JP2010278782A - Estimating device, encoder, decoder, estimating method, encoding method, decoding method, and program - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To shorten a calculating time by suppressing the number of constraint inequalities applied to linear programming. <P>SOLUTION: This estimating device first calculates a provisional solution z<SB>j</SB>by using an initial constraint. Next, the device creates a set of indexes j showing a column in which a matrix element is not zero, and classifies the set of j into a set S<SB>0</SB>and a set S<SB>1</SB>. Then, the device performs adjustment so that the number of j classified into S<SB>1</SB>is an odd number when an intercept of a linear equation is zero and that the number of j classified into S<SB>1</SB>is an even number when the intercept is one. As a result of the adjustment, it is determined whether the total sum of an added value obtained by adding distances between all provisional solutions z<SB>j</SB>corresponding to j that turns out to belong to S<SB>1</SB>and one, and an added value obtained by adding distances between all provisional solutions z<SB>j</SB>corresponding to j that turns out to belongs to S<SB>0</SB>and zero is one or larger. When the total sum is not one or larger as a result of the determination, a cutting inequality is extracted on the basis of inequality used for the determination and added to the set of constraint inequalities. <P>COPYRIGHT: (C)2011,JPO&INPIT

Description

  The present invention relates to an estimation device, an encoder, and a decoder.

  Encoding and decoding methods include encoding information into short representations, the purpose of “information compression” to guarantee the quality of the decoded information, and restoring the correct information even if noise is mixed in the transmitted information There is a purpose of "error correction". In this regard, a technique for optimizing an objective function using a linear equation on a binary finite field as a constraint is known to realize both objectives (Non-Patent Documents 1 to 4, etc.).

  By the way, when solving an integer programming problem that optimizes an objective function using a linear equation as a constraint condition by linear programming in real space, the linear equation is transformed into a set of linear inequalities in real space. Therefore, the encoder and decoder use this set of linear inequalities to perform objective function optimization, but the number of linear inequalities may increase exponentially, increasing the computational load. There has been concern about increasing memory requirements.

  For this reason, conventionally, for example, Non-Patent Document 6 proposes a method for adaptively adding linear inequalities. In this method, first, a provisional optimal solution is obtained using only the initial set of linear inequalities as constraints, and then a linear inequality that is not satisfied by the provisional optimal solution (a linear inequality that removes the provisional optimal solution). ) From the set. When the linear inequality can be searched, the linear inequality is added to the set, and a temporary optimal solution is obtained again using the new set as a constraint. In addition, various algorithms have been proposed (Non-Patent Documents 5 to 7 and the like).

J. Muramatsu and S. Miyake, “Hash property and coding theorems for sparse matrices and maximum-likelihood coding,”, http://arxiv.org/abs/0801.3878 J. Muramatsu and S. Miyake, “Hash property and fixed-rate universal coding theorems,” http://arxiv.org/abs/0804.1183 J. Muramatsu, “Secret key agreement from correlated source outputs using low density parity check matrices,” IEICE Trans. Fundamentals, vol. E89-A, no. 7, pp. 2036-2046, Jul. 2006. Jun Muramatsu and Shigeki Miyake, “Construction of Wiretap Channel Codes by Using Sparse Matrices,” Proceedings of 6th Shannon Theory Workshop, 2008. J. Feldman, M. J. Waingwright, and D. R. Karger, “Using linear programming to decode binary linear codes,” IEEE Trans. Inform. Theory, vol. IT-51, pp. 954-972, Mar. 2005. M. H. Taghavi N. and P. H. Siegel, “Adaptive methos for linear programming decoding,” IEEE Trans. Inform. Theory, vol. IT-54, pp.5396-5410, Dec. 2008. M. Miwa, T. Wadayama, I. Takumi, “A cutting plane method based on redundant rows for improving fractional distance,” http://arxiv.org/abs/0807.2701

  However, the conventional technique described above has a problem that the calculation time is still long. For example, in the conventional method of adding a linear inequality adaptively, the condition must be repeatedly checked in order to obtain a linear inequality that removes the provisional optimal solution, and the calculation time is still long.

  The disclosed technology has been made in view of the above, and an object thereof is to provide an estimation device, an encoder, a decoder, an estimation method, an encoding method, a decoding method, and a program capable of reducing the calculation time. And

In one aspect, an estimation apparatus, an encoder, and a decoder disclosed in this application are an estimation apparatus that optimizes an objective function on a binary finite field of 0 and 1, using a linear equation as a constraint, A set of constraint inequalities indicating that the above is less than or equal to 1 is used as an initial constraint, and if a resection inequality that removes the provisional optimal solution z _j is extracted, Using the set as a new constraint, a provisional solution calculation means for calculating the provisional solution z _j and a set of indexes j indicating columns in which the elements of the matrix are not 0 are created for each row of the matrix that is a coefficient of the linear equation. , the set of the indices j, the first set S ₀ if the distance between the tentative solution z _j and 1 corresponding to the index j exceeds the distance between the 0, the distance between the tentative solution z _j and 1 Below the distance from 0 And classifying means for classifying each of the second set S ₁ in the case, as the number of the classified index j in the second set S ₁ is an odd number in the case the intercept of the linear equation is zero, also When the intercept of the linear equation is 1, the first set S ₀ and the second set classified by the classification means so that the number of indexes j classified into the second set S ₁ is an even number. and adjusting means for adjusting the S _1, the adjusting means results adjusted by said second set corresponding to the index j became belong to S ₁ interim solution z _j sum value distance obtained by adding 1 and for all And whether or not the sum of the provisional solution z _j corresponding to the index j belonging to the first set S ₀ and the added value obtained by adding the distance to 0 is 1 or more Means and said determination A means for extracting the ablation inequality based on the inequality used for the determination when the result of the determination by the means determines that it is not 1 or more, and the ablation inequality extracted by the extraction means is a set of the constraint inequality And an additional means for adding to.

  According to one aspect of the estimation device, the encoder, the decoder, the estimation method, the encoding method, the decoding method, and the program disclosed in the present application, it is possible to shorten the calculation time.

FIG. 1 is a block diagram illustrating the configuration of the estimation apparatus 10 according to the first embodiment. FIG. 2 is a flowchart illustrating a processing procedure in the estimation apparatus 10 according to the first embodiment. FIG. 3 is a diagram for explaining an error correction code for a communication path with additive noise. FIG. 4A is a flowchart for explaining the first algorithm. FIG. 4B is a flowchart for explaining the first algorithm. FIG. 4-3 is a flowchart for explaining the first algorithm. FIG. 5A is a flowchart for explaining the second algorithm. FIG. 5B is a flowchart for explaining the second algorithm. FIG. 5C is a flowchart for explaining the second algorithm. FIG. 6 is a block diagram illustrating the configuration of the estimation apparatus 10 according to the second embodiment. FIG. 7 is a flowchart illustrating a processing procedure in the estimation apparatus 10 according to the second embodiment. FIG. 8 is a diagram illustrating an optimization problem that appears in encoding and decoding. FIG. 9 is a diagram for explaining an encoder and a decoder in a fixed-length distortionless code. FIG. 10 is a diagram for explaining an encoder and a decoder in a fixed-length undistorted code that can use auxiliary information. FIG. 11 is a diagram for explaining an encoder and a decoder in a general channel code. FIG. 12 is a diagram for explaining an encoder and a decoder in a general channel code. FIG. 13 is a diagram for explaining an encoder and a decoder in a distorted code. FIG. 14 is a diagram for explaining an encoder and a decoder in a distorted code. FIG. 15 is a diagram illustrating a simulation result.

  Hereinafter, embodiments of the estimation device disclosed in the present application will be described in detail. In addition, this invention is not limited by the following examples.

  The estimation apparatus 10 according to the first embodiment is an apparatus that optimizes the function f (z) on the binary finite field GF (2) using the linear equation Az = c as a constraint. A is an l-row n-column matrix, z is an n-dimensional column vector, and c is an l-dimensional column vector. Further, as will be described below, the estimation apparatus 10 according to the first embodiment uses an inequality that can remove a provisional optimal solution to a set Ineq (k) of linear inequalities in a real space obtained by modifying the linear equation Az = c. Use the method of adding only. Hereinafter, first, the configuration of the estimation apparatus 10 according to the first embodiment will be described with reference to FIG. 1, and then the processing procedure and algorithm in the estimation apparatus 10 according to the first embodiment will be described with reference to FIGS. 2 to 5-3. explain.

  Note that the matrix A does not have to be full rank, and as described in Non-Patent Document 7, a redundant row for cutting out the optimal solution near the zero vector having a non-integer component can be added in advance to the matrix. . The first algorithm and the second algorithm of the present invention can be used to obtain a redundant row for cutting out an optimal solution near the zero vector having a non-integer component. However, in the algorithm in the following embodiment, the second algorithm may be executed on an original matrix that does not include redundant rows.

[Configuration of Estimation Device 10 According to Embodiment 1]
FIG. 1 is a block diagram illustrating the configuration of the estimation apparatus 10 according to the first embodiment. As illustrated in FIG. 1, the estimation apparatus 10 according to the first embodiment includes a storage unit 11, an input unit 12, a provisional solution calculation unit 13, an ablation inequality extraction unit 14, and an output unit 15.

  The storage unit 11 stores various information used for the estimation process in the estimation device 10. Specifically, the storage unit 11 includes, as information registered in advance by a user who uses the estimation device 10 or the like, the matrix A shown in Formula (1-1) and the initial value shown in Formula (1-2). Store the set of constraint inequalities. Moreover, the memory | storage part 11 memorize | stores by storing the vector c shown to Numerical formula (1-3) from the input part 12. FIG. Further, the storage unit 11 stores the resection inequality extracted by the resection inequality extraction unit 14 in the estimation process in the estimation device 10 by being stored from the resection inequality extraction unit 14. These pieces of information stored in the storage unit 11 are mainly used for processing by the provisional solution calculation unit 13.

  The input unit 12 acquires a processing target signal for estimation processing in the estimation device 10. Specifically, the input unit 12 acquires a vector c, which is a processing target signal, from outside the estimation device 10 by communication or the like, and stores the acquired vector c in the storage unit 11. Note that the input unit 12 may acquire an objective function from outside the estimation device 10 and transmit the acquired objective function to the provisional solution calculation unit 13.

The provisional solution calculation unit 13 calculates a provisional optimum solution (referred to as “provisional solution” as appropriate) by an optimization method (linear programming). Specifically, the provisional solution calculation unit 13 includes a set of initial constraint inequalities registered in advance in the storage unit 11 and constraint inequalities updated by the excision inequality extraction unit 14 and stored in the storage unit 11 during the estimation process. The provisional solution is calculated using the set of and the calculated provisional solution is stored in the excision inequality extraction unit 14. The provisional solution is a vector z shown in Equation (1-4).

  The excision inequality extraction unit 14 extracts an inequality for excising the provisional solution. Specifically, the excision inequality extraction unit 14 uses the matrix A and the vector c stored in the storage unit 11 and the provisional solution calculated by the provisional solution calculation unit 13 for all rows of the matrix A. On the other hand, the first algorithm is applied to determine whether or not a resection inequality exists.

  When the excision inequality exists, the excision inequality extraction unit 14 extracts the excision inequality while extracting the row from which the provisional solution is excised from the matrix A, and stores the extracted excision inequality in the storage unit 11. At this time, the excision inequality extraction unit 14 adds the newly extracted excision inequality to the set of constraint inequalities stored in the storage unit 11 to update the set of constraint inequalities and to the provisional solution calculation unit 13. , Notify that the set of constraint inequalities has been updated.

  On the other hand, if no excision inequality exists, the excision inequality extraction unit 14 further determines whether or not all the components of the provisional solution are integers. If all of the provisional solutions are integers, the excision inequality extraction unit 14 determines that the provisional solution is the final solution. The output unit 15 is notified. When the provisional solution component includes a non-integer, the excision inequality extraction unit 14 applies the second algorithm to the matrix A and the vector c to generate the matrix A ′ (dash) and the vector c ′ (dash). To do.

  Then, the excision inequality extraction unit 14 applies the first algorithm to all the rows of the matrix A ′ using the matrix A ′ and the vector c ′ and the provisional solution calculated by the provisional solution calculation unit 13. Determine whether a resection inequality exists.

  When the excision inequality exists, the excision inequality extraction unit 14 extracts the excision inequality while extracting the row from which the provisional solution is excised from the matrix A ′, and stores the extracted excision inequality in the storage unit 11. At this time, the excision inequality extraction unit 14 adds the newly extracted excision inequality to the set of constraint inequalities stored in the storage unit 11 and updates the set of constraint inequalities. On the other hand, when there is no excision inequality, the excision inequality extraction unit 14 notifies the output unit 15 of the provisional solution including a non-integer as the final solution.

  The output unit 15 outputs the final solution of the estimation process in the estimation device 10. Specifically, the output unit 13 outputs the final solution notified from the excision inequality extraction unit 14.

[Processing procedure and algorithm in estimation apparatus 10 according to Embodiment 1]
Next, the processing procedure and algorithm in the estimation apparatus 10 according to the first embodiment will be described with reference to FIGS. FIG. 2 is a flowchart illustrating a processing procedure in the estimation apparatus 10 according to the first embodiment, and FIG. 3 is a diagram for explaining an error correction code for a communication path with additive noise. FIGS. 4-1 to 4-3 are flowcharts for explaining the first algorithm, and FIGS. 5-1 to 5-3 are flowcharts for explaining the second algorithm.

[Processing Procedure in Estimation Device 10]
As shown in FIG. 2, first, in the estimation apparatus 10, the input unit 12 acquires a vector c that is a signal to be processed (step S1).

  Next, the provisional solution calculation unit 13 sets a set of initial constraint inequalities previously registered in the storage unit 11 as a constraint condition (step S2), and calculates a provisional solution by an optimization method using the set constraint condition. (Step S3).

  Subsequently, the excision inequality extraction unit 14 uses the matrix A and the vector c stored in the storage unit 11 and the provisional solution calculated by the provisional solution calculation unit 13 for all rows of the matrix A. The first algorithm is applied to determine whether or not an excision inequality exists (step S4).

  When the excision inequality exists and is extracted (Yes at step S5), the excision inequality extraction unit 14 adds the newly extracted excision inequality to the set of constraint inequalities stored in the storage unit 11 to obtain the constraint condition. The set of inequalities is updated (step S6).

  On the other hand, when the excision inequality does not exist and is not extracted (No at Step S5), the excision inequality extraction unit 14 further determines whether or not all the components of the provisional solution are integers (Step S7). When all are integers (Yes at Step S7), the provisional solution is notified to the output unit 15 as the final solution (Step S8).

  On the other hand, when the component of the provisional solution includes a non-integer (No at Step S7), the excision inequality extraction unit 14 applies the second algorithm to the matrix A and the vector c, and determines the matrix A ′ and the vector c ′. Generate (step S9).

Then, the excision inequality extraction unit 14 applies the first algorithm to all the rows of the matrix A ′ using the matrix A ′ and the vector c ′ and the provisional solution calculated by the provisional solution calculation unit 13. Then, it is determined whether or not an excision inequality exists (step S10). Note that the order of the rows to which the first algorithm is applied is arbitrary. For example, in the order from the top row or the bottom row of the matrix, the processing is terminated when the first excision inequality that cuts the provisional solution is found, or in redundant row _a'i ablating solutions | Supp _(a'i) | is methods are conceivable such Request resection inequality smallest row. Here, the vector a ′ _i is one row in the matrix A ′.

  When the excision inequality exists and is extracted (Yes at step S11), the excision inequality extraction unit 14 adds the newly extracted excision inequality to the set of constraint inequalities stored in the storage unit 11, and the constraint condition The set of inequalities is updated (step S6).

  On the other hand, when the excision inequality does not exist and is not extracted (No at Step S11), the provisional solution including the non-integer is notified to the output unit 15 as the final solution. (Step S8).

  In addition, between step S2 and step S3 in FIG. 2, a process of deleting an inequality whose provisional solution does not satisfy the equal sign for the set of constraint inequalities registered in the recording unit 11 is executed. May be. By performing such processing, there is an effect of suppressing the number of constraint inequalities to which linear programming is applied.

[Maximum likelihood decoding using error correction code construction and linear programming]
By the way, before explaining the first algorithm, a construction method of an error correction code for a channel with additive noise using a matrix and a maximum likelihood decoding method using linear programming will be explained.

[Error correction code for channels with additive noise]
For example, a conventional method will be described by taking an error correction code for a communication path with additive noise as an example. An error correction code for a channel with additive noise is given in FIG. FIG. 3 is a diagram for explaining an error correction code for a communication path with additive noise. Hereinafter, as shown in Equation (1-5), GF (2) is a binary finite field, and an n-dimensional vector space is indicated as GF (2) ⁿ .

ここで、数式（１−７）は、全てＧＦ（２）上の要素である。

Assuming that the integer n is the input / output length of the communication path, the n-dimensional column vector on GF (2) is expressed by Equation (1-6).

Here, all the mathematical expressions (1-7) are elements on GF (2).

An l × n matrix A on GF (2) is expressed by Equation (1-8).

なお、零ベクトルの次元は特に明示しないが、ここではｌ次元のベクトルとする。 Also, a zero vector in which all components are 0 is expressed as Expression (1-9).

Note that the dimension of the zero vector is not particularly specified, but here it is an l-dimensional vector.

In the error correction code, a set satisfying the mathematical formula (1-10) is referred to as a “code” or a “codeword set”. When the rank of the matrix A is l, C is an n−1-dimensional vector space. C is a partial space of the n-dimensional vector space GF (2) ⁿ .

Since C is an n-1 dimensional vector space, if the message set is GF (2) ^n-1 , bijection from the set GF (2) ^n-1 to the set C exists. Actually, by obtaining the equation (1-12) using the n × (n−1) matrix G called “generation matrix” for the message vector shown in the equation (1-11), Match elements. The generator matrix G can be given so that Gm becomes an element of C for any message m.

In the error correction code of the communication path with additive noise, the message m is encoded into Equation (1-13) and input to the communication path.

When the noise is represented by Expression (1-14), the output of the communication path with additive noise is represented by Expression (1-15) and Expression (1-16). Here, (+) (plus sign in a circle) represents the addition of GF (2).

When Expression (1-16) is expressed as a vector component, Expression (1-17) is obtained.

The decoder that has received the vector y obtains Equation (1-18) that estimates the input signal vector x using a method called “maximum likelihood decoding”.

Assuming that the communication channel is stationary memoryless, the transition probability is expressed by Equation (1-19).

Here, Prob (z _j ) is a probability that a one-dimensional noise z _j is generated, and Equation (1-20) is assumed.

In the maximum likelihood decoding, an estimated x vector is obtained as in Expressions (1-21) and (1-22).

Here, in the first =, the input of the communication path satisfies the formula (1-23) and the formula (1-24) is used.

In the last =, the objective function can be maximized by increasing j that satisfies z _j = 0 as much as possible, that is, by reducing j that satisfies z _j = 1 as much as possible. By what can be achieved.

In the case of Expression (1-26), Expression (1-27) is similarly obtained.

[Maximum likelihood decoding using linear programming]
In the following, the maximum likelihood decoding method using the linear programming proposed in Non-Patent Document 5 will be described using Equation (1-28) as an example.

すなわち、

を求めておく。 The decoder calculates the mathematical expression (1-29) from the received vector y.

That is,

Ask for.

を制約条件として、線形の目的関数

を最小化する問題である。すなわち、数式（１−２８）は、数式（１−３３）と等価となる。

Equation (1-28) is a linear equation over GF (2)

Is a linear objective function

Is a problem of minimizing. That is, Equation (1-28) is equivalent to Equation (1-33).

  Hereinafter, the objective function is denoted as f (z), and maximization and minimization are simply referred to as “optimization”. The equation (1-28) is strictly an integer programming problem, not a linear programming problem in a real space. Hereinafter, the problem of optimizing the objective function f (z) with the linear equation on GF (2) as a constraint is simply referred to as “an integer programming problem with the linear equation on GF (2) as a constraint”.

  In solving an integer programming problem with a linear equation on GF (2) as a constraint by linear programming on real space, the linear equation on GF (2) is rewritten into a set of linear inequalities on real space.

に注目すると、線形方程式Ａｚ＝ｃは、ｌ個のＧＦ（２）上の方程式

の集合とみなせる。ここで、「・」は、ベクトルの内積を表す。ベクトルａ＝（ａ_１、・・・、ａ_ｊ、・・・、ａ_ｎ）に対して、集合｛１、２、・・・、ｎ｝の部分集合Ｓｕｐｐ（ａ_ｉ）（ａは行ベクトル）を数式（１−３６）のように定義する。

One row vector in A

Note that the linear equation Az = c is the equation over l GF (2)

Can be regarded as a set of Here, “·” represents an inner product of vectors. For a vector a = (a ₁ ,..., A _j ,..., A _n ), a subset Sup (a _i ) (a is a row vector) of the set {1, 2,. ) Is defined as Equation (1-36).

ただし、部分集合Ｓは、Ｃ_ｉ＝０のときは｜Ｓ｜が奇数であるものだけを、Ｃ_ｉ＝１のときは｜Ｓ｜が偶数であるものだけをとる。 Hereinafter, | S | represents the size of the set S, and S \ S ′ (dash) represents a difference set (a set including elements included in S but not included in S ′ (dash)). Formula (1-35), which is a linear equation on GF (2), is rewritten to Formula (1-37), which is a set of linear inequalities in real space.

However, the subset S takes only those in which | S | is odd when C _i = 0, and only those in which | S | is even when C _i = 1.

に変形（緩和）して、線形計画法を用いて解を求め、これを利用して目的関数ｆ（ｚ）の最適化を行う。 Thus, a linear equation on GF (2) is a set of linear inequalities on a real space.

Then, a solution is obtained using linear programming, and the objective function f (z) is optimized using this.

  Here, according to Non-Patent Document 5, when the components of z that give the optimal solution obtained by the linear programming method are all integers, the optimization of the integer programming problem with the linear equation on GF (2) as the constraint condition Proven to be the solution.

  However, in the above method, the following two problems are known.

First, there are 2 ^{| Supp (ai) | -1} inequalities included in the mathematical expression (1-37). This means that the number of inequalities increases exponentially with respect to the number of 1 included in the vector a _i , and causes an increase in the amount of calculation of linear programming and the required amount of memory.

  Also, the components of z that give the optimal solution obtained by linear programming are not always integers. When z includes a non-integer component, the optimal solution of the integer programming problem with the linear equation on GF (2) as a constraint is not obtained.

[Algorithm A]
As described above, when solving linear programming using all the inequalities included in Equation (1-38) as constraints, a large amount of calculation and memory are required. Find the optimal solution.

(1) k = 0. A provisional optimal solution is obtained by linear programming using only the set of initial inequalities represented by Expression (1-39) as a constraint expression.

(2) One or more inequalities that are not satisfied by the provisional optimal solution (excision inequalities for excising the provisional optimal solution) are searched from the mathematical expression (1-38). If not found, the provisional optimal solution is terminated as the final optimal solution.

(3) If a resection inequality for resecting the provisional optimal solution is found in step (2), the resection inequality added to Ineq (k) is Ineq (k + 1), and Ineq (k + 1) is a constraint expression. An optimal solution is obtained by the linear programming method described above, and this is set as a new provisional optimal solution.

(4) Increase the value of k by 1 and return to step (2).

[Algorithm B]
In algorithm A, it is not necessary to determine all of the inequalities _{Ineq (a i · z = c} i) in each i, find ablation inequalities that can ablate provisional optimum solution from _{Ineq (a i · z = c} i) It can be seen that it is sufficient. However, in the simple method, it is examined in order whether 2 ^{| Supp (ai) | -1} inequalities can cut off the provisional optimal solution, and the amount of calculation that increases exponentially remains. In Non-Patent Document 6, assuming that the constraint GF (2) equation is a _i · z = 0 (right side is 0), O (l _maxi | Supp (a _i ) | + nlog ₂ n The algorithm that can be calculated with the calculation amount of) is given as follows. This is called “algorithm B”. A provisional optimal solution including a non-integer component is defined as z.

(1) The index of the Supp (a _i ) is rearranged in descending order of the value of the z component corresponding to the index.

(2) With v = 1, let S (1) be a set consisting only of _j with the largest z _j . The maximum j uses the rearrangement obtained in step (1).

(3) If the mathematical formula (1-40) is not satisfied, this is ended as an ablation inequality in which z is excised.

(4) If the inequality satisfies the mathematical expression (1-40) in step (3), if the mathematical expression (1-41) is satisfied, it is reported that the inequality being cut is not found, and the process ends. .

(5) If the inequality satisfies equation (1-40) in step (3) and does not satisfy equation (1-41) in step (4), z _{j is} the largest in equation (1-42). S (v + 2) is obtained by adding two j to S (v). The maximum j is obtained using the rearrangement obtained in step (1). Then, the value of v is increased by 2.

満たしていなければ、切除している不等式は見つからないことを報告して終了する。 If S (v) satisfies Expression (1-43), the process returns to Step (3).

If not, report that the inequalities being excised are not found and exit.

[Algorithm C]
By the way, in the case where z includes a non-integer component, the problem that has not been obtained for the optimal solution of the integer programming problem with the linear equation on GF (2) as a constraint is as described above. Next, an adaptive procedure proposed in Non-Patent Document 6 for this problem will be described.

  In Non-Patent Document 5, an integer programming problem in which a row that can be written as a linear sum of a row vector set of a matrix A called “redundant row” is added to the matrix A, and a new linear equation on GF (2) is used as a constraint. It is proposed that the decoding performance can be improved by preparing and solving this by linear programming in real space. Specifically, the method is as follows.

A set of row vectors of the matrix A is represented by Expression (1-44).

Further, an arbitrary vector is represented by Equation (1-45) and Equation (1-46).

Then, Expression (1-47) is obtained from the vector c.

Then, the linear equation Az = c on GF (2) has the same value as the mathematical formula (1-48).

Similarly, by adding a plurality of redundant rows to the matrix A, an equation equivalent to Az = c can be created. Hereinafter, when the “redundant row” of the matrix A is referred to, the value on the right side of the corresponding equation (c _{1 + 1} in the above example) is also included. Further, when the set of inequalities determined by the mathematical expression (1-37) for the added redundant row can cut out the optimal solution including the non-integer component obtained by the linear programming, This is called a “redundant row that can cut out the optimum solution including the obtained non-integer components”.

  Non-Patent Document 6 proposes a method for obtaining a redundant row so that an optimal solution including a non-integer component obtained by linear programming can be removed by a set of equations. A specific method is given below and is referred to as “algorithm C”.

  As in Non-Patent Document 6, the equation Az = c is assumed, and an optimal solution including a non-integer component is z.

(1) Remove all columns in which the corresponding z _j is a non-integer from the j-th column of the matrix A, and make this a new matrix B.

(2) The index i of the row of the matrix B is randomly selected from {1, 2,..., L}, and i is recorded in order.

(3) Randomly select one j whose j-th component of the i-th row of the matrix B is not 0. However, j is different from j of the step (4) executed immediately before.

(4) Randomly selects i for which the i-th component of the j-th column of the matrix B is not 0. However, i is different from i in step (3) executed immediately before.

(5) If i selected in step (4) remains in the record, index recording (including i) until i appears in the past is set as a set of indexes for obtaining redundant rows. If i selected in step (4) does not remain in the record, the process returns to step (3).

(6) The redundant line a ′ obtained by adding all the rows of A corresponding to the set of indexes obtained in step (5).

(7) If an inequality that eliminates z from the equation a ′ · z = 0 obtained in step (6) can be created, this is added to the constraint inequality of linear programming. If not excised, the index record is cleared and the process returns to step (2). Algorithm B is used to determine whether an inequality that cuts z out of the equation can be created.

[Algorithm D]
Algorithm C is called when no ablation inequality is found from the inequality group created by the first matrix equation in step (2) of Algorithm A, and the ablation inequality obtained from the redundant row determined by algorithm C is: It is added in step (3) of algorithm A. In addition, Non-Patent Document 7 proposes a method using basic matrix transformation as a method for obtaining redundant rows. In this method, an optimal solution including a non-integer component near the vector 0 is obtained, and the optimal solution is removed assuming the equation Az = 0. This is called “algorithm D”.

そして、行列ＡでＱに含まれるインデックスの行だけを残した部分行列をＡ^Ｑとする。 (1) A subset Q of the set {1, 2,..., L} is defined as in Expression (1-49).

Then, leaving only the part matrix row of the index that is included in the Q in the matrix A and A ^Q.

(2) Formula (1-50) is obtained by rearranging the components of z in descending order.

そして、数式（１−５１）の置換と対応する行列Ａ^Ｑの行の置換を行う。行列Ａ^Ｑの行を置換する行列Πとする。 (3) Replace j ₁ , j ₂ ,..., J _n obtained in step (2) as shown in equation (1-51).

Then, the row of the matrix A ^Q corresponding to the substitution of the mathematical formula (1-51) is substituted. And matrix Π to replace the rows of the matrix A ^Q.

(4) A column matrix is applied to A ^Qを to obtain a staircase matrix (a matrix in which all elements below the diagonal components of the matrix are 0), and this is U.

複数のｉの候補があるときは、最も小さなｉと対応する冗長行を与える。 (5) An inverse matrix Π ^-1 of Π is obtained, and a redundant row is one in which the i-th row vector u _i of U ^{−１ -1} satisfies Equation (1-52).

When there are a plurality of i candidates, the redundant row corresponding to the smallest i is given.

  As described above, the algorithm D is used to add the redundant rows obtained by the algorithm D to the original matrix and expand the matrix in advance.

[Problems of Algorithms A to D]
By the way, the algorithms A to D described above have several problems.

  First, in algorithm B, it is necessary to repeatedly check equation (1-40).

  Secondly, in the algorithm B, the algorithm C, and the algorithm D, the equation Az = 0 is assumed as the constraint equation of GF (2), but it is necessary to be able to handle the general constraint equation Az = c.

  Third, algorithm C may not be able to find redundant rows and may continue to operate. Also, the same redundant row may be searched many times, and the redundant row is not found efficiently.

  Fourth, the method using the basic modification to the staircase matrix of the algorithm D may overlook that the redundant plane has the cut plane because the cut plane determination is too strict, and the redundant row is not efficiently found. .

  The above problems not only deteriorate the execution time of the linear programming method, but also possibly deteriorate the code performance without finally obtaining an integer solution. The first algorithm and the second algorithm described below have been made in view of the drawbacks of the above-described conventional methods, and have the effect of improving the probability that an integer solution can be obtained by linear programming. is there.

  The means for solving the above-described problems is composed of the following two parts.

  In order to solve the first problem, the first algorithm improves the algorithm B and more efficiently obtains a resection inequality that removes the provisional solution z from the equation Az = c on GF (2).

  In order to solve the second problem, the equation Az = c on GF (2) is transformed into the equivalent equation A hat z hat = 0.

  In order to solve the third problem or the fourth problem, the algorithm C or the algorithm D is improved and a redundant row to be cut out is efficiently obtained. At the same time, in order to solve the second problem, the general equation Az = c on GF (2) can be handled. The second algorithm corresponds to this.

  Hereinafter, the first algorithm and the second algorithm will be described in order.

[First algorithm]
In the first algorithm, a subset Supp (a) shown in Formula (1-36) (a is one row vector of the matrix A) is determined depending on whether the component z _{j of the} vector z exceeds 1/2. Classify into disjoint subsets S ₀ and S ₁ . In the first algorithm, one index j whose component z _{j of} vector z is closest to ½ is moved from _one of the subsets S ₀ and S ₁ to the other, so that the row of matrix A of vector c The relationship between the value c of the corresponding component and the even odd number of | S ₁ | is adjusted. In the first algorithm, whether or not there is an excision inequality for excising the provisional solution is determined by one inequality evaluation.

ここで、ｚ_ｊ＝１／２を満たすｊを部分集合Ｓ_０、Ｓ_１のどちらの集合に含めるようにするかは任意であるが、実施例１においては、部分集合Ｓ_０に含めるものとした。 Specifically, the excision inequality extraction unit 14 sets disjoint subsets S ₀ and S ₁ of the set {1, 2,..., N} as shown in Expression (1-53).

Here, it is arbitrary whether j that satisfies z _j = 1/2 is included in the subset S ₀ or S ₁ , but in the first embodiment, it is included in the subset S _0. did.

The excision inequality extraction unit 14 obtains Expression (1-54) when | S ₁ | is an even number when c = 0 and | S ₁ | is an odd number when c = 1.

ここで、右辺の部分集合Ｓ_０、Ｓ_１は、移動前の部分集合Ｓ_０、Ｓ_１である。なお、ｃ＝０かつ｜Ｓ_１｜が奇数、もしくは、ｃ＝０かつ｜Ｓ_１｜が偶数のときは、部分集合Ｓ_０、Ｓ_１について移動を行わない。 Then, the excision inequality extraction unit 14 moves the j ′ (dash) to the subset S ₁ if it is in the subset S ₀ , and moves it to the subset S ₁ if j ′ is in the subset S _1. move to the subset _{S 0.} That is, the moved subsets S ₀ and S ₁ are given by Expressions (1-55) and (1-56).

Here, the right side of the subset _S 0, _{S 1} is the pre-movement subset _S 0, _{S 1.} When c = 0 and | S ₁ | is an odd number, or c = 0 and | S ₁ | is an even number, the subsets S ₀ and S ₁ are not moved.

数式（１−５７）を満たしていない場合には、暫定解を切除する切除不等式が存在し、当該不等式が切除不等式であるとして抽出する。一方、数式（１−５７）を満たしている場合には、暫定解を切除する切除不等式が存在しないと判定する。 Then, the resection inequality extraction unit 14 determines whether or not the resection inequality exists by determining whether or not the mathematical expression (1-57) is satisfied.

If the mathematical formula (1-57) is not satisfied, an excision inequality for excising the provisional solution exists, and the inequalities are extracted as excision inequalities. On the other hand, when Expression (1-57) is satisfied, it is determined that there is no excision inequality for excising the provisional solution.

  This first algorithm will be described with reference to FIGS. The resection inequality extraction unit 14 applies the first algorithm to all the rows of the matrix A, and determines whether or not the resection inequality exists.

First, the excision inequality extraction unit 14 sets Supp (a) (a is a row vector) to Null (step S101). Here, the row vector a corresponds to one row or redundant row in the matrix A, or one row of the matrix A ′ (dash) transformed by applying the second algorithm to the matrix A. Here, a = (a ₁ ,..., A _j ,..., A _n ).

  Next, the excision inequality extraction unit 14 sets an initial value 1 to the index j (step S102), and determines whether the value of the index j is n or less (step S103).

When the value of index j is n or less (Yes at Step S103), the excision inequality extraction unit 14 determines whether or not the value of a _j is 0 (Step S104). Here, a _j is an element in the j-th column of the row vector a.

If the value of a _j is not 0 (No at Step S104), the excision inequality extraction unit 14 adds the value of Index j to the element of Supp (a) (Step S105), and then increases the value of Index j by one. (Step S106). On the other hand, if the value of a _j is 0 (Yes at step S104), the excision inequality extraction unit 14 increases the value of index j by one without adding the value of index j to the element of Supp (a). (Step S106).

  The excision inequality extraction unit 14 increases the value of the index j by one (step S106), and then returns to step S103 again to determine whether the value of the index j is n or less. That is, the excision inequality extraction unit 14 repeats the processing from step S103 to step S106 until the value of the index j exceeds n.

Now, when the value of the index j exceeds n in Step S103 (No in Step S103), the excision inequality extraction unit 14 continues with each other depending on whether or not the component z _{j of the} vector z exceeds 1/2. Classify into prime subsets S ₀ and S ₁ .

  Specifically, as shown in FIG. 4B, the excision inequality extraction unit 14 sets an initial value 1 to the index j, sets an initial value 1 to the index j ′ (dash) (step S107), and sets the index It is determined whether or not the value of j is n or less (step S108).

  When the value of the index j is n or less (Yes at Step S108), the excision inequality extraction unit 14 determines whether or not the index j is an element of Supp (a) (Step S109).

If the index j is an element of Supp (a) (step S109: Yes), the cut inequality extraction unit 14, the value of _{z j} is equal to or more than 0.5 (step S110), the _{z j} If the value is 0 or more and 0.5 or less (step S110: Yes), the index j is an element of the subset _{S 0} (step S111). On the other hand, when the value of z _j is not 0 or more and 0.5 or less (No at Step S110), the excision inequality extraction unit 14 sets the index j as an element of the subset S ₁ (Step S112).

Subsequently, the excision inequality extraction unit 14 determines whether z _j is closer to 0.5 than z _{j ′} (dash) (step S113). If they are close (Yes at step S113), the excision inequality extraction unit 14 replaces the value of j ′ with the value of index j (step S114), and then increases the value of index j by one (step S115). On the other hand, if it is not close (No at Step S113), the excision inequality extraction unit 14 increases the value of the index j by 1 without replacing the value of j ′ with the value of the index j (Step S115).

  The excision inequality extraction unit 14 increases the value of the index j by 1 (step S115), and then returns to step S108 again to determine whether the value of the index j is n or less. That is, the excision inequality extraction unit 14 repeats the processing from step S108 to step S115 until the value of the index j exceeds n.

When the value of the index j exceeds n in Step S108 (No in Step S108), the excision inequality extracting unit 14 subsequently sub-sets one index j whose component z _{j of the} vector z is closest to 1/2. By moving from _one of S ₀ and S ₁ to the other, the relationship between the value of the component c of the vector c and the even and odd numbers of | S ₁ | is adjusted.

  Specifically, as shown in FIG. 4C, the excision inequality extraction unit 14 determines whether or not the condition A is satisfied (step S116).

Here, the condition A is a condition that (b) the value of c is 0 and the number of elements included in S ₁ is an even number, (b) the value of c is 1, and the number of elements in S ₁ is an odd number, with the proviso that. Also, c uses c _i when the row vector a is the i-th row of the matrix A. When the row vector a is a redundant row, the corresponding c in the storage unit is used. If the row vector a is the i-th row of a new matrix A ′ (dash) obtained by applying the second algorithm to the matrix A, the corresponding c ′ _i is used. In FIG. 2, the corresponding value of c is referred to as a redundant row. That is, the excision inequality extraction unit 14 determines whether or not either of the conditions (a) and (b) is satisfied.

When it is determined that the condition A is satisfied (Yes at Step S116), the excision inequality extraction unit 14 moves j ′ to the subset S ₁ when j ′ (dash) is included in the subset S ₀ . j'is when included in the subset _{S 1} to move the j'to the subset _{S 0} (step S117). On the other hand, when it is determined that the condition A is not satisfied (No at Step S116), the excision inequality extraction unit 14 does not perform the process at Step S117.

  Next, the excision inequality extraction unit 14 determines whether or not Expression (1-57) is satisfied (step S118). That is, the excision inequality extraction unit 14 determines whether or not the inequality of the mathematical formula (1-57) is true.

  If it is determined that the result is not true (No at step S118), the excision inequality extraction unit 14 determines that there is an excision inequality that excises the provisional solution, and that the inequality determined to be true is the excision inequality. The algorithm processing is terminated (step S119). On the other hand, if it is determined to be true (No at step S118), the excision inequality extraction unit 14 determines that this line does not have an excision inequality that excises the provisional solution, and ends the first algorithm (step S120). .

[Algorithm B ′]
Subsequently, as a method for solving the second problem, a method of transforming the equation Az = c on GF (2) into the equation A hat z hat = 0 will be described.

これと同値な２つの方程式である数式（１−５９）、数式（１−６０）のように書き改めることができる。これを、Ａハットｚハット＝０、ｚ_ｎ＋１＝１とする。ここで（−）（丸の中にマイナス）はＧＦ（２）の減法を表す。

When the equation Az = c on GF (2) is expressed as Equation (1-58),

It can be rewritten as the following equations (1-59) and (1-60) which are two equations equivalent to this. This is A hat z hat = 0 and z _{n + 1} = 1. Here, (-) (minus in a circle) represents subtraction of GF (2).

Therefore, the linear programming problem can be applied using the variable vector as a z-hat obtained by incrementing the dimension of z by one, and the constraint condition as the mathematical formula (1-61). Here, the variable z _{n + 1} does not appear in the objective function f (z).

  By the way, applying algorithm B with such a modification is equivalent to replacing step (2) of algorithm B with the following.

(2 ′) If c _i = 1, v = 0 and S (0) is an empty set, and if c _i = 0, v = 1 and S (1) is a set consisting only of _j that maximizes z _j . The maximum j uses the rearrangement obtained in step (1). An algorithm B ′ is obtained after such replacement.

[Second algorithm]
In the second algorithm, the matrix A is subjected to basic transformation so that the row component of the matrix corresponding to the index j having a small value of | z _j −1/2 | Like that.

This second algorithm will be described with reference to FIGS. When the matrix A is given by Expression (1-62) and the vector c is given by Expression (1-63), the excision inequality extraction unit 14 converts the matrix A ′ (dash) into Expression (1-64). ).

  That is, the excision inequality extraction unit 14 creates a matrix A ′ as shown in FIG. 5A (step S201).

Next, the excision inequality extraction unit 14 obtains an array of the index j in which the index j of the vector z is rearranged in ascending order of the value of | z _j −1/2 | (step S202). If the values of the plurality of indexes j are the same order, for example, the ordering is performed such that the indices j are arranged in ascending order.

  Subsequently, the excision inequality extraction unit 14 sets an initial value 1 to k (step S203), sets an initial value 1 to i (step S204), and determines whether or not the value of i is 1 or less. (Step S205).

When the value of i is less than or equal to 1 (Yes at Step S205), the excision inequality extraction unit 14 determines whether or not the element a _{i, jk} (k is a subscript of j) of the matrix A is 0. (Step S206). When it is 0 (Yes at Step S206), the excision inequality extraction unit 14 increases the value of i by 1 (Step S207), and returns to the process at Step S205.

When all the elements a _{i, jk} of the matrix A are 0, the determination is always positive in step S206, and then the value of i exceeds the value of l in step S205 (No in step S205).

  Then, the excision inequality extraction unit 14 determines whether or not the value of k is 1 or less (step S208). If it is 1 or less (Yes in step S208), the value of k is increased by one. (Step S210), the process returns to Step S204 again.

  On the other hand, when the value of k exceeds the value of l (No at Step S208), the excision inequality extraction unit 14 separates the n + 1 column of the matrix A ′ as a vector c ′ (dash), and the second algorithm is The process ends (step S209).

In step S206, if the element a _{i, jk} of the matrix A is not 0 (No in step S206), the excision inequality extracting unit 14 performs row basic transformation on the matrix A ′ and performs the i′-th row. All the values of the elements a _{i, jk} are set to 0.

  That is, as shown in FIG. 5C, the excision inequality extraction unit 14 sets an initial value 1 to i ′ (step S211), and determines whether the value of i ′ is 1 or less (step S212). ).

When the value of i ′ is 1 or less (Yes at Step S212), the excision inequality extraction unit 14 determines that i ′ is not i (Step S213), and when i ′ is not i (Yes at Step S213). Then, it is determined whether or not the elements a _{i, jk} of the matrix A are not 0 (step S214).

If the element a _{i, jk} of the matrix A is not 0 (Yes at Step S214), the excision inequality extraction unit 14 determines that the vector (a _{i ′, 1} , a _{i ′, 2} ,..., A i _{′, n} , C _{i ′} ) and a vector (a _{i, 1} , a _{i, 2} ,..., A _{i, n} , c _i ) are _multiplied by a _{i ′, jk} / a _{i ′, jk} (k is j appended) A new vector (a _{i ′, 1} , a _{i ′, 2} ,..., A i _{′, n} , c _{i ′} ) is determined (step S215). As a result, the elements a _{i, jk} of the matrix A become 0.

  Then, the excision inequality extraction unit 14 increases the value of i ′ by one (step S216), and returns to the process of step S212 illustrated in FIG. In step S212, when the value of i ′ exceeds 1 (No in step S212), the excision inequality extraction unit 14 returns to the process of step S208 as shown in FIGS. 5-3 and 5-2.

If it is known in advance that the value of the objective function is an integer value, and the optimization problem is a minimization problem, the mathematical expression (1-65) is obtained after the provisional optimal solution z ′ is obtained. Excise z ′. Therefore, in Example 1, this constraint equation can be added when obtaining Ineq (k + 1). Here, the right side is an integer obtained by rounding up the decimal part of f (z ′).

Similarly, when it is known in advance that the optimization problem is a maximization problem and the value of the objective function becomes an integer value, after z ′ is obtained as a provisional optimal solution, 66) can be added to the constraint equation. Here, the right side is an integer obtained by discarding the decimal part of f (z ′).

[Effect of Example 1]
As described above, the estimation apparatus according to the first embodiment is an estimation apparatus that optimizes an objective function using a linear equation as a constraint on a binary finite field of 0 and 1. In addition, the estimation device uses a set of constraint inequalities indicating that they are greater than or equal to 0 and less than 1 as initial constraints, and if an ablation inequality that excises the provisional optimal solution z _j is extracted, The provisional solution z _j is calculated using a set of constraint inequalities to which the inequalities are added as a new constraint condition. Further, the estimation device creates a set of indexes j indicating columns in which the elements of the matrix are not 0 for each row of the matrix that is a coefficient of the linear equation, and sets the set of indexes j as a provisional solution corresponding to the index j. When the distance between z _j and 1 is greater than the distance with ₀ , the first set S ₀ is obtained. When the distance between the provisional solution z _j and 1 is less than 0, the second set S ₁ is obtained. Classify. In addition, the estimation apparatus is configured such that when the intercept of the linear equation is 0, the number of indices j classified into the second set S ₁ is an odd number, and when the intercept of the linear equation is 1. The classified first set S ₀ and second set S ₁ are adjusted so that the number of indexes j classified into the second set S ₁ is an even number. Then, as a result of the adjustment, the estimation device adds an added value obtained by adding the distances to 1 to all the provisional solutions z _j corresponding to the index j that belongs to the second set S ₁ , and the first set S ₀ . It is determined whether or not the sum of the provisional solution z _j corresponding to the index j to which it belongs belongs to the added value obtained by adding the distance to 0 is 1 or more. When it is determined that the result of the determination is not 1 or more, the estimation device extracts the inequality used for the determination as a resection inequality. Then, the estimation device adds the extracted excision inequality to the set of constraint inequality.

Further, the estimation apparatus according to the first embodiment corresponds to the index j in which the distance between the median between 0 and 1 and the provisional solution z _j is small when the excision inequality is not extracted. To change the row component of the matrix to zero. Further, the estimation device classifies the set of indexes j into a first set S ₀ and a second set S ₁ for each row of the transformed matrix.

For this reason, the estimation apparatus according to the first embodiment can reduce the calculation time. That is, the estimation apparatus according to the first embodiment can determine whether or not there is an inequality that cuts the provisional solution z _j by only one inequality determination (determination based on Expression (1-57)). At the same time, when there is an inequality that cuts the provisional solution z _j , the inequality can be given.

At this time, the estimation apparatus according to the first embodiment does not need to sort the index j in descending order of the value of the component of z _j as in algorithm B, and whether z _j is larger or smaller than 1/2, You only need to ask first if it is closest to 2. These can be performed with a computation time of O (n). Since the determination of the inequality can also be performed with the calculation time of O (n), the final calculation time is O (n) after all, and the first algorithm is superior to the algorithm B in calculation time. In addition, in the first algorithm, although an inequality that cuts z out of the equation a · z = c can be constructed, it is determined that the inequality cannot be constructed, and it does not stop. That is, whenever an inequality that cuts z out of the equation a · z = c is found, the first algorithm outputs one of the inequalities and ends.

Here, the point for obtaining whether z _j is larger or smaller than 1/2 will be further described. The estimation apparatus according to the first embodiment cuts the provisional solution z _j by performing the determination according to Expression (1-57). Whether or not there is an inequality can be determined by one inequality determination. In this case, it is meaningful to make the left side as small as possible. In other words, if the value on the left side is set as small as possible, there is no smaller value, and one inequality determination is sufficient.

Therefore, the estimation device determines whether z _j is larger or smaller than 1/2 in order to make the left side as small as possible. As can be seen from the left side, the first term on the left side is an added value obtained by adding the distances to 1 for all the provisional solutions z _j corresponding to the index j belonging to the second set S ₁ . The second term is an added value obtained by adding the distances to 0 for all the provisional solutions z _j corresponding to the index j belonging to the first set S ₀ . In other words, if the index j having a value close to 1 is classified in the second set S ₁ in advance, and the index j having a value close to 0 is classified in the first set S ₀ in advance, the entire left side As can be seen from FIG. The estimation apparatus according to the first embodiment uses this point.

  Further, according to the estimation apparatus according to the first embodiment, the equation Az = c on GF (2) is transformed into the equivalent equation A hat z hat = 0, whereby the more general equation Az = c is expressed as a constraint equation. Can be set as, and more optimization problems can be handled. In addition, the calculation time can be greatly reduced, and at the same time, the code performance can be greatly improved. In addition, the cutting plane can be obtained efficiently and the coding efficiency can be improved.

  The estimation apparatus according to the second embodiment uses a technique of adding redundant rows that can remove a provisional optimum solution in addition to the technique of the estimation apparatus according to the first embodiment. Hereinafter, first, the configuration of the estimation apparatus 10 according to the second embodiment will be described with reference to FIG. 6, and then the processing procedure in the estimation apparatus 10 according to the first embodiment will be described with reference to FIG.

[Configuration of Estimation Device 10 According to Second Embodiment]
FIG. 6 is a block diagram illustrating the configuration of the estimation apparatus 10 according to the second embodiment. As shown in FIG. 6, the estimation apparatus 10 according to the second embodiment is different from the first embodiment in that excision redundant rows are transmitted and received between the excision inequality extraction unit and the storage unit. That is, in the first embodiment, the estimation device 10 does not need to store the cut redundant row in the storage unit, but in the second embodiment, stores this.

[Processing Procedure by Estimating Device 10 According to Second Embodiment]
Next, a processing procedure in the estimation apparatus 10 according to the second embodiment will be described. As illustrated in FIG. 7, the estimation apparatus 10 according to the second embodiment also records a cut redundant row as necessary in step S <b> 6 and applies the first algorithm to the redundant row of the matrix A in step S <b> 4. This is different from the estimation apparatus 10 according to the first embodiment. FIG. 7 is a flowchart illustrating a processing procedure in the estimation apparatus 10 according to the second embodiment.

  Specifically, the estimation apparatus 10 according to the second embodiment has the following processing procedure.

(1) k ₁ = 0 and k ₂ = 0. A temporary optimal solution is obtained by linear programming using only the initial set of inequalities (formula (1-39)) as a constraint equation. Let A _{0 be the} initial matrix.

(2) Among the inequalities in Ineq (k ₁ ), leaving the inequalities that the provisional optimal solution satisfies as equalities and deleting the other inequalities will be referred to as Ineq (k ₁ +1). Then, the value of _{k 1} is increased by one. The process of step (2) is executed between step S3 and step S4 of FIG.

(3) The first algorithm is applied to all the rows of the matrix A _k2 and one or more inequalities for cutting out the provisional optimal solution are searched from Ineq (A _k2 z = c).

(4) If an inequality that cuts the provisional optimum solution is found in step (3), the inequality added to Ineq (k ₁ ) is set to Ineq (k ₁ +1). An optimal solution is obtained by linear programming using Ineq (k ₁ +1) as a constraint equation, and this is set as a new provisional optimal solution. Then, the value of _{k 1} Increase 1 returns to step (2).

(5) If the inequality cannot be found and all the components of the provisional optimal solution are integers, this is ended as the optimal solution.

(6) If the inequality cannot be found and the provisional optimal solution includes a non-integer component, the provisional optimal solution is removed by applying the second algorithm to the initial matrix A ₀ . Find redundant rows. If found, a new matrix obtained by adding the redundant row to the matrix A _k2 is defined as A _{k2 + 1} . And the value of _{k 2} is increased 1 returns to the step (2). If not found, the provisional optimal solution including non-integer components is terminated as the optimal solution.

  Here, step (2) has an effect of suppressing the number of constraint inequalities to which linear programming is applied. While this has the effect of reducing the amount of memory required for linear programming, it may also increase the code execution time, so step (2) does not execute anything and moves to the next step. It doesn't matter.

  So far, the estimation apparatus that optimizes the function f (z) using the linear equation Az = c as a constraint on the binary finite field GF (2) has been described as the first and second embodiments. In the first embodiment, a channel code with additive noise has been described as an optimization problem to which the present invention can be applied. Subsequently, in the third and subsequent embodiments, embodiments in which the present invention is applied to an encoder and a decoder will be described.

  Here, as shown in FIG. 8, the present invention can be applied to various optimization problems. FIG. 8 is a diagram illustrating an optimization problem that appears in encoding and decoding. The constraint expressions Az = c and Bz = m can be expressed by one constraint expression when one matrix in which A and B are vertically connected is considered. In the search problem, maximization of the objective function is considered with the success of search being 1 and the failure being 0. In the following, an embodiment in which the present invention is applied to an encoder and a decoder in a fixed-length undistorted code will be described as a third embodiment.

In the following, it is assumed that the matrix and vector elements belong to {0, 1}. Also, | · | represents the number of sets, and H (z) and H (z | y) are defined as Equations (2-1) and (2-2).

  The encoder and decoder in the fixed length distortionless code are given in FIG. FIG. 9 is a diagram for explaining an encoder and a decoder in a fixed-length distortionless code.

ここで、ｇ_Ａの右辺の実現に、本発明を適用することができる。 The encoder and the decoder share an appropriate l × n matrix A in advance. The encoder operates the matrix A on the n-dimensional information expressed by the vector z to obtain the l-dimensional vector c = Az and transmits it or stores it in a memory or the like. Meanwhile, the decoder, the vector c is read from the information such as the reception or memory, when the probability distribution mu _Z is known, by solving the optimization problem defined by the equation (2-3), original information z Try to restore.

Here, the realization of the right side of g _A, it is possible to apply the present invention.

ここで、数式（２−４）、数式（２−５）で定義される目的関数は、実数の和を考える。この場合も、数式（２−４）、数式（２−５）で定義される目的関数に、本発明を適用することができる。 When the probability distribution mu _Z is unknown, Equation (2-4), the formula (2-5), by solving the optimization problem defined by the equation (2-6) attempts to restore the original information z.

Here, the objective function defined by Equations (2-4) and (2-5) considers the sum of real numbers. Also in this case, the present invention can be applied to the objective function defined by the mathematical formulas (2-4) and (2-5).

  Next, the fourth embodiment is an embodiment in which the present invention is applied to a code of a communication path with additive noise. Note that the error correction code for the communication path with additive noise is given in FIG. 3 as described above.

  The encoder and the decoder share an appropriate l × n matrix A in advance. As already described in the first embodiment, the encoder uses a matrix G called a generator matrix to obtain a vector x from the message vector m by Equation (1-13), and uses this as a channel input. Here, a set of communication path inputs is expressed by Expression (1-10). The decoder obtains a vector c = Ay from the vector y that is the channel output.

For this case, the channel noise z, since the following relationship holds formula Az = c, in the same manner as in Example 3, when the noise of the probability distribution mu _z is known, is defined by the equation (2-3) We try to recover the noise z by solving the optimization problem. Here, the realization of the right side of g _A, it is possible to apply the present invention.

Even when noise probability distribution mu _z is unknown, in the same manner as in Example 3, equations (2-4), the formula (2-5), solving the optimization problem defined by the equation (2-6) Tries to restore the noise z.

Finally, the decoder estimates the channel input x from the restored noise z by the equation (3-1) (the minus sign in the circle is a subtraction of GF (2) ⁿ ), and the equation (1- The original message m is estimated from the relational expression 13).

  Next, the fifth embodiment is an embodiment in which the present invention is applied to an encoder and a decoder in a fixed-length undistorted code that can use auxiliary information.

  An encoder and decoder in a fixed-length distortionless code that can use auxiliary information are given in FIG. FIG. 10 is a diagram for explaining an encoder and a decoder in a fixed-length undistorted code that can use auxiliary information.

The encoder and the decoder share an appropriate l × n matrix A in advance. The encoder operates the matrix A on the n-dimensional information expressed by the vector z to obtain the l-dimensional vector c = Az and transmits it or stores it in a memory or the like. It is assumed that the decoder can receive auxiliary information y correlated with z in addition to the vector c read from information received or from memory. When the transition probability distribution μ _{Z | Y of the} communication path is known, the restoration of the communication path input z is attempted by solving the optimization problem defined by Equation (4-1).

ここで、数式（４−２）及び数式（４−３）の制約条件に現れるｚ_ｉのΣ（ｉ＝１〜ｎ）、目的関数に現れるｙ_ｉｚ_ｉのΣ（ｉ＝１〜ｎ）は、実数の和を表している。数式（４−２）及び数式（４−３）で定義される最適化問題に対して本発明を適用することができる。 When the transition probability distribution μ _{Z | Y of the} communication path is unknown, an optimization problem defined by Expression (4-2), Expression (4-3), and Expression (4-4) is determined.

Here, equations (4-2) and Equation (4-3) sigma of _{z i} appearing in constraints (i = 1 to n), the _y i _{z i} that appears in the objective function sigma (i = 1 to n) Represents the sum of real numbers. The present invention can be applied to the optimization problem defined by Equation (4-2) and Equation (4-3).

  Next, a sixth embodiment is an embodiment in which the present invention is applied to an encoder and a decoder in a general channel code.

  An encoder and a decoder in a general channel code are given in FIG. 11 and FIG. 11 and 12 are diagrams for explaining an encoder and a decoder in a general channel code.

The encoder and the decoder share an appropriate l _A × n matrix A, l _B × n matrix B, l _A dimensional vector c, wε {0, 1,..., N} in advance. The encoder solves the optimization problem expressed by Equation (5-1) from the 1 _B- dimensional information represented by the vector m and the vector c, thereby performing n-dimensional communication satisfying Az = c and Bz = m. The road input z is obtained.

Here, with respect to the matrices A and B, a [l _A + l _B ] × n matrix obtained by vertically connecting the matrices A ′ (dash), and a l _A + l _B- dimensional vector obtained by vertically connecting the column vectors c and m If c ′ (dash), the constraint conditions Az = c and Bz = m can be rewritten into one constraint condition A′z = c ′. This optimization problem becomes a quadratic programming problem. The present invention can be applied to the realization of _gAB .

The decoder attempts to decode from the received n-dimensional vector y. Here, the decoder uses Equation (4-1) when the channel transition probability distribution μ _{Z | Y} is known, and Equation (4-2), Equation (4-3), Equation when the channel transition probability distribution is unknown. The communication path input z is estimated using the mathematical expression defined in (4-4). The decoder then estimates the original message by determining Bz from the estimated z.

  Next, a seventh embodiment is an embodiment in which the present invention is applied to an encoder and a decoder in a distorted code.

  The encoder and decoder in the distorted code are given in FIGS. 13 and 14 are diagrams for explaining an encoder and a decoder in a distorted code.

The encoder and decoder share an appropriate l _A × n matrix A, l _B × n matrix B, l _A dimensional vector c and an appropriate real number γ in advance. Here, positive and negative γ is determined from the distribution mu _Y sources. Further, a distortion measure between the letters y and z is d (y, z).

The encoder obtains an n-dimensional vector z satisfying Az = c by solving an optimization problem defined by Equation (6-1) from the n-dimensional information y, and sets the l _B- dimensional vector m≡Bz. The code word is transmitted and stored in a memory or the like.

Note that Equation (6-1) may be replaced with Equation (6-2). This has the effect of facilitating later decoding.

Then, it is possible to apply the present invention to the realization of the right side of _{g A} formula (6-1) and Equation (6-2).

ここで、行列Ａ、Ｂに対して、これを縦に連結した［ｌ_Ａ＋ｌ_Ｂ］×ｎ行列をＡ´（ダッシュ）、列ベクトルｃ、ｍを縦に連結したｌ_Ａ＋ｌ_Ｂ次元ベクトルをｃ´（ダッシュ）とすれば、制約条件Ａｚ＝ｃ、Ｂｚ＝ｍは、一つの制約条件Ａ´ｚ＝ｃ´に書き改めることができる。そして、このｇ_Ａの右辺の実現に本発明を適用することができる。 The decoder restores the vector z by solving the optimization problem defined by Expression (6-3) from the vector m received or received from the memory or the like.

Here, with respect to the matrices A and B, a [l _A + l _B ] × n matrix obtained by vertically connecting the matrices A ′ (dash), and a l _A + l _B- dimensional vector obtained by vertically connecting the column vectors c and m If c ′ (dash), the constraint conditions Az = c and Bz = m can be rewritten into one constraint condition A′z = c ′. Then, it is possible to apply the present invention to the realization of the right side of the g _A.

  Next, Example 8 provides a comparison between the conventional algorithm and the algorithm of the present invention. First, an l × n matrix A is prepared. An n-dimensional vector z is given independently and randomly. At this time, the probability that the element of each dimension is 1 is P (Z = 1). The encoder calculates a one-dimensional codeword vector c≡Az for the information vector z.

In the decoder, the n-dimensional vector g _A (c) is calculated by Expression (7-1).

Here, the vector z = (z ₁ , z ₂ ,..., Z _n ). When this is obtained using linear programming, the constraint condition is expressed by equation (7-2), and the objective function f (z) is expressed by equation (7-3).

_A vector obtained by applying linear programming to g _A (c) is defined as z (c) (hat to z). However, when a vector obtained by using linear programming has a non-integer component, the component is rounded off to 0 or 1. A decoding error occurs when z (c) (hat to z) does not match z.

By the way, in this problem, after the provisional optimal solution z ′ (dash) is obtained, the inequality of Expression (7-4) cuts out z ′. Add.

When this experiment is performed on k independently generated vectors z, the decoding error probability can be estimated by Equation (7-5).

Here, z _i is an information vector generated in the i-th experiment, c _i is a codeword vector generated in the i-th experiment, and is defined as in Expression (7-6).

  FIG. 15 is a diagram illustrating a simulation result. The horizontal axis of the graph represents P (Z = 1), and the vertical axis represents Error. The experiment was conducted with k = 1000. Each graph corresponds to the following experimental results. Hereinafter, the first algorithm is referred to as “algorithm E”, and the second algorithm is referred to as “algorithm F”.

  In algorithm D, the upper limit of the number of redundant rows to be added in advance is set to 100. However, only four rows are actually added due to a problem of used memory.

  A + B ′ + C (1) is an algorithm for executing the algorithm A when executing the algorithm A and further obtaining the excision inequality. Further, in the following A + E + F (1), when obtaining one excision inequality, 80 redundant row candidates are prepared by basic modification, so the number of times of obtaining redundant row candidates is 80 × 1000.

  A + B ′ + C (2) is an algorithm for executing the algorithm A when executing the algorithm A and obtaining a redundant row to be cut off. In the following A + E + F (2), since 80 redundant row candidates are prepared by basic modification when obtaining one redundant row to be cut, the number of times of obtaining redundant row candidates is 80 × 100 times.

A + E + F (1) is the form described in the first embodiment. However, all excision inequalities made from the rows included in matrix A were added. In addition, the number of times that the inequality is added is 1000 times. A + E + F (2) is the form described in the second embodiment. However, all excision inequalities made from the rows included in A _k2 were added. In addition, the number of redundant rows added was set to 100 times.

  However, in A + B ′ + C (1) and A + B ′ + C (1), algorithm B ′ is used instead of algorithm B in step 2 of algorithm A and step 7 of algorithm C. Here, algorithm B ′ is used because algorithm B cannot be used because it assumes Az = 0 in the constraint equation. Also, the result of P (Z = 1) = 0.06 in A + B ′ + C (1), A + B ′ + C (2), A + E + F (1), and A + E + F (2) is not described. It is because it became 0.

  The simulation result of FIG. 15 shows that the decoding performance of the first and second embodiments is improved as compared with the conventional algorithm.

  The present invention can be similarly applied to the Slepian-Wolf problem, Wyner-Ziv problem, Gel'fand-Pinsker problem, and One-helps-one problem described in Non-Patent Document 1.

DESCRIPTION OF SYMBOLS 10 Estimation apparatus 11 Memory | storage part 12 Input part 13 Provisional solution calculation part 14 Excision inequality extraction part 15 Output part

Claims (19)

An estimation device for optimizing an objective function on a binary finite field of 0 and 1 with a linear equation as a constraint,
A set of constraint inequalities indicating zero or more and less than or equal to 1 is used as an initial constraint, and when a cut-off inequality that cuts the temporary optimal solution z _j is extracted, a cut-off inequality is added to the cut-off inequality A provisional solution calculating means for calculating a provisional solution z _j using a set of
For each row of the matrix that is a coefficient of the linear equation, a set of indexes j indicating columns in which the elements of the matrix are not 0 is created, and the set of the indexes j is set as provisional solutions z _j and 1 corresponding to the index j, Classifying means for classifying into the first set S ₀ when the distance of 0 exceeds the distance with 0, and into the second set S ₁ when the distance between the provisional solution z _j and 1 is less than the distance with 0; ,
When the intercept of the linear equation is 0, the number of indices j classified into the second set S ₁ is an odd number, and when the intercept of the linear equation is 1, the second Adjusting means for adjusting the first set S ₀ and the second set S ₁ classified by the classifying means so that the number of indexes j classified into the set S ₁ is an even number;
As a result of adjustment by the adjusting means, an added value obtained by adding the distances to 1 for all the provisional solutions z _j corresponding to the index j that belongs to the second set S ₁ , and the first set S ₀ Determination means for determining whether or not the sum of the provisional solution z _j corresponding to the index j to which it belongs belongs to the sum of the addition values obtained by adding the distance to 0 is 1 or more;
An extraction means for extracting the ablation inequality based on the inequality used for the determination when it is determined that the determination means does not result in 1 or more;
An estimation device comprising: an adding unit that adds the excision inequality extracted by the extraction unit to the set of constraint inequalities. If the excision inequality is not extracted by the extraction means, the matrix is divided into the matrix row corresponding to the index j whose distance between the median value between 0 and 1 and the provisional solution z _j is small. And further comprising a deformation means for deforming so that the component becomes zero,
2. The classification unit according to claim 1, wherein the classifying unit classifies the set of indexes j into a first set S ₀ and a second set S ₁ for each row of the matrix after the transformation by the transformation unit. Estimating device.   The estimation apparatus according to claim 1, wherein the information encoded by the code of the channel with additive noise is expressed as an objective function with a linear equation as a constraint on a binary finite field of 0 and 1 A decoder comprising a decoder for decoding by optimizing.   The estimation apparatus according to claim 1 or 2, wherein the information encoded by the fixed-length undistorted code is used to optimize an objective function on a binary finite field of 0 and 1 with a linear equation as a constraint. A decoder comprising: a decoder that performs decoding in (1).   The estimation apparatus according to claim 1, wherein information encoded by a fixed-length undistorted code that can use auxiliary information is converted into an objective function using a linear equation as a constraint on a binary finite field of 0 and 1 A decoder provided with a decoder for decoding by optimizing the above.   A code that encodes information to be sent to a communication path by optimizing an objective function on a binary finite field of 0 and 1 using a linear equation as a constraint on the estimation device according to claim 1 or 2. A coder provided with the device.   The decoding apparatus according to claim 1 or 2, wherein the information to be transmitted to the communication path is decoded by optimizing an objective function with a linear equation as a constraint on a binary finite field of 0 and 1 A decoder comprising the device.   3. The encoder according to claim 1 or 2, wherein an information function is encoded by a distorted code by optimizing an objective function with a linear equation as a constraint on a binary finite field of 0 and 1. The encoder characterized by comprising.   The estimation apparatus according to claim 1, wherein information encoded by a distorted code is decoded by optimizing an objective function on a binary finite field of 0 and 1 using a linear equation as a constraint condition A decoder comprising a decoder for converting to a decoder. An estimation method for optimizing an objective function with a linear equation as a constraint on a binary finite field of 0 and 1,
Computer
A set of constraint inequalities indicating zero or more and less than or equal to 1 is used as an initial constraint, and when a cut-off inequality that cuts the temporary optimal solution z _j is extracted, a cut-off inequality is added to the cut-off inequality A provisional solution calculation step of calculating a provisional solution z _j using a set of
For each row of the matrix that is a coefficient of the linear equation, a set of indexes j indicating columns in which the elements of the matrix are not 0 is created, and the set of the indexes j is set as provisional solutions z _j and 1 corresponding to the index j, A classification step for classifying into a first set S ₀ when the distance of 0 exceeds a distance with 0, and into a second set S ₁ when the distance between the provisional solution z _j and 1 is less than the distance with 0; ,
When the intercept of the linear equation is 0, the number of indices j classified into the second set S ₁ is an odd number, and when the intercept of the linear equation is 1, the second An adjusting step of adjusting the first set S ₀ and the second set S ₁ classified by the classification step so that the number of indexes j classified into the set S ₁ is an even number;
As a result of the adjustment in the adjustment step, an added value obtained by adding the distances to 1 for all the provisional solutions z _j corresponding to the index j belonging to the second set S ₁ , and the first set S ₀ A determination step of determining whether or not the sum of the provisional solution z _j corresponding to the index j to which it belongs belongs to the added value obtained by adding the distance to 0 is 1 or more;
As a result of the determination by the determination step, when it is determined that it is not 1 or more, an extraction step of extracting the excision inequality based on the inequality used for the determination;
And an additional step of adding the excision inequality extracted by the extraction step to the set of constraint inequalities. If the excision inequality is not extracted by the extraction step, the matrix is defined as a matrix row corresponding to an index j in which the distance between the median between 0 and 1 and the provisional solution z _j is small. And further comprising a deformation step for deforming the component to zero,
11. The classification according to claim 10, wherein the classification step classifies the set of indexes j into a first set S ₀ and a second set S ₁ for each row of the matrix after the transformation in the transformation step. Estimation method.   The estimation method according to claim 10 or 11, wherein the information encoded by the code of the channel with additive noise is expressed as an objective function with a linear equation as a constraint on a binary finite field of 0 and 1 A decoding method comprising: a decoder for decoding by optimization.   12. The estimation method according to claim 10 or 11, wherein information encoded by a fixed-length undistorted code is optimized on a binary finite field of 0 and 1 with a linear equation as a constraint condition. The decoding method characterized by including the decoder decoded by.   The estimation method according to claim 10 or 11, wherein information encoded by a fixed-length undistorted code for which auxiliary information can be used is an objective function with a linear equation as a constraint on a binary finite field of 0 and 1 The decoding method characterized by including the decoder which decodes by optimizing.   12. An encoding method for encoding information to be sent to a communication path by optimizing an objective function using a linear equation as a constraint on a binary finite field of 0 and 1 according to the estimation method according to claim 10 or 11 A coding method characterized in that the device comprises.   12. An estimation method according to claim 10 or 11, wherein information to be sent to a communication path is decoded by optimizing an objective function on a binary finite field of 0 and 1 with a linear equation as a constraint. A decoding method characterized in that the device includes.   The estimation method according to claim 10 or 11, wherein an encoder encodes information using a distorted code by optimizing an objective function on a binary finite field of 0 and 1 using a linear equation as a constraint. The encoding method characterized by including.   12. The estimation method according to claim 10 or 11, wherein information encoded by a distorted code is decoded by optimizing an objective function with a linear equation as a constraint on a binary finite field of 0 and 1 The decoding method characterized by including the decoder to convert.   The program which makes a computer perform the method as described in any one of Claims 10-18.

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