Key aspects of the tensor-matrix theory of analysis and processing of multichannel measuring signals in the classical and neural network approaches

Abstract

This presentation is devoted to the tensor-matrix theory of the traditional approach to multichannel signal parameters measurements based analytical model of signals and the tensor-matrix theory of Neural Networks for Mechanical Measurements.
This presents the basic concepts of matrix operations that can be used for ultrasonic, sonar, radar systems, wireless communications, and more systems with digital beamforming. It is intended for individuals in the field who wish to gain a general view of this area. For additional information, read the references on the final slides.

Full text

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
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PUBLIC RELEASE
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
PUBLIC RELEASE
1st part
The 10th International Symposium on Precision Mechanical
Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

Examples of the using multichannel measurement systems
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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
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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

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PUBLIC RELEASE
The Typical
Multichannel
Ultrasonic
System
with Digital
Beamforming
Secondary
Channels
U = QA
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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
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
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


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


The 10th International Symposium on Precision Mechanical
Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

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PUBLIC RELEASE
The matrix's
models of
two-
coordinates
ultrasonic
system with
digital
beamforming
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M
2
1
MT
2T
1T
M1
21
11
MR1R
M111
MR1R
M111
a
a
a
F00
0F0
00F
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00F
xQxQ
xQxQ
xQxQ
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U
0
0
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     
;
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FFF
FFF
F
MT2T1T
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M12111
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,F
a00
0a0
00a
QU T
M
2
i
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


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

(1)
(2)
The 10th International Symposium on Precision Mechanical
Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

 
i
ij BaBA ^
The example
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333231
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3231
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bbb
bbb
bbb
B
aa
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aa
A
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
333232323132333132313131
232222222122232122212121
131212121112131112111111
babababababa
babababababa
babababababa
BA ^
Face-
splitting
product
of
matrices
8
PUBLIC RELEASE
The 10th International Symposium on Precision Mechanical
Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

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PUBLIC RELEASE
Khatry-Rao
product of
matrices
(Transposed
Face-
splitting
product of
matrices)
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j
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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
PUBLIC RELEASE
The fundamental properties of matrix operations
 
TT
TBABA %^ 
      
DBCADCBA % % 
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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

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The fundamental
properties of the
proposed matrix
operations
(continue)
B
a
a
a
B
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p
2
1
p
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1
^
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TTTT baba ^
baba %
baba ^
a and b is k-vector
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TT % ^ % QFbQbFbFQinvec T
M
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,
aaaa
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aaaa
Ainvec
p3p63
1p4p52
2p5p41
3
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 
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
PUBLIC RELEASE
     
     
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,
a
a
a
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M
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
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% %
The matrix
model of 4 -
coordinates
ultrasonic
system with
digital
beamforming
     
     
     
     
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% %
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yVyVyV
yVyVyV
yVyVyV
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
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

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


- angular coordinates , frequencies and ranges of signals sources
The 10th International Symposium on Precision Mechanical
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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
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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),
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
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
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 RNTD- matrix and
The 10th International Symposium on Precision Mechanical
Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

15
PUBLIC RELEASE
The
factorization
of
Neudecker
derivation of
transposed
face-splitting
product
 
? ? ?
% %
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M
MM
1
1
11
yyy
q
Q
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q
Q
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FVQ
Y
FVQ
Y
P
 
T
M1M1M1 xxyyY  
 
 
,
F0
F0
F
2T
21
2
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

 
 
,
F
F
F
11T
111
1
1
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







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
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





 ??
The 10th International Symposium on Precision Mechanical
Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

16
PUBLIC RELEASE
The
modular
face-
splitting
product of
matrixes
 
ijij BABA ^@ 

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










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
PUBLIC RELEASE
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
PUBLIC RELEASE
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
PUBLIC RELEASE
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
PUBLIC RELEASE
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
PUBLIC RELEASE
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
PUBLIC RELEASE
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
PUBLIC RELEASE
The 10th International Symposium on Precision Mechanical
Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

24
PUBLIC RELEASE
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

25
PUBLIC RELEASE
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
PUBLIC RELEASE
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
PUBLIC RELEASE
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
PUBLIC RELEASE
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
PUBLIC RELEASE
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
PUBLIC RELEASE
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
PUBLIC RELEASE
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
PUBLIC RELEASE
The 10th International Symposium on Precision Mechanical
Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

33
PUBLIC RELEASE
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

34
PUBLIC RELEASE
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
PUBLIC RELEASE
The 10th International Symposium on Precision Mechanical
Measurement , 15 - 17 October, 2021, Qingdao, China, VTC

Advanced approach to description of Neural Network model
36
PUBLIC RELEASE
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
37
PUBLIC RELEASE
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
PUBLIC RELEASE
₿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
PUBLIC RELEASE
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
PUBLIC RELEASE
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