RELATIONSHIP EXTRACTION APPARATUS, RELATIONSHIP EXTRACTION METHOD, AND PROGRAM
A relationship extraction device includes a memory; and a processor configured to execute obtaining a set of data {x 0 , . . . , x T−1 }⊆X each having multiple elements and a set of data {y 0 =f(x 0 ), . . . , y T−1 =f(x T−1 )}⊆Y each having multiple elements, where f is any mapping; generating an approximate operator that approximates a Perron-Frobenius operator K satisfying Kφ 1 (x t )=φ 2 (y t ) for t=0, . . . , T−1, wherein φ 1 is a feature mapping with respect to a positive definite kernel function k 1 on X×X that takes C*-algebra values, and φ 2 is a feature mapping with respect to a positive definite kernel function k 2 on Y×Y that takes C*-algebra values; obtaining data x t and x s as targets of relationship extraction; and extracting a relationship between each element of x t and each element of x s by using the approximate operator.
1 . A relationship extraction device comprising:
a memory; and
a processor configured to execute
a obtaining a set of data {x 0 , . . . , x T−1 }□X each having multiple elements and a set of data {y 0 =f(x 0 ), . . . , y T−1 =f(x T−1 )}□Y each having multiple elements, where f is any mapping;
generating an approximate operator that approximates a Perron-Frobenius operator K satisfying Kφ 1 (x t )=φ 2 (y t ) for t=0, . . . , T−1, wherein φ 1 is a feature mapping with respect to a positive definite kernel function k 1 on X×X that takes C*-algebra values, and φ 2 is a feature mapping with respect to a positive definite kernel function k 2 on Y×Y that takes C*-algebra values;
obtaining data x t and x s as targets of relationship extraction; and
extracting a relationship between each element of x t and each element of x s by using the approximate operator.
2 . The relationship extraction device as claimed in claim 1 , wherein the extracting extracts the relationship for analyzing anomaly detection or causal estimation.
3 . The relationship extraction device as claimed in claim 1 , wherein the extracting extracts a C*-algebra value representing the relationship, by approximating an inner product <x t , x s > k defined on an RKHM (reproducing kernel Hilbert C*-module) with respect to the positive definite kernel function k 1 , by the approximate operator.
4 . The relationship extraction device as claimed in claim 1 , wherein in a case of X=Y, y t =x t+1 , k 1 =k 2 =k, and φ 1 =φ 2 =φ, the generating generates the approximate operator by using an operator {circumflex over ( )}K with which {circumflex over ( )}Kφ(x T−1 ) approximates φ(x T ), and an orthonormal projection from an RKHM with respect to a positive definite kernel function K to a space represented by a linear combination of φ(x t ) and C*-algebra values.
5 . The relationship extraction device as claimed in claim 1 , wherein in a case of X≠Y, the generating generates the approximate operator by using a linear mapping {circumflex over ( )}K with which {circumflex over ( )}Kφ 1 (x t ) approximates φ 2 (y t ), and an orthonormal projection from an RKHM with respect to the positive definite kernel function k 1 to a space represented by a linear combination of φ 1 (x t ) and C*-algebra values.
6 . A method of extracting relationship executed by a computer including a memory and a processor, the method comprising:
obtaining a set of data {x 0 , . . . , x T−1 }□X each having multiple elements and a set of data {y 0 =f(x 0 ), . . . , y T−1 =f(x T−1 )}□Y each having multiple elements, where f is any mapping;
generating an approximate operator that approximates a Perron-Frobenius operator K satisfying Kφ 1 (x t )=φ 2 (y t ) for t=0, . . . , T−1, wherein (pi is a feature mapping with respect to a positive definite kernel function k 1 on X×X that takes C*-algebra values, and φ 2 is a feature mapping with respect to a positive definite kernel function k 2 on Y×Y that takes C*-algebra values;
obtaining data x t and x s as targets of relationship extraction; and
extracting a relationship between each element of x t and each element of x s by using the approximate operator.
7 . A non-transitory computer-readable recording medium having computer-readable instructions stored thereon, which when executed, cause a computer to function as the relationship extraction device as claimed in claim 1 .