KR102361647B1 - Distributed matrix computing apparatus based on group algebra and method of same - Google Patents

Abstract

The present invention relates to group algebra based distributed matrix computing apparatus for efficiently processing distributed information based on a mathematical principle of group algebra, and a method thereof. The group algebra based distributed matrix computing method for performing group algebra based distributed matrix computing by a central device in a distributed computing environment including the central device and at least one edge device comprises: generating a plurality of first partial matrixes and a plurality of second partial matrixes by dividing a first target matrix and a second target matrix as product operation targets by a preset partition parameter; encoding the first partial matrixes and the second partial matrixes based on a group algebra to generate a first encoded partial matrix and a second encoded partial matrix; transmitting the first encoded partial matrix and the second encoded partial matrix to the at least one edge device to acquire a product operation result with respect to the first encoded partial matrix and the second encoded partial matrix from the at least one edge device; and applying a restoring condition to a received product operation result from the edge device to restore a product operation result of the first target matrix and the second target matrix.

Description

Apparatus and method for group theory-based dispersion matrix operation

The present invention relates to an apparatus and method for calculating a distribution matrix based on group theory for efficiently performing distributed information processing based on a mathematical principle called group algebra.

High-dimensional matrix operations are essential to enable applications called high intelligence, such as big data analysis, augmented reality, and tactile communication, in real life. Since it takes a long time to perform such a high-load operation in a single device, research on improving performance in the form of distributing and distributing it to several edge devices is active. This technology, also called distributed computing, is being presented as one of the representative keywords of 6G, and is emerging as one of the technologies that will accelerate the intelligence of networks or the Internet of Things.

In addition to these environmental changes, distributed computing is being studied for various calculations. Among them, in the case of dispersion matrix multiplication, which has a more complex structure than four rule calculations, which is the basis for high-dimensional calculations, the operation speed, accuracy, and load of the entire system vary depending on how the divided matrices are encoded in each edge device. In the case of dividing the two matrices to be multiplied into m by n sub-matrices and p by n sub-matrices, respectively, as disclosed in the paper [1], current research in academia requires at least pmn+p-1 edge devices. It is only known that the original desired matrix operation can be recovered. In addition, in distributed computing, if the operation method of a single device is brought as it is, it may not be possible to restore the original operation.

The above-mentioned background art is technical information that the inventor possessed for the derivation of the present invention or acquired in the process of derivation of the present invention, and cannot necessarily be said to be a known technique disclosed to the general public prior to the filing of the present invention.

Paper [1] Q. Yu, M. A. Maddah-Ali, and A. S. Avestimehr, “Straggler mitigation in distributed matrix multiplication: Fundamental limits and optimal coding,” IEEE Transactions on Information Theory, vol. 66, no. 3, pp. 1920-1933, 2020. Paper [2] H. Cohn and C. Umans, “A group-theoretic approach to fast matrix multiplication,” in Proc. 44th Annual IEEE Symp. Foundations of Computer Science, pp. 438-449, 2003.

개의 엣지 장치가 있어야 원래 원하는 행렬 연산을 복구할 수 있다고 까지만 알려져 있는 종래 기술과 비교 시에

개보다 더 적은 개수의 엣지 장치로 행렬 연산을 복구하는데 있다.One problem of the present invention is, at least

In comparison with the prior art, which is known only to be able to recover the original desired matrix operation only when there are edge devices,

It consists in recovering matrix operations with fewer edge devices than one.

One object of the present invention is to restore the original operation result by applying a restoration condition to the operation result received from the edge device in order to solve the problem of the prior art in which restoration of the original operation may be impossible when the operation method of a single device is brought as it is. is doing

The problem to be solved by the present invention is not limited to the above-mentioned problems, and other problems and advantages of the present invention not mentioned can be understood by the following description, and more clearly understood by the embodiments of the present invention will be In addition, it will be understood that the problems and advantages to be solved by the present invention can be realized by means and combinations thereof indicated in the claims.

A group theory-based distribution matrix calculation method according to an embodiment of the present invention is a method of performing a group algebra-based distribution matrix operation by a central device in a distributed computing environment including a central device and one or more edge devices, Generating a plurality of first submatrices and a plurality of second submatrices by dividing each of the first and second target matrices to be multiplied by a predetermined partition parameter; 2 Encoding the submatrices based on the group theory to generate a first encoded submatrix and a second encoded submatrix, and transmitting the first encoded submatrix and the second encoded submatrix to one or more edge devices, , obtaining a product operation result for the first encoded submatrix and the second encoded submatrix from one or more edge devices, and applying a restoration condition to the product operation result received from the edge device to obtain the first target matrix and the second It may include the step of restoring the product operation result of the two target matrices.

A group theory-based distribution matrix operation device according to an embodiment of the present invention is a device for performing group theory-based distribution matrix operation in a distributed computing environment including a central device and one or more edge devices, wherein the central device includes: A division unit for generating a plurality of first sub-matrices and a plurality of second sub-matrices by dividing each of the first and second target matrices to be multiplied by a predetermined partition parameter, the first sub-matrices and An encoder that encodes the second submatrix based on the group theory to generate a first encoded submatrix and a second encoded submatrix, and the first encoded submatrix and the second encoded submatrix to one or more edge devices A processing unit for transmitting and obtaining a product operation result for the first encoded submatrix and the second encoded submatrix from one or more edge devices, and a first target matrix by applying a restoration condition to the product operation result received from the edge device and a restoration unit that restores a result of a multiplication operation of the second target matrix.

In addition to this, another method for implementing the present invention, another system, and a computer-readable recording medium storing a computer program for executing the method may be further provided.

Other aspects, features and advantages other than those described above will become apparent from the following drawings, claims, and detailed description of the invention.

According to the present invention, by performing a group algebra-based dispersion matrix multiplication operation, the complexity of matrix operation is restored by restoring the dispersion matrix operation with pmn edge devices less than pmn+p-1 compared to the prior art. can lower

In addition, by applying a restoration condition to the operation result received from the edge device to restore the original operation result, it is possible to solve the problem of the prior art in which restoration of the original operation may not be possible when the operation method of a single device is brought as it is.

Effects of the present invention are not limited to those mentioned above, and other effects not mentioned will be clearly understood by those skilled in the art from the following description.

1 is an exemplary diagram of a distributed computing environment including a central device, an edge device, a user terminal, and a network connecting them according to the present embodiment.
FIG. 2 is a block diagram schematically illustrating a configuration of a central device including a group theory-based dispersion matrix computing device according to an embodiment of the present invention.
3 is a graph showing the number of edge devices required according to the matrix size derived from FIG. 2 .
4 is a flowchart illustrating a method for calculating a group theory-based dispersion matrix according to the present embodiment.

Advantages and features of the present invention, and a method for achieving them will become apparent with reference to the detailed description in conjunction with the accompanying drawings. However, the present invention is not limited to the embodiments presented below, it can be implemented in a variety of different forms, and should be understood to include all transformations, equivalents, and substitutes included in the spirit and scope of the present invention. . The embodiments presented below are provided so that the disclosure of the present invention is complete, and to completely inform those of ordinary skill in the art to which the present invention pertains to the scope of the invention. In describing the present invention, if it is determined that a detailed description of a related known technology may obscure the gist of the present invention, the detailed description thereof will be omitted.

The terms used in the present application are only used to describe specific embodiments, and are not intended to limit the present invention. The singular expression includes the plural expression unless the context clearly dictates otherwise. In the present application, terms such as “comprise” or “have” are intended to designate that a feature, number, step, operation, component, part, or combination thereof described in the specification exists, but one or more other features It should be understood that this does not preclude the existence or addition of numbers, steps, operations, components, parts, or combinations thereof. Terms such as first, second, etc. may be used to describe various elements, but the elements should not be limited by the terms. The above terms are used only for the purpose of distinguishing one component from another.

Hereinafter, embodiments according to the present invention will be described in detail with reference to the accompanying drawings. decide to do

1 is an exemplary diagram of a distributed computing environment including a central device, an edge device, a user terminal, and a network connecting them according to the present embodiment. Referring to FIG. 1 , the distributed computing environment 1 includes a single central device 100 , one or more edge devices 200:200_1,??,200_N, and one or more user terminals 300:300_1,??,300_N. and a network 400 .

The central device 100 may receive a job processing request signal from the user terminals 300:300_1,??,300_N. The central device 100 may request distributed processing to one or more edge devices 200:200_1,??,200_N for a job received from the user terminal 300:300_1,??,300_N. The central device 100 may collect distributed processing results from one or more edge devices 200:200_1,??,200_N, generate a job processing result signal, and transmit it to the user terminals 300:300_1,??,300_N. .

In this embodiment, the user terminal 300 may include a communication terminal capable of performing a function of a computing device (not shown), and includes a desktop computer, a smartphone, a tablet PC, a notebook computer, a smart TV, and a user-operated computer. Cell phones, personal digital assistants (PDAs), laptops, media players, micro servers, global positioning system (GPS) devices, e-book terminals, digital broadcast terminals, navigation devices, kiosks, MP3 players, digital cameras, home appliances and other mobile or non-mobile or non-electronic devices. It may be, but is not limited to, a mobile computing device. Also, the user terminal 300 may be a wearable terminal such as a watch, glasses, a hair band, and a ring having a communication function and a data processing function. The user terminal 300 is not limited to the above content, and a terminal capable of web browsing may be borrowed without limitation.

The network 400 may serve to connect the central device 100 , the edge device 200 , and/or the user terminal 300 . The network 400 may include, for example, wireless networks such as wireless LANs, CDMA, Bluetooth, and satellite communication, but the scope of the present invention is not limited thereto. Also, the network 400 may transmit/receive information using short-distance communication and/or long-distance communication. Here, the short-distance communication may include Bluetooth, radio frequency identification (RFID), infrared data association (IrDA), ultra-wideband (UWB), ZigBee, and wireless fidelity (Wi-Fi) technologies. Communication may include code division multiple access (CDMA), frequency division multiple access (FDMA), time division multiple access (TDMA), orthogonal frequency division multiple access (OFDMA), single carrier frequency division multiple access (SC-FDMA) technology. can

Network 400 may include connections of network elements such as hubs, bridges, routers, switches, and gateways. Network 400 may include one or more connected networks, eg, multiple network environments, including public networks such as the Internet and private networks such as secure enterprise private networks. Access to network 400 may be provided via one or more wired or wireless access networks. Furthermore, the network 400 may support an Internet of Things (IoT) network and/or 5G communication that exchanges and processes information between distributed components such as things.

와

를 곱하는 엣지 프로세싱 시스템을 고려할 수 있다. 본 실시 예에서는 중앙 장치(100)가

의 연산 결과를 목표하고, 이를 위해 행렬을 부분적으로 나눈 다음 이들을 부호화(encoding)하고 N개의 엣지 장치(200, 200_1,??,200_N)에게 나눠 연산을 하게 한 다음 연산 결과를 받아 복원하는 방식을 포함할 수 있다.Two matrices in the central device 100 in this embodiment

Wow

An edge processing system that multiplies by can be considered. In this embodiment, the central device 100

To achieve the operation result of may include

) 및 제2 부분행렬(

)로 나눌 수 있다.The specific method is as follows. First, the central device 100 converts the first target matrix and the second target matrix to be the target of the multiplication operation as follows to the first sub-matrix (

) and the second submatrix (

) can be divided into

) 및 제2 부분행렬(

)을 부호화한

와

를 생성하고, 각 엣지 장치(200, 200_1,??,200_N)에 서로 다른

와

를 전송할 수 있다.Then the central device 100 is a first sub-matrix (

) and the second submatrix (

) encoded

Wow

, and each edge device (200, 200_1,??, 200_N) has a different

Wow

can be transmitted.

와

를 연산할 수 있다.Each edge device (200, 200_1,??, 200_N) receives

Wow

can be calculated.

를 수신하여 다음 단계로 복원을 할 수 있다.

의 연결한 형태(concatenated version)의 행렬을 하기 수학식 2와 같이

로 표현할 수 있다.Then, the central device 100 calculates each edge device 200, 200_1,??, 200_N.

can be restored to the next step by receiving

A matrix of a concatenated version of

can be expressed as

꼴을 하기 수학식 3과 같은 열 벡터(column vector)로 정의할 수 있다.Also the submatrix products we want

The shape may be defined as a column vector as in Equation 3 below.

와

사이의 관계식을 수학식 4와 같이 알 수 있다.Because the central unit 100 knows how it has been coded,

Wow

The relation between the two can be found as in Equation (4).

를 각 엣지 장치(200, 200_1,??,200_N)로부터 수신한 결과인

에 곱하고,

를 획득하여 원래 원하고자 한

를 구할 수 있다.Therefore, the central device 100 is

is the result of receiving from each edge device (200, 200_1,??, 200_N)

multiplied by ,

by obtaining the original desired

can be obtained

In this embodiment, in order to focus on the core idea, well-known concepts related to group theory and representation theory are omitted. First, the triple product property is defined, and the thesis [2] We introduce only the embedding matrix multiplication into group algebra method proposed in .

에 대해 부분집합

의 오른 몫집합(right quotient set)을

로 표기한다. (즉,

). 만약 세 부분집합

이 각 원소

,

,

가 하기 수학식 5와 같은 조건을 만족하면, 우리는

가 삼중 곱 법칙(triple product property)을 만족한다고 부를 수 있다.By defining the triple product property, a given group

about a subset

The right quotient set of

marked with (In other words,

). If three subsets

each of these elements

,

,

If a satisfies the condition as in Equation 5 below, we get

can be said to satisfy the triple product property.

를 고려하는 것이다. 또한 두 개의 복소 행렬,

행렬

,

행렬

를 고려한다. 이를 통해 군 대수(group algebra) 부호화를 정의하면 하기 수학식 6과 같이 표현할 수 있다. Here, the method of encoding the matrix product as a group algebra is a method that satisfies the triple product property.

will consider Also two complex matrices,

procession

,

procession

consider If the group algebra encoding is defined through this, it can be expressed as in Equation 6 below.

의 군 원소(group element)

의 계수들로부터

행렬 곱의 값을 얻을 수 있다. 이 개념은 직관적으로 설명하면 DFT(discrete fourier transform)의 순환 컨벌루션(cyclic convolution) 개념을 임의의 군으로 일반화한 것으로 생각할 수 있다.If Equation 6 is obtained,

group element of

from the coefficients of

You can get the value of matrix multiplication. Intuitively, this concept can be thought of as a generalization of the cyclic convolution concept of a discrete fourier transform (DFT) to an arbitrary group.

이 서로소인 경우에, 여러 군 중 순환군(cyclic group)을 활용한 분산 행렬 곱 기법을 제안하고자 한다. 주어진

에 대해서, 순환군

를 고려할 수 있다. 여기서 하기 수학식 7과 같이 순환군

의 3개의 부분집합을 고려할 수 있다. next

In this case, we propose a dispersion matrix multiplication technique using a cyclic group among several groups. given

About the circulatory group

can be considered. Here, as in Equation 7 below, the cyclic group

We can consider three subsets of

는 삼중 곱 법칙(triple product property)을 만족함을 증명할 수 있다(증명은 관련 논문 [2]에 기재됨). 따라서 논문 [2]와 같이 순환군

를 활용하여 수학식 8과 같은 부호화를 고려할 수 있다.then

can prove that the triple product property is satisfied (the proof is described in the related paper [2]). Therefore, as in the paper [2], the cyclic group

Encoding as in Equation 8 can be considered by using

를 계산하면 수학식 9와 같이 표현할 수 있다.Here, the calculation result obtained by the central device 100 by the edge devices 200:200_1,??,200_N

can be expressed as in Equation 9.

행렬의 결과값들이 나오는 것을 확인할 수 있다. 실제로 이 연산들을 각 엣지 장치(200:200_1,??,200_N)에게 주고 연산시키기 위해서는

와

를 복소수값의 신호로 변환할 수 있어야 한다. 그룹과 벡터 공간 사이의 관계를 다루는 표현론(representation theory)을 활용하면 복소수 값으로 변환할 수 있다. 순환군에 대한 약분 불가능한 표현(기약 표현, irreducible representation)은 함수로 총

개 이며, 하기 수학식 10과 같이 표현할 수 있다.In Equation 8, the coefficient of the first line is the desired value of the central device 100

You can see that the result values of the matrix are coming out. In order to actually give these operations to each edge device (200:200_1,??,200_N)

Wow

should be able to convert to a complex-valued signal. Representation theory, which deals with the relationship between groups and vector spaces, can be used to convert them to complex values. An irreducible representation of a recursive group is

, and can be expressed as in Equation 10 below.

이고,

이며,

은 순환군으로써 임의의 정수(

)를

으로 나눈 나머지의 집합(

)이다.here,

ego,

is,

is an arbitrary integer (

)cast

The set of remainders divided by (

)to be.

와

에게 함수

를 씌운 다음에 각 엣지 장치(200:200_1,??,200_N)에게 전달하는 것으로는 나중에 중앙 장치(100)가 받았을 때 원 함수의 복원을 보장할 수 없다. Now I want to explain how to actually encode it. As mentioned earlier, simply choosing different promise expressions

Wow

function to

By passing ? to each edge device (200:200_1,??,200_N) after covering , restoration of the original function cannot be guaranteed when the central device 100 receives it later.

번째 엣지 장치(200_

)에게 두 개의 부분행렬

로 부호화하여 전송해주면,

개의 엣지 장치(200)의 개수만으로 신호를 복원할 수 있다. 여기서

는 군(group) 원소를 입력으로 받는 함수로 수학식 11과 같이 정의 된다.Therefore, we would like to present the restoration condition of the dispersion matrix product as follows. The central unit 100 is each

second edge device (200_

) to two submatrices

If it is encoded and transmitted as

A signal can be restored with only the number of edge devices 200 . here

is a function that receives a group element as an input, and is defined as in Equation 11.

이고,

이며,

은 순환군으로써 임의의 정수(

)를

으로 나눈 나머지의 집합(

)이고,

으로 나눈 나머지가 서로 달라야 한다. 즉,

이다. here,

ego,

is,

is an arbitrary integer (

)cast

The set of remainders divided by (

)ego,

The remainder of dividing by must be different. in other words,

to be.

FIG. 2 is a block diagram schematically illustrating a configuration of a central device including a group theory-based dispersion matrix computing device according to an embodiment of the present invention. In the following description, the part overlapping with the description of FIG. 1 will be omitted. Referring to FIG. 2 , the central device 100 includes a communication unit 110 , a division unit 120 , an encoding unit 130 , a processing unit 140 , a restoration unit 150 , a storage medium 160 , and a control unit 170 . may include

The communication unit 110 may provide a communication interface required to provide a transmission/reception signal between the central device 100 and the edge device 200 in the form of packet data in cooperation with the network 400 . Furthermore, the communication unit 110 serves to transmit information processed by the central device 100 to the edge device 200 or to allow the central device 100 to receive information processed by the edge device 200 can be done Here, the communication network is a medium that connects the central device 100 and the edge device 200 , and an access path so that the central device 100 can transmit and receive information after accessing the edge device 200 . It may include a path that provides In addition, the communication unit 110 may be a device including hardware and software necessary for transmitting and receiving signals such as control signals or data signals through wired/wireless connection with other network devices.

) 및 복수의 제2 부분행렬(

)을 생성할 수 있다.The division unit 120 divides each of the first target matrix and the second target matrix, which are the target of the multiplication operation, according to a preset partition parameter, and divides each of the plurality of first submatrices (

) and a plurality of second submatrices (

) can be created.

)는 제1 대상행렬에서 행에 대한 파티션 파라미터를 포함할 수 있다. 제2 파티션 파라미터(

)는 제1 대상행렬에서 열에 대한 파티션 파라미터 및 제2 대상행렬에서 행에 대한 파티션 파라미터를 포함할 수 있다. 제3 파티션 파라미터(

)는 제2 대상행렬에서 열에 대한 파티션 파라미터를 포함할 수 있다.Here, the partition parameter may include a first partition parameter to a third partition parameter. the first partition parameter (

) may include a partition parameter for a row in the first target matrix. second partition parameter (

) may include a partition parameter for a column in the first target matrix and a partition parameter for a row in the second target matrix. the third partition parameter (

) may include a partition parameter for a column in the second target matrix.

) 및 제2 파티션 파라미터(

)를 이용하여

로 분할한 복수의 제1 부분행렬(

)을 생성할 수 있다.The partitioning unit 120 divides the first partition parameter (

) and the second partition parameter (

) using

A plurality of first submatrices divided by (

) can be created.

) 및 제3 파티션 파라미터(

)를 이용하여

로 분할한 복수의 제2 부분행렬(

)을 생성할 수 있다.The division unit 120 divides the second partition parameter (

) and the third partition parameter (

) using

A plurality of second submatrices divided by (

) can be created.

) 및 제2 부분행렬(

)을 군론을 기반으로 부호화하여 수학식 8과 같은 제1 부호화한 부분행렬(

) 및 제2 부호화한 부분행렬(

)을 생성할 수 있다. The encoder 130 is a first sub-matrix (

) and the second submatrix (

) is encoded based on the group theory, and the first encoded submatrix (

) and the second encoded submatrix (

) can be created.

는 제1 부분행렬(

)에서 행의 인덱스일 수 있다.

는 제1 부분행렬(

)에서 열의 인덱스이고, 제2 부분행렬(

)에서 행의 인덱스일 수 있다.

는 제2 부분행렬(

)에서 열의 인덱스일 수 있다.

는 순환군을 결정하는 파라미터(parameter)로써, 자유롭게 설정할 수 있다.

는 순환군에서 하나의 원소일 수 있다. 여기서, 군(group)이라 함은, 결합 법칙과 항등원과 각 원소의 역원을 가지는 이항 연산을 갖춘 집합(대수구조)일 수 있다. 또한, 순환군이라 함은 하나의 원소에 의해 생성되는 군(예를 들어,

:

으로 나눈 나머지의 집합)일 수 있다.In Equation 8,

is the first submatrix (

) can be the index of the row in

is the first submatrix (

) is the index of the column in the second submatrix (

) can be the index of the row in

is the second submatrix (

) can be the index of the column.

is a parameter that determines the circulation group, and can be freely set.

may be one element in the cyclic group. Here, a group may be a set (algebraic structure) having an associative law, an identity circle, and a binary operation having an inverse of each element. In addition, the cyclic group refers to a group produced by one element (eg,

:

is the set of remainders divided by ).

) 및 제2 부호화한 부분행렬(

)을 생성할 수 있다.The encoder 130 encodes a different first encoded submatrix for each edge device 200:200_1, ..., 200_N.

) and the second encoded submatrix (

) can be created.

), 제2 파티션 파라미터(

) 및 제3 파티션 파라미터(

)에 대한 순환군을 이용하여 제1 부분행렬(

) 및 제2 부분행렬(

)을 각각 군 대수화(group algebra)하여 제1 부호화한 부분행렬(

) 및 제2 부호화한 부분행렬(

)을 생성할 수 있다.The encoder 130 is a first partition parameter (

), the second partition parameter (

) and the third partition parameter (

) using the recursive group for the first submatrix (

) and the second submatrix (

) by group algebra, the first encoded submatrix (

) and the second encoded submatrix (

) can be created.

) 및 제2 부호화한 부분행렬(

) 생성 시에, 제1 부호화한 부분행렬(

) 및 제2 부호화한 부분행렬(

)에 순환군의 기약 표현을 적용하여 전압 또는 전류 형태의 복소수값으로 변환시킬 수 있다. 순환군의 기약 표현은, 각 엣지 장치(200:200_1,…,200_N)마다 상이한 표현으로 적용될 수 있다. The encoding unit 130 is a first encoded sub-matrix (

) and the second encoded submatrix (

) at the time of generation, the first encoded submatrix (

) and the second encoded submatrix (

) can be converted into a complex value in the form of voltage or current by applying the reduced expression of the cyclic group. The reduced expression of the cyclic group may be applied as a different expression for each edge device 200:200_1, ..., 200_N.

) 및 제2 부호화한 부분행렬(

)은 복소수값(전류/전압 형태)가 아니므로, 엣지 장치(200:200_1,…,200_N)로 전송이 불가능하다. 따라서 부호화부(130)는 제1 부호화한 부분행렬(

) 및 제2 부호화한 부분행렬(

)에 대하여, 전송할 엣지 장치(200:200_1,…,200_N) 마다 서로 다른 순환군의 기약 표현을 적용하여 복소수값으로 변환시킬 수 있다.Here, the first encoded submatrix (

) and the second encoded submatrix (

) is not a complex value (current/voltage form), so it is impossible to transmit to the edge devices 200:200_1, ..., 200_N. Accordingly, the encoder 130 performs the first encoded sub-matrix (

) and the second encoded submatrix (

), it can be converted into a complex value by applying a different reduced representation of a cyclic group for each edge device 200:200_1, ..., 200_N to be transmitted.

) 및 제2 부호화한 부분행렬(

)을 수학식 12와 같이 표현할 수 있다.The encoder 130 uses Equations (8) and (10) to apply the reduced expression of the cyclic group to convert the first encoded sub-matrix (

) and the second encoded submatrix (

) can be expressed as in Equation 12.

)와, 제1 파티션 파라미터(

)와, 제2 파티션 파라미터(

)와, 제3 파티션 파라미터(

)를 곱한

개 일 수 있다.Here, the irreducible representation of the recursive group is a function, and the parameter that determines the recursive group (

) and the first partition parameter (

) and the second partition parameter (

) and the third partition parameter (

) multiplied by

can be a dog

) 및 제2 부호화한 부분행렬(

)을 엣지 장치(200:200_1,…,200_N)로 전송할 수 있다. 처리부(140)는 수학식 12에서의

및

를

번째 엣지 장치(200_

)로 전송할 수 있다. The processing unit 140 converts the first encoded sub-matrix (

) and the second encoded submatrix (

) to the edge devices 200:200_1, ..., 200_N. The processing unit 140 in Equation 12

and

cast

second edge device (200_

) can be transmitted.

및

를 수신하여 곱 연산을 수행하여 곱 연산 결과(

)를 생성하여 중앙 장치(100)로 전송하면, 중앙 장치(100)는 곱 연산 결과(

)를 획득할 수 있다. Edge devices 200: 200_1, ..., 200_N are sent from the central device 100

and

Receive and perform a multiplication operation to obtain the product operation result (

) is generated and transmitted to the central device 100, the central device 100 generates a multiplication result (

) can be obtained.

)는 수학식 9를 이용하여 하기 수학식 13과 같이 표현할 수 있다.The product operation result ( ) that the central device 100 obtains from the edge devices 200:

) can be expressed as in Equation 13 below using Equation 9.

)에 복원 조건을 적용하여 제1 대상행렬 및 제2 대상행렬의 곱 연산 결과(

)를 복원할 수 있다.The restoration unit 150 receives the product operation result (200:200_1, ..., 200_N) from the edge device (

) by applying the restoration condition to the product operation result of the first target matrix and the second target matrix (

) can be restored.

) 및 제2 부호화한 부분행렬(

)을, 어느 한 엣지 장치(예를 들어, 엣지 장치(200_

))로 전송하면, 제1 파티션 파라미터(

)와, 제2 파티션 파라미터(

)와, 제3 파티션 파라미터(

)를 곱한

개의 엣지 장치(200)를 이용하여 제1 대상행렬 및 제2 대상행렬의 곱 연산 결과를 복원 가능하다는 것을 포함할 수 있다. Here, the restoration condition is the first encoded submatrix (

) and the second encoded submatrix (

), any one edge device (eg, edge device 200_

)), the first partition parameter (

) and the second partition parameter (

) and the third partition parameter (

) multiplied by

It may include that it is possible to restore a product operation result of the first target matrix and the second target matrix using the edge device 200 .

)를 대입하여 획득한 복소항(

)의 지수(

)를 소정의 값(

)으로 나눈 나머지가 각 엣지 장치(200)별로 상이한 조건을 포함할 수 있다. The restoration condition is an element of the cyclic group (

) obtained by substituting the complex term (

) of the exponent (

) to a given value (

) may include different conditions for each edge device 200 .

)에 대하여, 수학식 11에 기재된

을 만족하면, 제1 대상행렬 및 제2 대상행렬의 곱 연산 결과(

)를 복원 가능하다고 판단할 수 있다. 이는 각 엣지 장치(200:200_1,…,200_N)에 어떤 기약 표현을 적용했는지 알고 있음에 기인한 것이며,

는 각 엣지 장치(200:200_1,…,200_N)마다 서로 다르다.The restoration unit 150 satisfies the restoration condition, that is, the product operation result (

), as described in Equation 11

If is satisfied, the result of the product of the first target matrix and the second target matrix (

) can be determined to be recoverable. This is due to knowing which contract expression is applied to each edge device (200:200_1, ..., 200_N),

is different for each edge device 200:200_1, ..., 200_N.

)에 대하여 열 벡터(

)를 생성할 수 있다.When the restoration condition is satisfied, the restoration unit 150 uses Equation 3 to obtain the product operation result (

) with respect to the column vector(

) can be created.

와

사이의 관계식을 이용하여, 각 엣지 장치(200, 200_1,??,200_N)로부터 수신한 곱 연산 결과인

및 부호화 과정을 행렬

로 표현하였을 때, 이에 대한 역행렬

에 기반하여,

를 획득하고 원래 원하고자 하는 곱 연산 결과(

)를 복원할 수 있다.Since the restoration unit 150 knows how the central device 100 encodes, the same as in Equation 4

Wow

Using the relational expression between

and matrix encoding process

When expressed as , the inverse matrix

based on,

to obtain the desired multiplication result (

) can be restored.

The storage medium 160 may temporarily or permanently store data processed by the central device 100 or temporarily or permanently store an operation result received from the edge devices 200:200_1,??,200_N. Here, the storage medium 160 may include a magnetic storage medium or a flash storage medium, but the scope of the present invention is not limited thereto.

)를 생성하기 위해, 행렬을 부분적으로 나눈 다음 이들을 부호화하고 N개의 엣지 장치(200, 200_1,??,200_N)에게 나눠 연산을 하게 한 다음 연산 결과를 받아 복원하도록 전체의 동작을 제어할 수 있다.The control unit 170 is a kind of central processing unit and may control the operation of the entire central unit 100 . The control unit 170 is a target operation result (

), the matrix is partially divided and then encoded, and the N edge devices 200, 200_1,?? .

In this embodiment, the control unit 170 may include all kinds of devices capable of processing data, such as a processor. Here, the 'processor' may refer to, for example, a data processing device embedded in hardware having a physically structured circuit to perform a function expressed as a code or command included in a program. As an example of the data processing apparatus embedded in the hardware as described above, a microprocessor, a central processing unit (CPU), a processor core, a multiprocessor, an application-specific integrated (ASIC) circuit) and a processing device such as a field programmable gate array (FPGA), but the scope of the present invention is not limited thereto.

3 is a graph showing the number of edge devices required according to the matrix size derived from FIG. 2 . In the following description, descriptions of parts overlapping with those of FIGS. 1 and 2 will be omitted.

Referring to FIG. 3 , the performance of the present embodiment was measured with the minimum number of edge devices 200 that ensure that the normalized error of the reconstructed matrix is smaller than a predetermined threshold. That is, the number of edge devices 200 satisfying Equation 14.

Here, F may mean a Frobenius Norm operation. The above-mentioned theoretical values and this paper [1] were also compared with the theoretical values.

The simulation environment is as follows. The error reference value is 10 ^-10 , and the original matrix to be calculated is a 50p×50m matrix A and a 50p×50n matrix B with arbitrary values. , in order to use the same computing power for each edge device 200, all simulations were multiplied by matrices of the same size.

From the results, it can be seen that the method according to the present embodiment requires a smaller number of edge devices 200 than the conventional method both theoretically and experimentally. In addition, it can be seen that in order to lower the error below a certain level, a larger number of edge devices 200 are actually required than the theoretically obtained value in the thesis [1], and it can be seen that the difference increases as the size of the matrix increases. have. However, since the error is sufficiently low in the method proposed in this embodiment, it can be confirmed that there is no significant difference from the theoretically required number of edge devices 200 .

4 is a flowchart illustrating a method for calculating a group theory-based dispersion matrix according to the present embodiment. In the following description, descriptions of parts overlapping with those of FIGS. 1 to 3 will be omitted.

) 및 복수의 제2 부분행렬(

)을 생성한다.Referring to FIG. 4 , in step S410 , the central device 100 divides each of a first target matrix and a second target matrix, which are objects of the product operation, by a preset partition parameter, and divides each of the plurality of first target matrices as in Equation 1 submatrix (

) and a plurality of second submatrices (

) is created.

) 및 제2 파티션 파라미터(

)를 이용하여

로 분할한 제1 부분행렬(

)을 생성할 수 있다. 또한 중앙 장치(100)는 제2 대상행렬 대하여 제2 파티션 파라미터(

) 및 제3 파티션 파라미터(

)를 이용하여

로 분할한 제2 부분행렬(

)을 생성할 수 있다. Here, the central device 100 determines the first partition parameter (

) and the second partition parameter (

) using

The first submatrix divided by (

) can be created. In addition, the central device 100 provides a second partition parameter (

) and the third partition parameter (

) using

The second submatrix divided by (

) can be created.

) 및 제2 부분행렬(

)을 군론을 기반으로 부호화하여 수학식 8과 같이 제1 부호화한 부분행렬(

) 및 제2 부호화한 부분행렬(

)을 생성한다.In step S420, the central device 100 is a first sub-matrix (

) and the second submatrix (

) is encoded based on the group theory, and the first encoded submatrix (

) and the second encoded submatrix (

) is created.

), 제2 파티션 파라미터(

) 및 제3 파티션 파라미터(

)에 대한 순환군을 이용하여 제1 부분행렬(

) 및 제2 부분행렬(

)을 각각 군 대수화(group algebra)하여 제1 부호화한 부분행렬(

) 및 제2 부호화한 부분행렬(

)을 생성할 수 있다.The central device 100 determines the first partition parameter (

), the second partition parameter (

) and the third partition parameter (

) using the recursive group for the first submatrix (

) and the second submatrix (

) by group algebra, the first encoded submatrix (

) and the second encoded submatrix (

) can be created.

) 및 제2 부호화한 부분행렬(

)을 생성할 수 있다. 제1 부호화한 부분행렬(

) 및 제2 부호화한 부분행렬(

)은 복소수값(전류/전압 형태)가 아니므로, 엣지 장치(200)로 전송이 불가능하다. 따라서 중앙 장치(100)는 제1 부호화한 부분행렬(

) 및 제2 부호화한 부분행렬(

)에 대하여, 전송할 엣지 장치(200) 마다 서로 다른 순환군의 기약 표현을 적용하여 복소수값으로 변환시킬 수 있다. 순환군의 기약 표현은, 각 엣지 장치(200:200_1,…,200_N)마다 상이한 표현으로 적용될 수 있다. 또한, 순환군의 기약 표현(irreducible representation)은 함수로, 순환군을 결정하는 파라미터(

)와, 제1 파티션 파라미터(

)와, 제2 파티션 파라미터(

)와, 제3 파티션 파라미터(

)를 곱한

개 일 수 있다.The central device 100 is a first encoded sub-matrix (

) and the second encoded submatrix (

) can be created. The first encoded submatrix (

) and the second encoded submatrix (

) is not a complex value (current/voltage form), so it is impossible to transmit to the edge device 200 . Therefore, the central device 100 is the first encoded sub-matrix (

) and the second encoded submatrix (

), it can be converted into a complex value by applying a reduced representation of a different cyclic group for each edge device 200 to be transmitted. The reduced expression of the cyclic group may be applied as a different expression for each edge device 200:200_1, ..., 200_N. In addition, the irreducible representation of a recursive group is a function, a parameter that determines the recursive group (

) and the first partition parameter (

) and the second partition parameter (

) and the third partition parameter (

) multiplied by

can be a dog

) 및 제2 부호화한 부분행렬(

)을 엣지 장치(200)로 전송한다.In step S430, the central device 100 applies the reduced representation of the recursive group to convert the first encoded sub-matrix (

) and the second encoded submatrix (

) to the edge device 200 .

) 및 제2 부호화한 부분행렬(

)을 수신한다.In step S440, the edge device 200 converts the first encoded sub-matrix (

) and the second encoded submatrix (

) is received.

) 및 제2 부호화한 부분행렬(

)에 대한 곱 연산을 수행하여 생성한다.In step S450, the edge device 200 applies the reduced representation of the recursive group to convert the first encoded sub-matrix (

) and the second encoded submatrix (

) by performing a multiplication operation on

)를 중앙 장치(100)로 전송한다.In step S460, the edge device 200 performs a multiplication operation result (

) to the central device 100 .

)를 수신한다.In step S470, the central device 100 transmits the product operation result (

) is received.

)에 복원 조건을 적용하여 제1 대상행렬 및 제2 대상행렬의 곱 연산 결과(

)를 복원한다.In step S480, the central device 100 receives the product operation result (

) by applying the restoration condition to the product operation result of the first target matrix and the second target matrix (

) is restored.

) 및 제2 부호화한 부분행렬(

)을, 어느 한 엣지 장치(예를 들어, 엣지 장치(200_

))로 전송하면, 제1 파티션 파라미터(

)와, 제2 파티션 파라미터(

)와, 제3 파티션 파라미터(

)를 곱한

개의 엣지 장치(200)를 이용하여 제1 대상행렬 및 제2 대상행렬의 곱 연산 결과를 복원 가능하다는 것을 포함할 수 있다. 복원 조건은, 기약 표현에 순환군의 원소(

)를 대입하여 획득한 복소항(

)의 지수(

)를 소정의 값(

)으로 나눈 나머지가 각 엣지 장치(200)별로 상이한 조건을 포함할 수 있다. 중앙 장치(100)는 복원 조건을 만족하는 경우, 수학식 3을 이용하여, 각 엣지 장치(200)로부터 수신한 곱 연산 결과(

)에 대하여 열 벡터(

)를 생성할 수 있다. 중앙 장치(100)는 자신이 어떻게 부호화를 수행하였는지 알고 있기 때문에 수학식 4와 같은

와

사이의 관계식을 이용하여, 각 엣지 장치(200, 200_1,??,200_N)로부터 수신한 곱 연산 결과인

및 부호화 과정을 행렬

로 표현하였을 때, 이에 대한 역행렬

에 기반하여,

를 획득하고 원래 원하고자 하는 곱 연산 결과(

)를 복원할 수 있다.Here, the restoration condition is the first encoded submatrix (

) and the second encoded submatrix (

), any one edge device (eg, edge device 200_

)), the first partition parameter (

) and the second partition parameter (

) and the third partition parameter (

) multiplied by

It may include that it is possible to restore a product operation result of the first target matrix and the second target matrix using the edge device 200 . The restoration condition is an element of the cyclic group (

) obtained by substituting the complex term (

) of the exponent (

) to a given value (

) may include different conditions for each edge device 200 . When the restoration condition is satisfied, the central device 100 uses Equation 3 to obtain the product operation result (

) with respect to the column vector(

) can be created. Since the central device 100 knows how it has performed encoding, the same as in Equation 4

Wow

Using the relational expression between

and matrix encoding process

When expressed as , the inverse matrix

based on,

to obtain the desired multiplication result (

) can be restored.

The above-described embodiment according to the present invention may be implemented in the form of a computer program that can be executed through various components on a computer, and such a computer program may be recorded in a computer-readable medium. In this case, the medium includes a hard disk, a magnetic medium such as a floppy disk and a magnetic tape, an optical recording medium such as CD-ROM and DVD, a magneto-optical medium such as a floppy disk, and a ROM. , RAM, flash memory, and the like, and hardware devices specially configured to store and execute program instructions.

Meanwhile, the computer program may be specially designed and configured for the present invention, or may be known and used by those skilled in the computer software field. Examples of the computer program may include not only machine code generated by a compiler, but also high-level language code that can be executed by a computer using an interpreter or the like.

In the specification of the present invention (especially in the claims), the use of the term "above" and similar referential terms may be used in both the singular and the plural. In addition, when a range is described in the present invention, each individual value constituting the range is described in the detailed description of the invention as including the invention to which individual values belonging to the range are applied (unless there is a description to the contrary). same as

The steps constituting the method according to the present invention may be performed in an appropriate order unless the order is explicitly stated or there is no description to the contrary. The present invention is not necessarily limited to the order in which the steps are described. The use of all examples or exemplary terms (eg, etc.) in the present invention is merely for the purpose of describing the present invention in detail, and the scope of the present invention is limited by the examples or exemplary terms unless limited by the claims. it is not going to be In addition, those skilled in the art will appreciate that various modifications, combinations, and changes may be made in accordance with design conditions and factors within the scope of the appended claims or their equivalents.

Therefore, the spirit of the present invention should not be limited to the above-described embodiments, and the scope of the spirit of the present invention is not limited to the scope of the scope of the present invention. will be said to belong to

100: central unit
200: edge device
300: user terminal
400: network

Claims (19)

A method of performing a group algebra-based distributed matrix operation by the central device in a distributed computing environment including a central device and one or more edge devices, the method comprising:
generating a plurality of first submatrices and a plurality of second submatrices by dividing each of the first target matrix and the second target matrix to be multiplied by a predetermined partition parameter;
generating a first encoded submatrix and a second encoded submatrix by encoding the first submatrix and the second submatrix based on group theory;
Transmitting the first coded submatrix and the second coded submatrix to the one or more edge devices, and multiplying the first coded submatrix and the second coded submatrix from the one or more edge devices obtaining a result; and
Restoring the product operation result of the first target matrix and the second target matrix by applying a restoration condition to the product operation result received from the edge device,
The generating of the plurality of first sub-matrices and the plurality of second sub-matrices comprises:
generating the first sub-matrix by dividing the first target matrix according to a first partition parameter and a second partition parameter among the preset partition parameters; and
generating the second sub-matrix by dividing the second target matrix according to the second partition parameter and the third partition parameter among the preset partition parameters;
The first partition parameter includes a partition parameter for a row in the first target matrix, and the second partition parameter includes a partition parameter for a column in the first target matrix and a partition parameter for a row in the second target matrix Including, wherein the third partition parameter includes a partition parameter for a column in the second target matrix,
A method of calculating the variance matrix based on the group theory. delete The method of claim 1,
The generating of the first coded sub-matrix and the second coded sub-matrix comprises:
The first encoding is performed by group algebraing the first sub-matrix and the second sub-matrix using a recursive group for the first partition parameter, the second partition parameter, and the third partition parameter, respectively. generating a sub-matrix and the second encoded sub-matrix,
A method of calculating the variance matrix based on the group theory. The method of claim 1,
The generating of the first coded sub-matrix and the second coded sub-matrix comprises:
When generating the first coded submatrix and the second coded submatrix, an irreducible representation of a recursive group is applied to the first coded submatrix and the second coded submatrix to obtain a complex-valued signal. comprising the step of converting to
A method of calculating the variance matrix based on the group theory. 5. The method of claim 4,
The reduced expression of the cyclic group is applied as a different expression for each edge device,
A method of calculating the variance matrix based on the group theory. 5. The method of claim 4,
The reduced expression of the circulatory group is,
Corresponding to the number obtained by multiplying the first partition parameter, the second partition parameter, and the third partition parameter by a function,
A method of calculating the variance matrix based on the group theory. 5. The method of claim 4,
In the step of restoring a product operation result of the first target matrix and the second target matrix,
The restoration conditions are:
In the reduced expression, the elements of the cyclic group (

) obtained by substituting the complex term (

) of the exponent (

) to a given value (

) and the remainder includes different conditions for each edge device,
A method of calculating the variance matrix based on the group theory. 5. The method of claim 4,
In the step of restoring a product operation result of the first target matrix and the second target matrix,
The restoration conditions are:
When the central device transmits the first encoded submatrix and the second encoded submatrix to which the reduced representation of the cyclic group is applied to any one of the edge devices, the first partition parameter and the second partition parameter And, by using the number of edge devices multiplied by the third partition parameter, it is possible to restore the product operation result of the first target matrix and the second target matrix,
A method of calculating the variance matrix based on the group theory. The method of claim 1,
Restoring the product operation result of the first target matrix and the second target matrix comprises:
The result of the multiplication operation received from the edge device and the encoding process are displayed in a matrix

When expressed as , the inverse matrix

Comprising the step of restoring the product operation based on
A method of calculating the variance matrix based on the group theory. A computer-readable recording medium storing a computer program for executing the method of any one of claims 1 and 3 to 9 using a computer. A device for performing a group algebra-based distributed matrix operation in a distributed computing environment including a central device and one or more edge devices,
the central device,
a division unit generating a plurality of first sub-matrices and a plurality of second sub-matrices by dividing each of the first and second target matrices to be multiplied by a predetermined partition parameter;
an encoder for encoding the first submatrix and the second submatrix based on group theory to generate a first encoded submatrix and a second encoded submatrix;
Transmitting the first coded submatrix and the second coded submatrix to the one or more edge devices, and multiplying the first coded submatrix and the second coded submatrix from the one or more edge devices a processing unit for obtaining a result; and
and a restoration unit configured to restore a product operation result of the first target matrix and the second target matrix by applying a restoration condition to the product operation result received from the edge device,
The division part,
generating the first sub-matrix by dividing the first target matrix according to a first partition parameter and a second partition parameter among the preset partition parameters;
and generate the second sub-matrix by dividing the second target matrix according to the second partition parameter and the third partition parameter among the preset partition parameters;
The first partition parameter includes a partition parameter for a row in the first target matrix, and the second partition parameter includes a partition parameter for a column in the first target matrix and a partition parameter for a row in the second target matrix Including, wherein the third partition parameter includes a partition parameter for a column in the second target matrix,
A group theory-based dispersion matrix calculator. delete 12. The method of claim 11,
The encoding unit,
The first encoding is performed by group algebraing the first sub-matrix and the second sub-matrix using a recursive group for the first partition parameter, the second partition parameter, and the third partition parameter, respectively. configured to generate a sub-matrix and the second coded sub-matrix,
A group theory-based dispersion matrix calculator. 12. The method of claim 11,
The encoding unit,
When generating the first coded submatrix and the second coded submatrix, an irreducible representation of a recursive group is applied to the first coded submatrix and the second coded submatrix to obtain a complex-valued signal. configured to convert to
A group theory-based dispersion matrix calculator. 15. The method of claim 14,
The reduced expression of the cyclic group is applied as a different expression for each edge device,
A group theory-based dispersion matrix calculator. 15. The method of claim 14,
The reduced expression of the circulatory group is,
Corresponding to the number obtained by multiplying the first partition parameter, the second partition parameter, and the third partition parameter by a function,
A group theory-based dispersion matrix calculator. 15. The method of claim 14,
In the restoration unit,
The restoration conditions are:
In the reduced expression, the elements of the cyclic group (

) obtained by substituting the complex term (

) of the exponent (

) to a given value (

) and the remainder includes different conditions for each edge device,
A group theory-based dispersion matrix calculator. 15. The method of claim 14,
In the restoration unit,
The restoration conditions are:
When the central device transmits the first encoded submatrix and the second encoded submatrix to which the reduced representation of the cyclic group is applied to any one of the edge devices, the first partition parameter and the second partition parameter And, by using the number of edge devices multiplied by the third partition parameter, it is possible to restore the product operation result of the first target matrix and the second target matrix,
A group theory-based dispersion matrix calculator. 12. The method of claim 11,
The restoration unit,
The result of the multiplication operation received from the edge device and the encoding process are displayed in a matrix

When expressed as , the inverse matrix

configured to restore the product operation based on
A group theory-based dispersion matrix calculator.

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