the kernel matrix with ϕi(x′) =

√

λi φi(x′) (on a subset of size M ≪ N).

• Estimate in primal:

minw,b

1

2wT w + γ

1

2

N∑

i=1

(yi − wT ϕ(xi)− b)2

Sparse representation is obtained: w ∈ RM with M ≪ N and M ≪ h.

[Suykens et al., 2002; De Brabanter et al., CSDA 2010]

32

Random Fourier Features

• Proposed by [Rahimi & Recht, 2007].

• It requires a positive definite shift-invariant kernel K(x, y) = K(x− y).One obtains a randomized feature map z(x) : Rd → R

2D so that

z(x)Tz(y) ≃ K(x− y).

• Compute the Fourier transform p of the kernel K:

p(ω) =1

2π

∫

exp(−jωT∆)K(∆)d∆

Draw D iid samples ω1, ..., ωD ∈ Rd from p.

Obtain z(x) =√

1D[cos(ω

T1 x)... cos(ω

TDx) sin(ω

T1 x)... sin(ω

TDx)]

T .

33

Deep neural-kernel networks using random Fourier features

Use of Random Fourier Features [Rahimi & Recht, NIPS 2007] to obtainan approximation to the feature map in a deep architecture

[Mehrkanoon & Suykens, Neurocomputing 2018]

34

Example: electricity load forecasting

20 40 60 80 100 120 140 1600.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Actual LoadFS−LSSVM

Hour

Norm

alize

dLoad

20 40 60 80 100 120 140 1600.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Actual LoadLinear

Hour

Norm

alize

dLoad

(a) (b)

20 40 60 80 100 120 140 1600.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Actual LoadFS−LSSVM

Hour

Norm

alize

dLoad

20 40 60 80 100 120 140 1600.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Actual LoadLinear

Hour

Norm

alize

dLoad

(c) (d)

[Espinoza, Suykens, Belmans, De Moor, IEEE CSM 2007]

1-hour ahead

24-hours ahead

Fixed-size LS-SVM ր տ Linear ARX model

35

Robustness

35

Outliers and robustness

xx

x

x xx

xx

x

y

breakdown?

?Robust statistics: Bounded derivative of loss function, bounded kernel

[Huber, 1981; Hampel et al., 1986; Rousseeuw & Leroy, 1987]

36

Weighted versions and robustness

Convex cost function

convexoptimiz.

SVM solution

Weighted version with

modified cost function

robuststatistics

LS-SVM solution

SVM Weighted LS-SVM

• Weighted LS-SVM: minw,b,e

1

2wTw + γ

1

2

N∑

i=1

vie2i s.t. yi = wTϕ(xi) + b+ ei, ∀i

with vi determined from eiNi=1 of unweighted LS-SVM [Suykens et al., 2002].

Robustness and stability [Debruyne et al., JMLR 2008, 2010].

• SVM solution by applying iteratively weighted LS [Perez-Cruz et al., 2005]

37

Example: robust regression using weighted LS-SVM

−5 0 5−4

−3

−2

−1

0

1

2

3

4

function estimation using LS−SVMγ=0.14185,σ2=0.047615

RBF

LS−SVM

data

Real function

x

y

−5 0 5−4

−3

−2

−1

0

1

2

3

4

function estimation using LS−SVMγ=95025.4538,σ2=0.66686

RBF

LS−SVM

data

Real function

x

y

using LS-SVMlab v1.8 http://www.esat.kuleuven.be/sista/lssvmlab/

38

Generative models

38

Generative Adversarial Network (GAN)

Generative Adversarial Network (GAN) [Goodfellow et al., 2014]Training of two competing models in a zero-sum game:

(Generator) generate fake output examples from random noise(Discriminator) discriminate between fake examples and real examples.

source: https://deeplearning4j.org/generative-adversarial-network

39

GAN: example on MNIST

MNIST training data:

GAN generated examples:

source: https://www.kdnuggets.com/2016/07/mnist-generative-adversarial-model-keras.html

40

Restricted Boltzmann Machines (RBM)

• Markov random field, bipartite graph, stochastic binary unitsLayer of visible units v and layer of hidden units hNo hidden-to-hidden connections

• Energy:

E(v, h; θ) = −vTWh− cTv − aTh with θ = W, c, a

Joint distribution:

P (v, h; θ) =1

Z(θ)exp(−E(v, h; θ))

with partition function Z(θ) =∑

v

∑

h exp(−E(v, h; θ))

[Hinton, Osindero, Teh, Neural Computation 2006]

41

RBM and deep learning

RBM

p(v, h) p(v, h1, h2, h3, ...)

[Hinton et al., 2006; Salakhutdinov, 2015]

42

in other words ...

”sandwich”

E = −vTWh

”deep sandwich”

E = −vTW 1h1 − h1TW 2h2 − h2

TW 3h3

43

RBM: example on MNIST

MNIST training data:

Generating new images:

source: https://www.kaggle.com/nicw102168/restricted-boltzmann-machine-rbm-on-mnist

44

Convolutional Deep Belief Networks

Unsupervised Learning of Hierarchical Representations with Convolutional Deep Belief

Networks [Lee et al. 2011]

45

Restricted kernel machines

45

Restricted Kernel Machines (RKM)

• Kernel machine interpretations in terms of visible and hidden units(similar to Restricted Boltzmann Machines (RBM))

• Restricted Kernel Machine (RKM) representations for

– LS-SVM regression/classification– Kernel PCA– Matrix SVD– Parzen-type models– other

• Based on principle of conjugate feature duality (with hidden featurescorresponding to dual variables)

• Deep Restricted Kernel Machines (Deep RKM)

[Suykens, Neural Computation, 2017]

46

Kernel principal component analysis (KPCA)

−1.5 −1 −0.5 0 0.5 1−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

linear PCA

−1.5 −1 −0.5 0 0.5 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

kernel PCA (RBF kernel)

Kernel PCA [Scholkopf et al., 1998]:take eigenvalue decomposition of the kernel matrix

K(x1, x1) ... K(x1, xN)... ...

K(xN , x1) ... K(xN , xN)

(applications in dimensionality reduction and denoising)

47

Kernel PCA: classical LS-SVM approach

• Primal problem: [Suykens et al., 2002]: model-based approach

minw,b,e

1

2wTw − 1

2γ

N∑

i=1

e2i s.t. ei = wTϕ(xi) + b, i = 1, ..., N.

• Dual problem corresponds to kernel PCA

Ω(c)α = λα with λ = 1/γ

with Ω(c)ij = (ϕ(xi)− µϕ)

T (ϕ(xj)− µϕ) the centered kernel matrix

and µϕ = (1/N)∑N

i=1ϕ(xi).

• Interpretation:1. pool of candidate components (objective function equals zero)2. select relevant components

48

From KPCA to RKM representation (1)

Model:

e =W Tϕ(x)objective J= regularization term Tr(W TW )- (1λ) variance term

∑

i eTi ei

↓ use property eTh ≤ 12λe

Te+ λ2h

Th

RKM representation:

e =∑

j hjK(xj, x)

obtain J ≤ J(hi,W )solution from stationary points of J :∂J∂hi

= 0, ∂J∂W = 0

49


• Objective

J =η

2Tr(W TW )− 1

2λ

N∑

i=1

eTi ei s.t. ei =W Tϕ(xi), ∀i

≤ −N∑

i=1

eTi hi +λ

2

N∑

i=1

hTi hi +η

2Tr(W TW ) s.t. ei =W Tϕ(xi), ∀i

= −N∑

i=1

ϕ(xi)TWhi +

λ

2

N∑

i=1

hTi hi +η

2Tr(W TW ) , J

• Stationary points of J(hi,W ):

∂J

∂hi= 0 ⇒ W Tϕ(xi) = λhi, ∀i

∂J

∂W= 0 ⇒ W =

1

η

∑

i

ϕ(xi)hTi

50


• Elimination of W gives the eigenvalue decomposition:

1

ηKHT = HTΛ

where H = [h1...hN ] ∈ Rs×N and Λ = diagλ1, ..., λs with s ≤ N

• Primal and dual model representations

(P )RKM : e =W Tϕ(x)ր

Mց

(D)RKM : e =1

η

∑

j

hjK(xj, x).

51

Deep Restricted Kernel Machines

51

Deep RKM: example

v

h(1)x ϕ1(x)

yy

e(1)ϕ2(h

(1))e(2)h(2)

ϕ3(h(2))

e(3)h(3)

Deep RKM: KPCA + KPCA + LSSVM [Suykens, 2017]

Coupling of RKMs by taking sum of the objectives

Jdeep = J1 + J2 + J3

Multiple levels and multiple layers per level.

52

in more detail ...

v

h(1)x ϕ1(x)

yy

e(1)ϕ2(h

(1))e(2)h(2)

ϕ3(h(2))

e(3)h(3)

Jdeep = −N∑

i=1

ϕ1(xi)TW1h

(1)i +

λ12

N∑

i=1

h(1)i

Th(1)i +

η12Tr(W T

1 W1)

−N∑

i=1

ϕ2(h(1)i )TW2h

(2)i +

λ22

N∑

i=1

h(2)i

Th(2)i +

η22Tr(W T

2 W2)

+

N∑

i=1

(yTi − ϕ3(h(2)i )TW3 − bT )h

(3)i − λ3

2

N∑

i=1

h(3)i

Th(3)i +

η32Tr(W T

3 W3)

53

Primal and dual model representations

e(1) =W T1 ϕ1(x)

(P )DeepRKM : e(2) =W T2 ϕ2(Λ

−11 e(1))

ր y =W T3 ϕ3(Λ

−12 e(2)) + b

Mց e(1) = 1

η1

∑

j h(1)j K1(xj, x)

(D)DeepRKM : e(2) = 1η2

∑

j h(2)j K2(h

(1)j ,Λ−1

1 e(1))

y = 1η3

∑

j h(3)j K3(h

(2)j ,Λ−1

2 e(2)) + b

The framework can be used for training deep feedforward neural networksand deep kernel machines [Suykens, 2017].

(Other approaches: e.g. kernels for deep learning [Cho & Saul, 2009], mathematics of

the neural response [Smale et al., 2010], deep gaussian processes [Damianou & Lawrence,

2013], convolutional kernel networks [Mairal et al., 2014], multi-layer support vector

machines [Wiering & Schomaker, 2014])

54

Training process

0 100 20010 -1

100

101

102

103

104

105

106

iteration step

Jdeep,P

stab

Objective function (logarithmic scale) during training on the ion data set:

• black color: level 3 objective only

• Jdeep for cstab = 1, 10, 100 (blue, red, magenta color) in stabilization term

55

Generative RKM

55

RKM objective for training and generating (1)

• RBM energy function

E(v, h; θ) = −vTWh− cTv − aTh

with model parameters θ = W, c, a

• RKM ”super-objective” function (for training and for generating)

J(v, h,W ) = −vTWh+ λ2h

Th+ 12v

Tv + η2Tr(W

TW )

Training: clamp v → Jtrain(h,W )Generating: clamp h,W → Jgen(v)

[Schreurs & Suykens, ESANN 2018]

56


• Training: (clamp v)

Jtrain(hi,W ) = −N∑

i=1

vTi Whi +λ

2

N∑

i=1

hTi hi +η

2Tr(WTW )

Stationary points:

∂Jtrain∂hi

= 0 ⇒WTvi = λhi, ∀i∂Jtrain∂W = 0 ⇒W = 1

η

∑Ni=1 vih

Ti

Elimination of W :1

ηKHT = HT∆,

where H = [h1, . . . , hN ] ∈ Rs×N , ∆ = diagλ1, . . . , λs with s ≤ N

the number of selected components and Kij = vTi vj the kernel matrixelements.

57


• Generating: (clamp h,W )

Estimate distribution p(h) from hi, i = 1, ..., N (or assumed normal).Obtain a new value h⋆.Generate in this way v⋆ from

Jgen(v⋆) = −v⋆TWh⋆ +

1

2v⋆

Tv⋆

Stationary points:∂Jgen∂v⋆

= 0

This givesv⋆ =Wh⋆

58

Dimensionality reduction and denoising: linear case

• Given training data vi = xi with X ∈ Rd×N , obtain hidden features

H ∈ Rs×N :

X =WH = (1

η

N∑

i=1

xihTi )H =

1

ηXHTH

• Reconstruction error: ‖X − X‖2

xi G(·) hi F (·) xi

59

Dimensionality reduction and denoising: nonlinear case (1)

• New datapoint x⋆ generated from h⋆ by

ϕ(x⋆) =Wh⋆ = (1

η

N∑

i=1

ϕ(xi)hTi )h

⋆

• Multiplying both sides by ϕ(xj) gives:

K(xj, x⋆) =

1

η(

N∑

i=1

K(xj, xi)hTi )h

⋆

On training data:

Ω =1

ηΩHTH

with H ∈ Rs×N ,Ωij = K(xi, xj) = ϕ(xi)

Tϕ(xj).

60

Dimensionality reduction and denoising: nonlinear case (2)

• Estimated value x for x⋆ by kernel smoother:

x =

∑Sj=1 K(xj, x

⋆)xj∑S

j=1 K(xj, x⋆)

with K(xj, x⋆) (e.g. RBF kernel) the scaled similarity between 0 and

1, a design parameter S ≤ N (S closest points based on the similarityK(xj, x

⋆)).

[Schreurs & Suykens, ESANN 2018]

61

Explainable AI: latent space exploration (1)

hidden units: exploring the whole continuum:

-0.15 -0.1 -0.05 0 0.05 0.1

H1

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

H2

0

1

6

h(1,1) = -0.11 h(1,1) = -0.06 h(1,1) = -0.01 h(1,1) = 0.04 h(1,1) = 0.09

h(1,2) = -0.12 h(1,2) = -0.06 h(1,2) = 0 h(1,2) = 0.06 h(1,2) = 0.12

h(1,3) = -0.11 h(1,3) = -0.05 h(1,3) = 0.01 h(1,3) = 0.06 h(1,3) = 0.12

[figures by Joachim Schreurs]

62

Explainable AI: latent space exploration (2)

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

C

B

D

A

Yale Face database - generated faces from different regions A,B,C,D

[Winant, Schreurs, Suykens, BNAIC 2019]

63

Tensor-based RKM for Multi-view KPCA

min 〈W,W〉−N∑

i=1

⟨

Φ(i),W⟩

hi+λN∑

i=1

h2i with Φ(i) = ϕ[1](x[1]i )⊗...⊗ϕ[V ](x

[V ]i )

[Houthuys & Suykens, ICANN 2018]

64

Generative RKM (Gen-RKM) (1)

Train:

Generate:

[Pandey, Schreurs & Suykens, 2019, arXiv:1906.08144]

65

Gen-RKM (2)

The objective

Jtrain(hi,U ,V ) =∑N

i=1

(

−φ1(xi)TUhi − φ2(yi)TV hi +

λ2h

Ti hi

)

+η12 Tr(U

TU ) + η22 Tr(V

TV )

results for training into the eigenvalue problem

(1

η1K1 +

1

η2K2)H

T = HTΛ

with H = [h1...hN ] and kernel matrices K1,K2 related to φ1, φ2.


66

Gen-RKM (3)

Generating data is based on a newly generated h⋆ and the objective

Jgen(φ1(x⋆), ϕ2(y

⋆)) = −φ1(x⋆)TV h⋆−φ2(y⋆)TUh⋆+1

2φ1(x

⋆)Tφ1(x⋆)+

1

2φ2(y

⋆)Tφ2(y⋆)

giving

φ1(x⋆) =

1

η1

N∑

i=1

φ1(xi)hTi h

⋆, φ2(y⋆) =

1

η2

N∑

i=1

φ2(yi)hTi h

⋆.

For generating x, y one can either work with the kernel smoother or workwith an explicit feature map using a (deep) neural network or CNN.


67

Gen-RKM (4)

Fx Fy

H

X Y

U⊤ U V V ⊤

φ1(·) ψ1(·) ψ2(·) φ2(·)

Gen-RKM schematic representation modeling a common subspace H between two data

sources X and Y. The φ1, φ2 are the feature maps (Fx and Fy represent the feature-

spaces) corresponding to the two data sources. While ψ1, ψ2 represent the pre-image

maps. The interconnection matrices U, V model dependencies between latent variables

and the mapped data sources.


68

Gen-RKM: implicit feature map

Obtain

kx⋆ =1

η1K1H

⊤h⋆, ky⋆ =1

η2K2H

⊤h⋆,

with kx⋆ = [k(x1,x⋆), . . . , k(xN ,x

⋆)]⊤.

Using the kernel-smoother:

x = ψ1 (φ1(x⋆)) =

∑nrj=1 k1(xj,x

⋆)xj∑nr

j=1 k1(xj,x⋆)

, y = ψ2 (φ2(y⋆)) =

∑nrj=1 k2(yj,y

⋆)yj∑nr

j=1 k2(yj,y⋆)

,

with k1(xi,x⋆) and k2(yi,y

⋆) the scaled similarities between 0 and 1; nris the number of closest points based on the similarity defined by kernels k1and k2.

69

Gen-RKM: explicit feature map

Parametrized feature maps: φθ(·), ψζ(·) (e.g. CNN and transposed CNN).

Overall objective function, using a stabilization mechanism [Suykens, 2017]:

minθ1,θ2,ζ1,ζ2

Jc = Jtrain +cstab2 J 2

train

+cacc2N

(

∑Ni=1

[

L1(x⋆i , ψ1ζ1

(φ1θ1(x⋆i ))) + L2(y

⋆i , ψ2ζ2

(φ2θ2(y⋆i )))

])

with reconstruction errors

L1(x⋆i , ψ1ζ1

(φ1θ1(x⋆i ))) =

1N‖x⋆i − ψ1ζ1

(φ1θ1(x⋆i ))‖22,

L2(y⋆i , ψ2ζ2

(φ2θ2(y⋆i ))) =

1N‖y⋆i − ψ2ζ2

(φ2θ2(y⋆i ))‖22

and with Φx = [φ1(x1), . . . , φ1(xN)],Φy = [φ2(y1), . . . , φ2(yN)], U, Vfrom

[

1η1ΦxΦ

⊤x

1η1ΦxΦ

⊤y

1η2ΦyΦ

⊤x

1η2ΦyΦ

⊤y

]

[

UV

]

=

[

UV

]

Λ.

Hence, joint feature learning and subspace learning.

70

Gen-RKM: examples (1)

MNIST Fashion-MNIST

Generated samples from the model using CNN as explicit feature map in the kernel function.

The yellow boxes show training examples and the adjacent boxes show the reconstructed

samples. The other images (columns 3-6) are generated by random sampling from the

fitted distribution over the learned latent variables.


71

Gen-RKM: examples (2)

CIFAR-10 CelebA

Generated samples from the model using CNN as explicit feature map in the kernel function.

The yellow boxes show training examples and the adjacent boxes show the reconstructed

samples. The other images (columns 3-6) are generated by random sampling from the

fitted distribution over the learned latent variables.


72

Gen-RKM: multi-view generation (1)

CelebA

Multi-view generation on CelebA dataset showing images and attributes


73

Gen-RKM: multi-view generation (2)

MNIST: Implicit feature maps with Gaussian kernel + generation by kernel-smoother

MNIST: Explicit feature maps using Convolutional Neural Networks

CIFAR-10: Explicit feature maps using CNNs + Transposed CNNs

Multi-view Generation (images and labels) using implicit and explicit feature maps


74

Gen-RKM: latent space exploration (1)

Exploring the learned uncorrelated-features by traversing along the eigenvectors

Explainability: changing one single neuron’s hidden feature changes the hair color while

preserving face structure! [Pandey, Schreurs & Suykens, 2019, arXiv:1906.08144]

75


MNIST reconstructed images by bilinear-interpolation in latent space


76


CelebA reconstructed images by bilinear-interpolation in latent space


77

Future challenges

• efficient algorithms and implementations for large data

• extension to other loss functions and regularization schemes

• multimodal data, tensor models, coupling schemes

• models for deep clustering and semi-supervised learning

• choice kernel functions, invariances and symmetry properties

• deep generative models

• optimal transport

• synergies between neural networks, deep learning and kernel machines

78

Conclusions

• function estimation: parametric versus kernel-based

• primal and dual model representations

• neural network interpretations in primal and dual

• RKM: new connections between RBM, kernel PCA and LS-SVM

• deep kernel machines

• generative models

79

Acknowledgements (1)

• Current and former co-workers at ESAT-STADIUS:

C. Alzate, Y. Chen, J. De Brabanter, K. De Brabanter, B. De Cooman,L. De Lathauwer, H. De Meulemeester, B. De Moor, H. De Plaen, Ph.Dreesen, M. Espinoza, T. Falck, M. Fanuel, Y. Feng, B. Gauthier, X.Huang, L. Houthuys, V. Jumutc, Z. Karevan, R. Langone, F. Liu, R.Mall, S. Mehrkanoon, G. Nisol, M. Orchel, A. Pandey, P. Patrinos, K.Pelckmans, S. RoyChowdhury, S. Salzo, J. Schreurs, M. Signoretto, Q.Tao, F. Tonin, J. Vandewalle, T. Van Gestel, S. Van Huffel, C. Varon,Y. Yang, and others

• Many other people for joint work, discussions, invitations, organizations

• Support from ERC AdG E-DUALITY, ERC AdG A-DATADRIVE-B, KULeuven, OPTEC, IUAP DYSCO, FWO projects, IWT, iMinds, BIL, COST

80


81


NEW: ERC Advanced Grant E-DUALITYExploring duality for future data-driven modelling

82

Thank you

83

deep learning, neural networks and kernel machines ... · generalization, deep learning and kernel...

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