Relationship Extraction Using C*-Algebra RKHM Framework
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Solution Overview
Problem
Current methods, such as RKHS and vv-RKHS, are limited in representing relationships between multiple data elements, with RKHS only handling single elements and vv-RKHS measuring proximity in complex values, leading to inefficiencies in extracting relationships between data items.
Innovation Solution
A relationship extraction device using the RKHM framework generates an approximate Perron-Frobenius operator to extract relationships between elements by mapping feature functions from positive definite kernel values, enabling the representation of relationships between multiple elements.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If vv-RKHS is used to represent relationships among multiple elements, then the capability to approximate relationships among multiple elements is improved, but the complexity of measuring proximity increases due to requiring n2 complex numbers
Solution Approach 1:
The patent changes the parameter space from complex numbers to C*-algebra values. By using C*-algebra which generalizes both complex numbers and matrices, the method represents proximity relationships using algebraic structures with conjugation and norm operations, reducing the complexity from n2 complex numbers to C*-algebra valued functions in RKHM space.
Solution Approach 2:
The patent transitions from measuring proximity in the original data space to measuring proximity in the transformed RKHM feature space. By mapping data to RKHM using C*-algebra valued kernel functions, the proximity measurement is performed in a higher-dimensional functional space where relationships among multiple elements can be captured more efficiently.
2Ease of operation
If RKHS is used for data analysis, then the handling of single element data is improved, but the capability to describe relationships among multiple elements is lost
Solution Approach 1:
The patent creates a unified framework (RKHM) that generalizes both single-element RKHS analysis and multi-element relationship analysis. By using C*-algebra valued kernel functions, the same mathematical framework can handle both single element data (when C*-algebra reduces to complex numbers) and multi-element relationship data (when full C*-algebra structure is utilized), making the method universally applicable.
Data Source
AI summary
A relationship extraction device includes a memory; and a processor configured to execute obtaining a set of data {x0, . . . , xT−1}⊆X each having multiple elements and a set of data {y0=f(x0), . . . , yT−1=f(xT−1)}⊆Y each having multiple elements, where f is any mapping; generating an approximate operator that approximates a Perron-Frobenius operator K satisfying Kφ1(xt)=φ2(yt) for t=0, . . . , T−1, wherein φ1 is a feature mapping with respect to a positive definite kernel function k1 on X×X that takes C*-algebra values, and φ2 is a feature mapping with respect to a positive definite kernel function k2 on Y×Y that takes C*-algebra values; obtaining data xt and xs as targets of relationship extraction; and extracting a relationship between each element of xt and each element of xs by using the approximate operator.