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Key aspects of the tensor-matrix theory of

analysis and processing of multichannel

measuring signals in the classical and

neural network approaches

VADYM I. SLYUSAR

Doctor of Sciences, Professor,

Principal Research Fellow of Central Research Institute of Armaments

and Military Equipment of Ukraine’s Armed Forces

16 October, 2021, Qingdao, China, VTC

PUBLIC RELEASE

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

https://youtu.be/3OEI0uMhXfs

Introduction

2

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Vadym Slyusar (born 15 October 1964, Poltava region, Ukraine) -

Soviet and Ukrainian scientist, Professor (2005), Doctor of

Technical Sciences (2000), Honored Scientist and Technician

of Ukraine (2008).

The founder of tensor-matrix theory of digital antenna arrays,

N-OFDM and other theories in fields of radar systems, smart

antennas for wireless communications and digital

beamforming, has 68 patents and 905 publications in this

areas.

The chief of a research department in Central Research Institute

of Armaments and Military Equipment of Armed Forces of

Ukraine

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

The tensor-matrix theory of the traditional

approach to multichannel signal

parameters measurements based

analytical model of signals

3

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1st part

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

Examples of the using multichannel measurement systems

4

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Fig.1. Schematic diagram of multiple signal superposition of

the angular time grating (Zhonghua Gao, Fangyan Zheng,

Study on the Method of Error Separation and Compensation

based on Multiple Signal Superposition... , 2016.

doi: 10.1117/12.2211599)

Fig.2. Configuration of the weld joint with two plates and

the flexible transducer array (Zhenying Xu, Yuanxia

Wang, Han Du, Wei Fan, Ultrasonic wave focusing on

flexible array sensors in weld detection, 2019,

doi: 10.1117/12.2548742)

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

5

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A MATRIX TECHNIQUE FOR DIGITAL SIGNALS

PROCESSING HAVE A FOLLOWING ADVANTAGES:

•the compactness of mathematical models of physical systems;

• the best presentation of essence of signal processing

algorithms;

• computer time economy.

The matrix means is especially advantageous for the digital

multichannel systems of data processing !

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

6

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The Typical

Multichannel

Ultrasonic

System

with Digital

Beamforming

Secondary

Channels

U = QA

M

2

1

MR2R1R

M22212

M12111

R

2

1

a

a

a

A;

)x(Q)x(Q)x(Q

)x(Q)x(Q)x(Q

)x(Q)x(Q)x(Q

=Q ;

U

U

U

U

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

7

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The matrix's

models of

two-

coordinates

ultrasonic

system with

digital

beamforming

M

2

1

MT

2T

1T

M1

21

11

MR1R

M111

MR1R

M111

a

a

a

F00

0F0

00F

F00

0F0

00F

xQxQ

xQxQ

xQxQ

xQxQ

U

0

0

;

FFF

FFF

FFF

F

MT2T1T

M22212

M12111

,F

a00

0a0

00a

QU T

M

2

i

(1)

(2)

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

i

ij BaBA ^

The example

333231

232221

131211

3231

2221

1211

bbb

bbb

bbb

B

aa

aa

aa

A

333232323132333132313131

232222222122232122212121

131212121112131112111111

babababababa

babababababa

babababababa

BA ^

Face-

splitting

product

of

matrices

8

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The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

9

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Khatry-Rao

product of

matrices

(Transposed

Face-

splitting

product of

matrices)

23

13

23

22

12

22

21

11

21

23

13

13

22

12

12

21

11

11

b

b

a

b

b

a

b

b

a

b

b

a

b

b

a

b

b

a

BA%

The example

j

ij BaBA %

232221

131211

232221

131211

bbb

bbb

B

aaa

aaa

A

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

STLMCKAB

TKSCMBLA

% ^

10

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The fundamental properties of matrix operations

TT

TBABA %^

DBCADCBA % %

DBCADCBA ^ ^

DBCADCBA %^

Theorem 1

Theorem 2

Theorem 3

Theorem 4

""

denotes element-wise multiplication (Hadamard product )

""

denotes Kronecker product

Theorem 5

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

11

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The fundamental

properties of the

proposed matrix

operations

(continue)

B

a

a

a

B

a00

0a0

00a

p

2

1

p

2

1

^

TTTT baba ^

baba %

baba ^

a and b is k-vector

TT % ^ % QFbQbFbFQinvec T

M

,

aaaa

aaaa

aaaa

Ainvec

p3p63

1p4p52

2p5p41

3

T

p 1-p 2-p 3-p 4-p 5-p654321 aaaaaaaaaaa a =A

M

invec

denote the inversion vectorization operator (new)

The example where

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

12

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,

a

a

a

zSzSzS

zSzSzS

zSzSzS

FFF

FFF

FFF

M

2

1

MD2D1D

M22212

M12111

MT2T1T

M22212

M12111

% %

The matrix

model of 4 -

coordinates

ultrasonic

system with

digital

beamforming

% %

MN2N1N

M22212

M12111

MR2R1R

M22212

M12111

yVyVyV

yVyVyV

yVyVyV

xQxQxQ

xQxQxQ

xQxQxQ

- angular coordinates , frequencies and ranges of signals sources

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

13

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Estimation of parameters of M signals sources using maximum

likelihood method

denoted the matrix trace operation,

min,PAUPAUtrL

~

The measuring procedure in four-coordinate variant is reduced to

maximization of expression

T

1

TPPPPG,GUUtrL

T

1

TTTT PSSFFVVQQPG

tr

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

14

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The Cramer-

Rao bound

- the Neudecker derivate of matrix P ( P is the function of matrix Y),

Y

P

)1(AA

Y

P

P)(A

Y

P

Y

P

)P(APP

1

=I

RNTD

*

TT

T*T

2

Pvec

YvecY

P

For 4 - coordinates ultrasonic system with digital beamforming

vec

- denote vectorization ( stacking columns of a matrix to form a vector).

2

is the noise dispersion,

where

RNTD

1

is identity RNTD- matrix and

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

15

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The

factorization

of

Neudecker

derivation of

transposed

face-splitting

product

? ? ?

% %

M

M

MM

1

1

11

yyy

q

Q

QQ

q

Q

QQ

FVQ

Y

FVQ

Y

P

T

M1M1M1 xxyyY

,

F0

F0

F

2T

21

2

,

F

F

F

11T

111

1

1

For example: 3 coordinates and M signals sources

;

FF

F

FF

F

V

y

V

V

V

y

V

V

MM

M

M

11

1

1

M

M

M

M

1

1

1

1

??

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

16

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The

modular

face-

splitting

product of

matrixes

ijij BABA ^@

3222312232213121

2222212222212121

1222112212211121

3212311232113111

2212211222112111

1212111212111111

2221

1211

aaaa

aaaa

aaaa

aaaa

aaaa

aaaa

AA

AA

=A

332232223122332132213121

232222222122232122212121

132212221122132112211121

331232123112331132113111

231222122112231122112111

131212121112131112111111

2221

1211

bbbbbb

bbbbbb

bbbbbb

bbbbbb

bbbbbb

bbbbbb

BB

BB

=B

,

BABA

BABA

=BA

22222121

12121111

^^

^^

@

The example

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

17

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The modular transposed face-splitting product of matrixes

ijij BABA % ?

The example

TT BA

BA

T @

?

% %

% %

% %

?

32

32

31

31

22

22

21

21

12

12

11

11

3231

2221

1211

3231

2221

1211

BABA

BABA

BABA

BB

BB

BB

AA

AA

AA

Theorem 6

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

18

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The fundamental properties of a modular variants of

face-splitting products

Theorem 9

i

ikjiikjijkikikjiji MBKAPM KB A ?@

i

ikjiikjijkikikjiji MBKAPMKB A ^ @

Theorem 7

Theorem 8

.MBKAPM KBA

i

ikjiikjijkikikjiji

% ?

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

19

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The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

Digital

Beamforming

U(f)

f

FFT Filters

V

H

A

A

VH

VH

FF

FF

?

?

VH

VH

QQ

QQ

VH

HV

q

q

U

V

H

U

U

Dual polarization case

V

H

HA

A

VHVH

VHV

q

q

FQFQ

FQQ

VH

VH F

- XPI (Cross Polarization

Isolation)

VH

q

HV

q

and

H ,V – horizontal and vertical polarizations

20

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The matrix's model of multiple sections of 4 - coordinates system

with digital beamforming

g - the number of

section

MNg1Ng

M1g11g

MN11N1

M11111

MRg1Rg

M1g11g

MR11R1

M11111

yVyV

yVyV

yVyV

yVyV

V

~

xQxQ

xQxQ

xQxQ

xQxQ

Q

~ ,

ASFVQU ? ? ?

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

21

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The matrix's model of multiple positions 4 - coordinates system

MNg1Ng

M1g11g

MN11N1

M11111

MRg1Rg

M1g11g

MR11R1

M11111

yVyV

yVyV

yVyV

yVyV

V

~

xQxQ

xQxQ

xQxQ

xQxQ

Q

~ ,

g - the position

number

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

22

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Lower Cramer-

Rao bound of

magnitudes

estimation

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

LCRB

A

A

MSE

""

denotes element-wise (Hadamard product )

Penetrating face product of matrices

n

BA

2

BA

1

BA

n

BABA = $ $

n

2

1

BA

BA

BA

=BA $

The example:

O R

BA $

,

aa

aa

aa

3231

2221

1211

A

3

2

1

B

B

B

B

3233231331

2232221321

1231211311

3223231231

2222221221

1221211211

3213231131

2212221121

1211211111

baba

baba

baba

baba

baba

baba

baba

baba

baba

,

bb

bb

bb

bb

bb

bb

bb

bb

bb

323313

223213

123113

322312

222212

122112

321311

221211

121111

23

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The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

24

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The matrix's model of 3 - coordinates ultrasonic system with digital

beamforming for no identical channels (1 source)

t21 FQFQFQa=FQaU $

)y (x, Q)y (x, Q)y (x, Q

)y (x, Q)y (x, Q)y (x, Q

)y (x, Q)y (x, Q)y (x, Q

= Q

RRR2R1

2R2221

1R1211

RRTT1R

RT1T11

1RR11R

1R1111

FF

FF

FF

FF

=F

- matrix of the directivity characteristics

of primary channels in azimuth and

elevation angle planes (can not be

factorized)

- matrix of the frequency

responses

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

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The generalized penetrating product

ig2i1iij BBBAB

~

A $ ^

^

^

PG2P1P

G22221

G11211

PT2P1P

T22221

T11211

BBB

BBB

BBB

~

AAA

AAA

AAA

B

~

A

.

$ $

$ $

$ $

PG1PPTPG1P1P

G221T2G22121

G111T1G11111

A

BBABBA

BBABBA

BBABB

The example:

The definition:

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

26

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The generalized transposed penetrating product

%

% ~

AAA

AAA

AAA

B

~

A

PT2P1P

T22221

T11211

$ $ $

$ $ $

%

PG

G2

G1

PT

2P

22

12

2P

1P

21

11

1P

PG

G2

G1

T1

2P

22

12

12

1P

21

11

11

PG2P1P

G22221

G11211

B

B

B

A

B

B

B

A

B

B

B

A

B

B

B

A

B

B

B

A

B

B

B

A

BBB

BBB

BBB

~

Gj

j2

j1

ij

B

B

B

AB

~

A

$ %

The definition:

The example:

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

27

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The matrix's model of 4-coordinates system with digital beamforming

for no identical channels (1 source)

)y (x, Q)y (x, Q)y (x, Q

)y (x, Q)y (x, Q)y (x, Q

)y (x, Q)y (x, Q)y (x, Q

= Q

RRR2R1

2R2221

1R1211

- matrix of the directivity

characteristics of primary

channels in azimuth and

elevation angle planes (can

not be factorized)

aFSFSFSQaF

~

SQU D21

$ $ $ $ ^ $

zSzS

zSzS

zSzS

zSzS

=S

RRDD1R

RD1D11

1RR11R

1R1111

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

28

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The matrix's model of multistatic system with digital beamforming

for no identical channels (1 source)

,a

FF

FF

~

SS

SS

~

Q

Q

Q

= U

TP1P

T111

DP1P

D111

P

2

1

^^

,

)y (x, Q)y (x, Q

)y (x, Q)y (x, Q

)y (x, Q)y (x, Q

)y (x, Q)y (x, Q

)y (x, Q)y (x, Q

)y (x, Q)y (x, Q

Q

Q

Q

RRPR1P

2RP21P

1RP11P

RR1R11

2R1211

1R1111

P

2

1

,

zSzS

zSzS

=S

RRdpdp1R

Rdp1dp11

dp

.

FF

FF

=F

RRtptp1R

Rtp1tp11

tp

,aFSQU tpdppdtp

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

The tensor-matrix theory of Neural Networks

for Mechanical Measurements

29

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2nd part

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

Embedded Machine Learning for Predictive Maintenance

The building of incidence matrices for every engine mode or for every

partial network inside the engine structure

30

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The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

The co-occurrence matrix for the analysis of triple combinations

31

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The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

Elements of the matrix C characterize the frequency of

occurrence of specific triplets of parameter ranges in the

studied sequence of modes

From the matrix C it follows that the sets of working parameter ranges (X2Y1Z3), (X1Y2Z4), and (X1Y4Z2) occur once.

32

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The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

33

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The co-occurrence

matrix for the

analysis of

4 combinations of

pictures

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

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Tensor Sketch

* Ahle, Thomas Almost Optimal Tensor Sketch. Researchgate (3 Sept. 2019).

*

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

Penetrating face product of matrices - the part of tensor-matrix

theory of Neural Networks

35

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The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

Advanced approach to description of Neural Network model

36

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Few properties of penetrating face product of matrices are:

where □ denotes the face-splitting product of matrices,

is the symbol of Kronneker product, 1T – vector-row of ones.

If c is a vector then c □ A= A □ c=c ◙ A= A ◙ c.

3231

2221

1211

aa

aa

aa

323313

223213

123113

322312

222212

122112

321311

221211

121111

bb

bb

bb

bb

bb

bb

bb

bb

bb

A ◙ B =

323323133132232312313213231131

223222132122222212212212221121

123121131112212112111211211111

babababababa

babababababa

babababababa

B = A =

n

BA

2

BA

1

BA

n

BABA = $ $

A ◙ B = B ◙ A, A □ A= A ◙ (A 1T),

•The matrix A can be considered as an input matrix of picture pixels.

•Every block of matrix B corresponds to a block of weight coefficients for few

neurons in one layer of the Neural Network.

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

To description of Convolutional Neural Networks

Softmax((A B)×1+d) or ReLU(1T(A B)1+d),

where × is the conventional product of matrices, 1 – a vector of ones, d is a vector or a

scalar Softmax((A B)[×]1+d) or Tanh (1T(A B)+d),

where [×] is the blocked conventional product of matrices, 1 – a block vector of ones

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In particular, the multiplication can have one of the forms

•a vector-row 1T(A ◙ B);

•a vector (A ◙ B)×1, where × is the conventional product of matrices, 1 – a vector of

ones;

•a matrix (A ◙ B) [×]1, where [×] is the blocked conventional product of matrices, 1 – a

block vector of ones;

•a scalar 1T(A ◙ B)1.

The convolutional neural networks model based on the penetrating product of

matrices must be multiplied by a vector of ones.

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

AlexNet Neural Network model

38

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₿2= ReLU [1T [×] (A1 [◙] B1) [×] 1]

₿3 = max pool [B2]

₿4= ReLU [1T [×] (A2 [◙] ₿3) [×] 1]

₿3 = max pool [P [◙] B2]

OR

₿4= ReLU [1T [×] (A2 [◙] ₿3) [×] 1]

₿5 = max pool [B4]

₿6= ReLU [1T[×] (A3 [◙] B5) [×] 1]

₿7= ReLU [1T[×] (A4 [◙] B6) [×] 1] ₿8= ReLU [1T[×] (A5 [◙] B7) [×] 1]

B10= ReLU [A7ReLU [A6 dropout(B9)]]

B11= Sofmax [A8 B10]

₿9 = max pool [B8]

A – a block-matrix of

weight coefficients;

B - data matrix

[◙] - penetrating

Kronecker

product

(Slide No. 41 )

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

TensorFlow example

39

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tf.multiply(M, V)

𝟏𝟎

𝟐𝟎

𝟑𝟎

×

𝟏 𝟐

𝟐 𝟑

𝟑 𝟒

𝟑 𝟒

𝟒 𝟓

𝟓 𝟔

=

=

𝟏 𝟐

𝟐 𝟑

𝟑 𝟒

𝟑 𝟒

𝟒 𝟓

𝟓 𝟔

×

𝟏𝟎

𝟐𝟎

𝟑𝟎

=

=

𝟏𝟎 𝟐𝟎

𝟒𝟎 𝟔𝟎

𝟗𝟎 𝟏𝟐𝟎

𝟑𝟎 𝟒𝟎

𝟖𝟎 𝟏𝟎𝟎

𝟏𝟓𝟎 𝟏𝟖𝟎

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

Convolution Zoo of Neural Networks

40

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Operation symbol Context of operations

A ◙∑ B

Penetrating product

+

the summation of all matrix elements within each

block

A ◙M B Penetrating product + MaxPull operation within each block

A ◙∑M B Penetrating product + the row-wise summation of all matrix elements

within each block + MaxPull operation within resulting column of

elements within each block

A ◙∑TM B Penetrating product + the column-wise summation of all matrix

elements within each block

+ MaxPull

operation within resulting row of

elements within each block

A ◙2FMB Penetrating product + the 2D Fast Fourier Transform (FFT) within each

block + MaxPull operation within resulting block of elements

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

41

PUBLIC RELEASE

,

The block generalized penetrating product

222212221211

122112121111

21 AA=A AAAA

AAAA

222212221211

122112121111

21 BB=B BBBB

BBBB

221211

121111

=AA

AA

222212

122112

221211

121111 AA

AA

BB

BB

222212

122112 BB

BB

=

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

42

PUBLIC RELEASE

,

The block generalized column-wise penetrating product

222212221211

122112121111

21 AA=A AAAA

AAAA

222212221211

122112121111

21 BB=B BBBB

BBBB

221211

121111

=AA

AA

222212

122112

221211

121111 AA

AA

BB

BB

222212

122112 BB

BB

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

43

PUBLIC RELEASE

,

The penetrating Kronecker product

PTPP

T

T

AAA

AAA

AAA

21

22221

11211

PGPP

G

G

BBB

BBB

BBB

21

22221

11211

[◙]

T

AAA 11211

PGPP

G

G

BBB

BBB

BBB

21

22221

11211

[◙]

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

44

PUBLIC RELEASE

,

The block penetrating Kronecker product

A [[◙]] B=[Aij [◙] Bij]= [[Abc○Bmr]ij],

222212221211

122112121111

=A AAAA

AAAA

322312321311

222212221211

122112121111

=B

BBBB

BBBB

BBBB

A[[◙]]B =

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

45

PUBLIC RELEASE

References

1. Slyusar V. I. (1997) New operations of matrices product for applications of radars, in Proc. Direct and

Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-97), Lviv, September 15-17,

1997, P. 73-74 (in Russian).

2. Slyusar V. I. (1997) Analytical model of the digital antenna array on a basis of face-splitting matrixs

products, in Proc. ICATT - 97, Kyiv, p. 108 – 109.

3. Slyusar, V. I. (1998) End matrixs products in radar applications. Radioelectronics and Communications

Systems, Vol. 41, no. 3.

4. Slyusar, V. (1999). A Family of Face Products of Matrices and its Properties. Cybernetics and systems

analysis c/c of Kibernetika i sistemnyi analiz. Consultants bureau (USA), 3(35), 379–384. DOI:

10.1007/BF02733426.

5. Slyusar, V. I. (2003) Generalized face-products of matrices in models of digital antenna arrays with

nonidentical channels. Radioelectronics and Communications Systems, Vol. 46; Part 10, pages 9-17.

6. Slyusar V. (2021). Neural Networks Models based on the tensor-matrix theory. Problems of the

development of promising micro- and nanoelectronic systems (MNS-2021), 23–28. DOI:

10.31114/2078-7707-2021-2-23-28.

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

THANK YOU

FOR YOUR ATTENTION !

46

swadim@ukr.net

www.slysuar.kiev.ua

https://scholar.google.com.ua/citations?hl=ru&user=wSegaWsAAAAJ

https://orcid.org/0000-0002-2912-3149

https://www.scopus.com/authid/detail.uri?authorId=7004240035

https://www.researchgate.net/profile/Vadym-Slyusar

The 10th International Symposium on Precision Mechanical

Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

Video version of this presentation: https://youtu.be/3OEI0uMhXfs