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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

Authors:
  • Central Scientific Research Insitute of Armaments and Military Equipment of Armed Forces of Ukraine

Abstract and Figures

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.
Content may be subject to copyright.
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
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
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
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
37
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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
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]= [[AbcBmr]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), 379384. 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), 2328. 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
... The importance of antenna arrays in different technical fields has led to the intensive development of the mathematical theory of transmitting and receiving antenna arrays of various types, which is presented in many publications [Slyusar, 1999[Slyusar, , 2021. This developing mathematical theory provides, in particular, new, previously unknown approaches in the matrix-tensor analysis of complex systems not only for antenna arrays technology but also for mathematical natural science [Minochkin et al., 2011]. ...
... The article is based, in particular, on the results previously published by the author on the universal rules of stochastic organization of genomic DNAs, discovered through matrix analysis of nucleotide sequences in single-stranded DNAs of eukaryotic and prokaryotic genomes [Petoukhov, 2019[Petoukhov, , 2020a[Petoukhov, -c, 2021a. The formalisms of algebra-matrix representations of these universal rules of genomes drew the author's attention to their analogy with the formalisms of the tensor-matrix theory of digital antenna arrays [Slyusar, 1999[Slyusar, , 2021. This particular analogy led to a general question about the possible existence of analogies between inherited physiological systems, which use photonic and other wave interrelations, and the amazing emergent properties of antenna arrays. ...
... The problems arising here about Smart Antennas, sometimes containing hundreds and thousands of information processing channels, have led to the development of new tools for matrix analysis that have not previously been encountered in other areas of science and technology. First of all, we are talking about the tensor-matrix theory of antenna arrays by the Ukrainian scientist V. Slyusar, who proposed some new operations with matrices [Slyusar, 1999[Slyusar, , 2021. These new operations in the theory of digital antenna arrays (Smart Antennas) are closely related to the Hadamard product of matrices, which, as shown above, turned out to be adequate for describing the universal rules for the stochastic organization of genomic DNA (Figs. ...
Article
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The article is devoted to the possibilities of considering the evolution of biological systems in connection with the unique emergent properties of antenna arrays, that is, systems of mutually matched antennas widely used in technology. Materials are presented in favor of the proposition that the evolution of biosystems can be formally considered as the evolution of systems of bio-antenna arrays and their energy-information wave activity, which participates in biological computation and contributes to the unification of body parts into a coherent whole. The use of digital antenna arrays in technology is based on their tensor-matrix theory. The author discovers a structural analogy of this theory with the tensor-matrix features of genetic coding systems, as well as algebraic modeling of the universal rules for the stochastic DNA organization of the genomes of higher and lower organisms. This analogy is just one of the facts presented in the article in favor of the usefulness of borrowing knowledge from modern antenna technology to consider the evolution of biosystems. The described new approach may exist along with other known approaches in evolutionary biology.
... Следует отметить, что применение в выражении (4) операций умножения на блок-строку и блок-вектор единиц является лишь данью традиции использования обычных матричных операций. Формальное выполнение таких умножений существенно увеличивает количество вычислений, поэтому чтобы избежать усложнения обработки, предлагается использовать свёрточное проникающее (символ ◙∑ [3]) или свёрточное прямое проникающее ([◙∑]) произведения, в которых операция поблочного Адамарова умножения дополняется суммированием всех результирующих элементов внутри отдельно взятого блока. С учётом такой модификации выражение (4) можно переписать в виде: ...
... Аналіз відеопотоку в системах моніторингу -перспективний напрямок розвитку моделей, методів штучного інтелекту і машинного навчання [1][2][3][4]. Технології машинного навчання успішно застосовуються в багатьох галузях. За допомогою комп'ютерного зору проводять розвідку об'єктів моніторингу (ОМ) шляхом аналізу аерофотознімків і відеопотоків. ...
... Одним з таких варіантів може бути використання згорткових нейронних мереж (Convolutional Neural Networks, CNN). Однак тільки в останнє десятиліття CNN знайшли широке застосування в розпізнаванні об'єктів в потоковому відео [2,3]. Невирішеним залишається завдання процесу автоматизації обробки відеоданих, особливо в БПАК. ...
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In this paper, the tensor-matrix model of LeNet5 is proposed.
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The versions of the mathematical formalization of neural hypernetworks based on the family of penetrating face products of matrices and tensors expanded to the block matrices are considered. As an example, the matrix A in the penetrating face product of matrix A and block matrix B can be considered as a picture pixels matrix on the input of a neural network. In this case, every block of matrix B corresponds to a block of weight coefficients for a few neurons in one layer of the neural network. Further steps of data processing in the considered neural network can be varied depending on the structure and type of layers of the neural network. In the case of convolutional neural networks the result of penetrating face products of matrices A and B has to be multiplied by a vector of one’s 1. This multiplication can produce a scalar, a vector-row, a vector, or a matrix. The result of such multiplication can be used as argument of an activation function. For the data processing in hierarchies of neural hypernetworks clusters, the generalized face-splitting products of matrices and block versions of these multiplications can be used. The operation of block penetrating Kronecker product of matrices has been introduced to simulate the input layer of a neural hypernetwork which processes multiple video streams from several video cameras in different spectral ranges in parallel by a set of several neural networks.
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Under consideration of multi coordinate digital antenna arrays (DAA) there is the problem of compact matrix record of the reception channels. The known mathematical apparatus does not allow to use habitually for the perception structures of matrixes, describing directivity characteristics of the antenna elements and amplitude-frequency characteristics of the filters. For the solution of the given problem, it is offered to operate with a special type of the product of matrixes, named by the author as "face-splitting".
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The paper describes new matrix operations for compact notation of responses in radio engineering systems, which use the technology of digital formation of radiation patterns of antenna arrays in the case of nonidentical reception channels.
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This paper is the first publication about concept of face-splittiing product of matrices and his properties, which dated on December 1996. In this article was used the old version of English translation the origin term "торцевое произведение", which was introduced by Vadym Slyusar. In next publications was used translation as face-splitting product.
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This article continued the description of properties of the face-splitting product of matrices, which was proposed by Vadym Slyusar in 1996.
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The new concept of face-splitting and transposed face-splitting matrix products is determined; its main characteristics and modifications of the new types of products for module matrices are considered