TENSORY w rozpoznawaniu wzorcó · Short introduction to the tensor mathematics. Tensor decompositions as a way of multi-dimensional data analysis. ... (a valence) of a tensor Bishop

TENSORY w rozpoznawaniu wzorców

Bogusław Cyganek

Akademia Górniczo-Hutnicza Al. Mickiewicza 30, 30-059 Kraków, Poland

[email protected]

(C) ML PAN, Warszawa, 12 listopada 2013

An overview:

2

What are tensors and why tensors in pattern recognition.

Short introduction to the tensor mathematics.

Tensor decompositions as a way of multi-dimensional data

analysis.

Higher-Order Singular Value Decomposition (HOSVD).

Applications in pattern recognition.

Implementation and practical issues.

Conclusions.

3

Short intro to tensors

Tensor definitions:

4

Tensors can be especially useful in the following two situations:

1. Description of physical laws independent of the coordinate systems;

2. Representation of multidimensional data, in which each factor is governed by

a separate index of a tensor.

We have three equivalent definitions of a tensor:

1. A multilinear function which takes the p-dimensional vector space and

q-dimensional its dual, into the space of real values ;

2. A mathematical object described by the indexed values ti..jm..n, in which there

is p covariant and q contravariant indices. These values transform in

accordance with the transformation laws with a change of the base of the

spaces;

3. A multi-dimensional array of data with N=p+q independent indices.

Tensor definitions:

5

Let W and W* be the vector space and its dual, respectively. A tensor over the

base W is a multilinear scalar function f with arguments from W and W*

* *: p times q times

f W W W W

The number of arguments p and q denote a covariant and a contravariant degrees

(a valence) of a tensor

Bishop R.L., Goldberg S.I.: Tensor Analysis on Manifolds. Dover, 1980.

Origin of tensor transformations, basic properties:

6

THEOREM:

A tensor is uniquely determined by its values on the basis and its dual basis. The

values are products of the tensor and the elements of the base and the dual base.

THEOREM:

A dimension of a tensor of p covariant and q contravariant indices is dp+q, where

d=dim(W) denotes dimension of the vector space W

EXAMPLE:

Scalaras and vectors are both special cases of a more general object called a

tensor of order n. In a 3D Euclidean space we have:

d=3

a scalar – a tensor of a 0-th order – has 30=1 element

a vector – a tensor of a 1-st order – has 31=3 elements

a matrix – a tensor of a 2-nd order – has 32=9 elements

However, what is important are the TRANSFORMATION LAWS !!

Origin of tensor transformations, index notations:

7

A vector x in the L dimensional space with basis bi can be expressed as

1

Li i

i i

i

x x

x b b

The Einstein’s notation

the contravariant (upper) index position

the covariant (lower) index position

Summation can only take place over „dummy” indices in different positions (i.e.

one subscript, the other superscript).

The tensor algebra is constructed mostly on two mathematical concepts:

• The linear operators.

• Change of the coordinate system (the Jacobian).


8

let us assume that the base b linearly transforms into b’ as follows

Ai

j j i b b

the elements Aij form a square matrix A for which we assume that it is invertible,

so the change of base is possible in two directions

A j

i j i b b

A’ij are elements of A-1.

Let us now find coefficients of a vector x in this new base b’:

A A

i

i i j i j

i i j j i

x

x x x

x b b b

Ai i j

jx x

when the new base (primed) is obtained from the original one by multiplication with the

matrix A, coordinates of a vector x’ in this new (primed) space are related to the original

vector x by its inverse, i.e. A-1,


9

For instance in the simplest 1D case, when going from one space U into U’, the

covariant components of a tensor T transform as

k

k

ii TT '

' α ki

k

i TT '' α

Coefficients of the direct and indirect transformation of the coordinates spaces

αj

j

i i

x

x

The ordered table of ij is called a coordinate

systems transformation matrix, called also a

Jacobian matrix J=[ij].


10

For instance in the simplest 1D case, when going from one space U into U’, the

covariant components of a tensor T transform as

k

k

ii TT '

' α ki

k

i TT '' α

Coefficients of the direct and indirect transformation of the coordinates spaces

Today tensors in computer science (computer vision, image processing,

data mining, medicine, pattern recognition):

• Multi-view correspondance.

• Image synthesis.

• MRI, analysis, filtering and visualization.

• Intertia tensors for object representation.

• Local structure in multi-dimensional signals.

• Multi-dimensional data processing.

Tensors in mathematics and physics, then psychometrics and chemistry.

11

Tensors for multi-dimensional data processing:

Tensors can be relaxed from their physical meaning and can be viewed as

multidimensional arrays of data, in which each dimension separately corresponds

usually to a different feature of described objects or phenomena.

This point of view is “parallel” to the previous one, rather than “instead”, since even

with explicitly provided n covariant and m contravariant indices, a tensor in a given

base is unanimously described by nm real (or complex) values.

This way we obtain a multilinear approach which constitutes an extension to the

classical two-dimensional (matrix) analysis. Such an approach was recently

undertaken to tackle problems of data mining, pattern recognition, neuroscience and

psychometrics, chemistry, signal processing, computer vision and many more


12

An image and its three color channels as an example of a 3D tensor


13

A hyperspectral image with selected regions of different reflectance properties.

Series of hyperspectral images naturally form 3D tensors with two spatial

dimensions (x,y) and one spectral λ.

400 2400

Wavelength (nm)

Re

fle

cta

nce

0

1

400 2400

Wavelength (nm)

Re

fle

cta

nce

0

1

400 2400

Wavelength (nm)

Re

fle

cta

nce

0

1

x

y

l


14

A tensor T of P dimensions of values N1, N2, …, NP, is denoted as follows

1 2 PN N N

A single element t of T is addressed providing its precise position by a series of

indices n1, n2, …, nP,

1 2 1 2P Pn n n n n nt

1 1 2 21 , 1 , , 1

P Pn N n N n N

in the equivalent function-like notation

1 2 1 2

, , ,Pn n n P

t n n n

Tensors for multi-dimensional data processing – an example:

15

A 333 tensor and possible representation with three frontal 2D slices.

0

0

12 0

4 1

-132

N1

N2

N3

-2

4

0 0

10 0

012

1

4

1 -1

0 4

125

0

0

12 0

4 1

-132

12 0 0 1 1 1 0 0 2

0 4 1 4 0 4 4 10 0

2 3 1 5 2 1 2 1 0

1

2

3

Tensors for multi-dimensional data processing – an example:

16

12 0 0 1 1 1 0 0 2

0 4 1 4 0 4 4 10 0

2 3 1 5 2 1 2 1 0

1

2

3

123

1,2,3 0t

133

1,3,3 2t

312

3,1,2 5t

A middle slice of T can be written using the colon notation

1 1 1

:,:,2 4 0 4

5 2 1

:,2,3 0 10 1

A specific vector cut out from T is known also as a fiber

17

Basic concepts of the multilinear algebra – tensor flattening:

Tensor flattening (known also as tensor unfolding or matricization):

Three types of k-mode vectors (fibers) for a three dimensional tensors: 1-mode (column)

fibers, 2-mode (row) fibers, and 3-mode (tube) fibers

18

N1=3

N2=4

N 3=2

N1=3

N1=3

N 3=2

N2=4

N2=4

N3=2

N1=3

N3=2

N2=4

N1=3

N 3=2

N2=4

N1=3

N1=3

N3=2

N2=4

N1=3

N 3=2

N2=4

N2=4 N2=4

T(1)

T(2)

T(3)

Flattening of the N1N2N3 tensor T to obtain the matrix

T(1) of dimensions N1N2N3, T(2) – N2N1N3, and T(3) – N3N1N2


19

1

12 1 0 0 1 0 0 1 2

0 4 4 4 0 10 1 4 0

2 5 2 3 2 1 1 1 0

T

2

12 0 2 1 4 5 0 4 2

0 4 3 1 0 2 0 10 1

0 1 1 1 4 1 2 0 0

T

3

12 0 0 0 4 1 2 3 1

1 1 1 4 0 4 5 2 1

0 0 2 4 10 0 2 1 0

T

-2

4

0 0

10 0

012

1

4

1 -1

0 4

125

0

0

12 0

4 1

-132


20


1

3 2

1

23

1

3 2

1

3 2

1

23

1

3 2

Two types of permutations of the three indices 1,2, and 3. Grayed circle denotes a

starting index. Forward permutations (adding 1) are 1-2-3, 2-3-1, and 3-1-2 . Backward

permutations (subtracting 1) from the top to the bottom rows are 1-3-2, 2-1-3, and 3-2-1

Forward Backward

21

Basic concepts of the multilinear algebra – video tensor flattening:

Nk

Nr

Nf

Nc

..

..

..

Nr

Nk

Nc

Nf

Image 1

red channel

Image 1

green channel

Image 1

blue channel

Image 2

red channel

Image f

blue channel...

A color video stream can be

represented as a four dimensional

tensor. An example of such a tensor

of dimensions 32024043

(columns Nk rows Nr frames Nf

color-channels Nc)

A practical way of tensor flattening - the forward mode - which maps images to the

memory in the scanning order

Basic concepts of the multilinear algebra – tensor products:

22

• Outer product of tensors.

• Contracted product of tensors.

• Inner product of tensors (all contracted indices)

1 2

1 2 1 2

1 21 1 1

,C

C C

C

K K K

k k k k k kk k k

s t

Orthogonal tensors

0

Basic concepts of the multilinear algebra – product examples:

23

1 2 3 0 0 8

0 1 7 2 11 11

2 1 1 2 0 4

0 2 4 1 1 0

1 23 3

2 4 8

, 10 23 7

30 17 21K K

1 2 1 2

1 2 1 2

2 2 2 2

3 2 1 2 1 21 1 1 1

3,2 , ,3 , ,2

0 8 2 11 4 11 2 17

k k k kk k k k

s t k k k k

0Their inner product

Their contructed product for the first common indices K1=2 and K2=2 yields a 3x3 matrix:

24

Basic concepts of the multilinear algebra - k-mode product:

The k-mode product k 1 2 PN N N

kQ NM k

M

1 2 1 1k k PN N N Q N N

1 2 1 1 1 2 1 11 2 1 1 1

k

k k P k k k P kk k Pk

N

n n n qn n k n n n n n n qnn n n qn nn

t mM

of a tensor and a matrix

The k-mode product can be equivalently expressed in terms of the flattened matrices

k M k k

S MT

1 2 TT SVD V S D

For example:

1 V S SV 2 TS D SDsince

25

-2

4

0 0

10 0

012

1

4

1 -1

0 4

125

0

0

12 0

4 1

-132

N1=3

N2=4

N 3=2

1

2

1 0

2 1

100

0

2

1 0

4 1

-1100

0

0

12

-10

023

A

B

C

477726 88

8718156 190

695512 50

-6 -18 -8 -20

-45

-19

iA

iB

iC

An example of a 1-mode, 2-mode, and 3-mode products of a 333 tensor T with

the three matrices A, B, and C. The resulting tensor has dimensions 342 imposed

by dimensions of the three matrices

Basic concepts of the multilinear algebra - k-mode product example:

1 2 3 2 1 3A B C B A C

26

-2

4

0 0

10 0

012

1

4

1 -1

0 4

125

0

0

12 0

4 1

-132

N1=3

N2=4

N 3=2

1

2

1 0

2 1

100

0

2

1 0

4 1

-1100

0

0

12

-10

023

A

B

C

477726 88

8718156 190

695512 50

-6 -18 -8 -20

-45

-19

iA

iB

iC

An example of a 1-mode, 2-mode, and 3-mode products of a 333 tensor T with

the three matrices A, B, and C. The resulting tensor has dimensions 342 imposed

by dimensions of the three matrices


1 2 3 2 1 3A B C B A C

27

-2

4

0 0

10 0

012

1

4

1 -1

0 4

125

0

0

12 0

4 1

-132

0

2

1 0

4 1

-1100

023

B

iB

*


*

2

12 24 0 36 1 1 11 1 0 2 2 0

0 17 39 8 4 12 4 12 4 48 100 32

2 17 31 12 5 19 19 19 2 8 10 8

B

28

Basic concepts of the multilinear algebra – ranks of a tensor:

Rank-1 of a tensor.

A P-dimensional tensor T has rank 1 if it is an outer product of P vectors s1,

…, sP, as follows

1 2 Ps s s

Rank of a tensor

The rank of a P-dimensional tensor T is the minimal number of rank-1

tensors that produce T in their linear combination

The k-rank of a tensor

k kR rank

is a dimension of the vector space spanned by its k-mode vectors

29

Tensor Decompositions

30

Types of tensor decompositions:

30

Higher Order Singular Value Decomposition (HOSVD, ~Tucker)

Rank-1 (Canonical decomposition, CANDECOMP/PARAFAC or

CP)

Best Rank-Rk

Nonnegative matrix and tensor decompositions

31

Visualization of the HOSVD for a 3D tensor

Types of tensor decompositions – HOSVD:

S1

S2

S3

N1

N2

N3

N1

N1

N2

N2

N3

N3

N1

N2

N3

Nr

Nr

1 1 2 2 3 3S S S

32

Types of tensor decompositions – Rank-1 decomposition:

N1

N2

N3

...

R components

N1

N3

N2

1

11s

12s

13s

2

21s

22s

23s

R

1

Rs

2

Rs

3

Rs

N1

N2

N3

Visualization of the rank-1 decomposition of a 3D tensor

1

Rr r r

r

a b c

T can be expressed as a pure sum of rank-1 tensors

Efficient linear discriminant analysis LDA (Fisher Bauckhage ...)

33

Types of tensor decompositions – Best rank-Rk decomposition:

Visualization of the rank-Rk decomposition of a 3D tensor

S1

S2

S3

N1

N2

N3

R1

N1

R2

N2

N3

R3

R1

R2

R3

1 1 2 2 3 3S S S

1 1

1

N RS P PN R

PS

Similar to HOSVD but different properties !

1 1rank R

P Prank R

desired ranks

34

Higher Order Singular Value Decomposition

(HOSVD, ~Tucker)

35

Basic concepts of the multilinear algebra – tensor decompositions:

Can we do the SVD for 3D cubes (3D tensors)?

Assuming a 3D tensor, of dimensions spanning N1N2N3, can it be written as a

sum of rank-one tensors

1 2 3

1 1 1

N N Np q r

pqrp q r

v a b cTucker

decomposition

36

1 2 m n PN N N N N A tensor:

Basic concepts of the multilinear algebra – HOSVD:

can be decomposed to

1 1 2 2 P P S S S

Sk denotes a mode matrix, which is a unitary matrix of dimensions NkNk spanning

the column space of the matrix T(k) obtained from the mode-n flattening of T;

1 2 m n PN N N N N

is a core tensor of the same dimensions as T.

37

Visualization of the HOSVD for a 3D tensor


S1

S2

S3

N1

N2

N3

N1

N1

N2

N2

N3

N3

N1

N2

N3

Nr

Nr

1 1 2 2 3 3S S S

38

Properties of the core tensor:


kn a kn bTwo sutensors

obained by fixing the nk index to a, or b respectively, are orthogonal

0k kn a n b

Subtensors can be ordered according to their norms

1 20

k k k Pn n n N

k

k

n a a

is the a-mode singular value of T

Each i-th vector of the matrix Sk is the i-th k-mode singular vector

2

1

39


1 1 2 2 P P S S S

1

PNh

h P Ph

s 1 1 2 2 1 1h P P S S S

the basis tensors

shP are columns of the unitary matrix SP.

N1

N2

N3

s1

...

N3 components

N1

N3

N2

s2

N1

N3

N2

s

N1

N3

N2

N3

1 2N3

Th are orthogonal. Hence, Th constitute a basis. This result allows

construction of classifiers based on the HOSVD decomposition!

Pattern recognition in the HOSVD spanned multilinear spaces:

40

Pattern recognition with HOSVD boils down to testing a distance of a given test pattern Px to its projections in each of the spaces spanned by the set of the bases Th.

This can be done by computing the following minimization problem

2

, 1

minih

Ni i

x h hi c h

P c

where cih are the coordinates of Px in the space spanned by Th

i, N≤NP denotes a

number of chosen dominating components. Due to the orthogonality of the tensors

Thi, the above reduces to the maximization of the following parameter

2

1

ˆ ˆ,N

i

i h xh

P

.,. denotes the scalar product of the tensors, Px and Thi are normalized. Returned is a

class i for which the corresponding i from is the largest .

Pattern recognition in multilinear spaces – practical issues

41

The higher N, the better fit, though at an expense of computation time. The

original tensor Ti of a class i is obtained from the available exemplars of the

prototype patterns for that class (which can be of different number).

2

1

ˆ ˆ,N

i

i h xh

P

Pattern recognition in multilinear spaces – practical issues

42

The higher N, the better fit, though at an expense of computation time. The

original tensor Ti of a class i is obtained from the available exemplars of the

prototype patterns for that class (which can be of different number).

Where to take the prototype tensors from?

For example, the patterns can be cropped from the training images(road signs)

which are additionally rotated in a given range (e.g. ±12° with a step of 2°) with

additionally added normal noise. Such a strategy allows each pattern to be trained

with different number of prototypes.

2

1

ˆ ˆ,N

i

i h xh

P

43

A Word on

Nonnegative Decompositions

44

Nonnegative matrix (tensor) factorization:

...

N

L

D

...

R

B

L×N

M

C

L×R R×N

...di ciD BC

2

2 2

, ,min min min

F F FD D

D BC BCD D D BC D BC

0, 0,B Csubject to:

Lee D.D, Seung H.S.: Learning the parts of objects by non-negative matrix

factorization. Nature, Vol. 401, pp. 788-791, 1999.

Solution – the

multiplicative update rules:

,

T

nrnr nr T

nr

b bDC

BCC

,

T

rmrm rm T

rm

c cB D

B BC

45

Nonnegative matrix (tensor) factorization:

46

Tucker L.R.: Some mathematical notes of three-mode factor analysis. Psychometrika, 31,

1966, pp. 279-311.

Harshman, R. A.: Foundations of the PARAFAC procedure: Models and conditions for an

“explanatory” multimodal factor analysis. UCLA Papers in Phonetics, 16, pp. 1-84, 1970.

Lathauwer de L.: Signal Processing Based on Multilinear Algebra. PhD dissertation,

Katholieke Universiteit Leuven, 1997.

Lathauwer de L., Moor de, B., Vandewalle J.: On the Best Rank-1 and Rank-(R1, R2, …,

RN) Approximation of Higher-Order Tensors, Vol. 21, No. 4, pp. 1324-1342, 2000.

Literature on tensors and their decompositions:

Lathauwer de L., Moor de, B., Vandewalle J.: On the Best Rank-1 and Rank-(R1, R2, …,

RN) Approximation of Higher-Order Tensors, Vol. 21, No. 4, pp. 1324-1342, 2000.

Kolda T.G., Bader B.W.: Tensor Decompositions and Applications. SIAM Review, 2008.

47

PATTERN RECOGNITION

APPLICATIONS

48

Human Face Recognition

Human face recognition:

49

Images of different persons in different lighting and pose conditions build up a tensor.


50

After the HOSVD decomposition:

1 2 3 4 5faces views illum expressions pixelsU U U U U

The first mode matrix Ufaces, of dimensions 2828, spans the space of human face

parameters, the 55 mode Uviews spans the space of viewpoints, the two 33

matrices Uilluminations and Uexpressions – the spaces of illumination and face expression

parameters, respectively.

The last matrix Upixels has dimensions 79437943 and orthonormally spans the space

of images. Each column of this matrix is an eigenimage (eigenface)

Vasilescu M.A., Terzopoulos D.: Multilinear analysis of image ensembles:

TensorFaces. Proceedings of Eurpoean Conference on Computer Vision, pp. 447-

460, 2002


51

A training image for the f-th face, v-th view, i-th illumination, and e-th expression

can be represented as the following 1111Ipixel tensor

, , , T T T T

5 1 2 3 4 exp

f v i e f v i e

pixels faces views illum ressd U u u u u


52



, , , T T T T

5 1 2 3 4 exp

f v i e f v i e


Now we have a new test image dx which parameters, such as person, view, etc. are

unknown. We can assume that it can also be expressed in the same way as were

all the training images d

T T T T

5 1 2 3 4 expx pixels faces views illum ressd U u u u u


53



, , , T T T T

5 1 2 3 4 exp

f v i e f v i e





T T T T

5 1 2 3 4 expx pixels faces views illum ressd U u u u u unknown ....


54



, , , T T T T

5 1 2 3 4 exp

f v i e f v i e





T T T T

5 1 2 3 4 expx pixels faces views illum ressd U u u u u unknown ....

A reconstruction of mode parameters of a test image can be represented as follows

exp

2

, , ,

ˆ ˆ ˆ ˆ, , , argminfaces views illum ress

faces views illum express t xu u u u

u u u u d d Solution:

tensor

factorization

55

Road Signs Recognition in

Digital Images

What we try to achieve?

56

A computer system for recognition of the road signs which fulfills:

Accuracy (high accuracy rate).

Flexibility (operates in real traffic conditions).

Response time (real-time response).

Extensibility (easy to add new prototype signs).

Further assumption:

In this research we focus on

the circular signs

(most demanding than

rectangular & triangular)

– these are prohibition and

obligation signs.

Road signs recognition problem – simple or not?

57

Traffic scenes are acquired in different

viewpoints, distances,

weather & lighting conditions, ...

Exemplary application – road signs recognition:

58

The architecture of the system for road signs recognition with HOSVD:

SIGN DETECTION

WITH AN ADAPTIVE

WINDOW TECHNIQUE

AND BASED ON

CHARACTERISTIC

COLORS

Sign cropping,

extraction of the

blue channel and

registration

Answer

”No right turn”CLASSIFICATION

IN THE TENSOR

SPACE

BASE TENSORS

FOR EACH SIGN

HOSVD

DETECTION MODULE

TRAINING MODULE

Deformable

prototypes

generator

Important factors:

• accuracy

• response time (training can take longer)

59

The data base – where take the prototypes for training from?

1. Take enough real examples (pictures taken of different

signs, in different conditions).

2. Take the formal specification – just one exemplar per sign

(formal regulations).


60

A generator of the geometrically deformed prorotype patterns

We generate rotated versions of

each of the prototypes, as well as we

add some Gaussian noise

In result we obtain a tensor T

of the deformed prototypes


61

1 1 2 2 P P S S S

Consecutive ”layers” of the core tensorZ

ENERGY FACTOR

1

PNh

h P Ph

s

1 1 2 2 1 1h P P S S S

Base tensors(!)

shP are columns of the unitary matrix SP.


62 Input color image (RGB) After color detection


63 Morphological filtering Proposed objects

(an adaptive window growing method)


64 Properly found object (fuzzy rules) The sign in the original image


65 Blue channel of the RGB image Registered test pattern

ˆxP input pattern


”No

right

turn”

66

Exemplary application – road signs recognition results:

Speed of 15-25 frames/s of resolution 640480.

The obtained accuracy on the group of the prohibition signs reached 95%

(daily conditions).

A threshold =0.85 was set below which the system answers “don’t know”. Such

a situation arises e.g. if a pattern is provided which the system was not trained for.

Error rate – prohibition signs Error rate – obligation signs

67

A word on implementations

68

The HOSVD algorithm:

1. For each k=1, …, P do: a. Flatten tensor T to obtain T

k ;

b. Compute Sk from the SVD decomposition

of the flattening matrix Tk

Tk=S

kVkDkT

2. Compute the core tensor from the formula:

1 1 2 2

T T T

P P S S S

Experimental setup – implementation issues:

The HOSVD algorithm relies on successive application of the matrix SVD

decompositions applied to each of the flattened versions T(k) of the original tensor T.

In result the Sk matrices are obtained.

The problem – how to avoid data copying for different modes of T(k) in HOSVD?

69

The HOSVD algorithm:

1. For each k=1, …, P do: a. Flatten tensor T to obtain T

k ;

b. Compute Sk from the SVD decomposition

of the flattening matrix Tk

Tk=S

kVkDkT

2. Compute the core tensor from the formula:

1 1 2 2

T T T

P P S S S

Experimental setup – implementation issues:

The HOSVD algorithm relies on successive application of the matrix SVD

decompositions applied to each of the flattened versions T(k) of the original tensor T.

In result the Sk matrices are obtained.

The problem – how to avoid data copying for different modes of T(k) in HOSVD?

Can be parallelized (Cyganek, 2013)

A class hierarchy for efficient tensor representation

70

TImageFor

# fData : pixel type = val

Pixel Type

+ GetPixel( matrix_index ) : PixelType

N

1..*

Composite

relation

TFlatTensorFor

# fTensorMode : int

Element

Type

TFlatTensorProxyFor

Element

Type

- fMotherTensor : TFlatTensorFor &

+ SetPixel( matrix_index ) : PixelType

HIL Library

+ GetElement( TensorIndex ) : ElType

+ SetElement( TensorIndex, ElType )

# Offset_ForwardCyclic

( TensorIndex, MatrixIndex, Mode )

# Offset_BackwardCyclic

( TensorIndex, MatrixIndex, Mode )

# fIndexVector: vector



Action delegation to

the mother tensor

+ GetPixel( matrix_index ) : PixelType

+ SetPixel( matrix_index ) : PixelTypeMatrix data access

with mapped matrix

index to the proxy

mode

Matrix

representation

# Offset_BackwardCyclic

( MatrixIndex, TensorIndex, Mode )

# Offset_ForwardCyclic

( MatrixIndex, TensorIndex, Mode )

Proxy Design Pattern

71

TensorAlgebraFor

Best_Rank_R_DecompFor

T

S_Matrix_Initializer

+ operator()( const FlatTensor & T,

const RankVector & ranks,

SingleMatrixVector & S_vector ) = 0 : bool

T

OrphanInitializedMatrices_S_UniformRandomGenerator

T

+ operator()( const FlatTensor & T, const RankVector & ranks, SingleMatrixVector & S_vector ) : bool

T

# fMatrix_Initializer_Obj : S_Matrix_Initializer< T, ACC > *

11

+ FindDominantSubspace(

Compute_SVD & svd_calculator,

const typename FlatTensor::DataMatrix & S_hat,

typename FlatTensor::DataMatrix & S,

int requested_tensor_Rank_k,

int tensor_index_k );

+ operator() ( const FlatTensor & T,

const RankVector & requested_ranks,

typename SingleMatrixVector & S_vector,

const AccumulatorType epsilon = 1e-6,

const int max_iter_counter = 1000,

int * num_of_iterations = 0 ) : FlatTensor_AP

1

TFlatTensorFor

# fTensorMode : int




T

+ TFlatTensorFor(

const TensorElem_IndexVector & indexVec,

int tensor_mode,

ECyclicMode cyclicMode = kForward )

N

HOSVD

+ operator()( const TFlatTensorFor< U > & T,

SingleMatrixVector & S_vector )

: FlatTensor_AP

T

Software framework for tensor representation and decomposition:

DeRecLib available on the Internet for free ...

72

TensorAlgebraFor

Best_Rank_R_DecompFor

T

S_Matrix_Initializer

+ operator()( const FlatTensor & T,

const RankVector & ranks,

SingleMatrixVector & S_vector ) = 0 : bool

T

OrphanInitializedMatrices_S_UniformRandomGenerator

T

+ operator()( const FlatTensor & T, const RankVector & ranks, SingleMatrixVector & S_vector ) : bool

T

# fMatrix_Initializer_Obj : S_Matrix_Initializer< T, ACC > *

11

+ FindDominantSubspace(

Compute_SVD & svd_calculator,

const typename FlatTensor::DataMatrix & S_hat,

typename FlatTensor::DataMatrix & S,

int requested_tensor_Rank_k,

int tensor_index_k );

+ operator() ( const FlatTensor & T,

const RankVector & requested_ranks,

typename SingleMatrixVector & S_vector,

const AccumulatorType epsilon = 1e-6,

const int max_iter_counter = 1000,

int * num_of_iterations = 0 ) : FlatTensor_AP

1

TFlatTensorFor

# fTensorMode : int




T

+ TFlatTensorFor(

const TensorElem_IndexVector & indexVec,

int tensor_mode,

ECyclicMode cyclicMode = kForward )

N

HOSVD

+ operator()( const TFlatTensorFor< U > & T,

SingleMatrixVector & S_vector )

: FlatTensor_AP

T

Software framework for tensor representation and decomposition:

• Object-oriented design (templates, C++)

• Software attached to the recent book: ”Object Detection

and Recognition in Digital Images ”, Wiley, 2013

DeRecLib available on the Internet for free ...

73

Other applications ...

74

Other applications of multi-dimensional pattern recognition:

Chemistry.

Smilde A., Bro R., Geladi P.: Multi-way Analysis: Applications in the Chemical

Sciences. Wiley, 2004.

75


Web link analysis.

Kolda T.G., Bader B.W., Kenny J.P.: Higher-Order Web Link Analysis Using

Multilinear Algebra. Sandia Report, 2005

Chemistry.



76


Web link analysis.



Signal filtering.

Muti D., Bourennane S.: Survey on tensor signal algebraic filtering. Signal

Processing 87, pp. 237-249, 2007.

Chemistry.



77


Ensembles of tensor based classifiers:

Cyganek B.: Ensemble of Tensor Classifiers Based on the Higher-Order

Singular Value Decomposition. Springer, Lecturer Notes in Computer Science

7209, pp. 578–589, 2012

Web link analysis.



Signal filtering.

Muti D., Bourennane S.: Survey on tensor signal algebraic filtering. Signal

Processing 87, pp. 237-249, 2007.

Chemistry.



78

Conclusions

79

Conclusions:

Tensors provide an efficient tool for multi-dimensional data

processing, pattern recognition (image processing).

Tensor decompositions allow a view into data contents.

Nonnegative decompositions allow easy interpretation of the

factoring components.

The HOSVD decomposition can be effectively used for pattern

recognition (face, road signs, handwritten chars, chemistry, ...).

Very good accuracy and fast response.

Effective computer implementation (software, parallel operation).

80

Thank you!

Documents

TENSORY w rozpoznawaniu wzorcó · Short introduction to the tensor mathematics. Tensor decompositions as a way of multi-dimensional data analysis. ... (a valence) of a tensor Bishop