Geometric Algebra Data Encoding for Ordered Relationships
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Solution Overview
Problem
Current data processing techniques struggle with encoding high-dimensional data, particularly in heterogeneous spaces with multiple manifolds, as they assume homogenous spaces and lose ordered relationships, making it difficult to distinguish between statements like 'the man bit the dog' and 'the dog bit the man'.
Innovation Solution
A geometric algebra approach is used to encode data relationships, allowing for multiple manifolds in heterogeneous spaces with unique attitudes, orientations, and stances, using a quarter rotation operation in Clifford Algebra and Geometric Algebra to preserve order and distinguish between semantic classes.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of operation
If traditional vector space approaches and proximity matrices are used to process data, then data processing can be performed using standard methods, but ordered relationships between data are lost and statements like 'the man bit the dog' become indiscernible from 'the dog bit the man'
Solution Approach 1:
The patent applies asymmetry by using directed edges in hypergraphs to represent ordered relationships between data elements. Instead of symmetric distance matrices where d(i,j) = d(j,i), the invention uses asymmetric adjacency structures where the relationship from element i to element j is distinct from the relationship from j to i, thereby preserving the directional information necessary to distinguish 'the man bit the dog' from 'the dog bit the man'
Solution Approach 2:
The patent transitions from traditional vector space (2nd order tensors) to hypergraph representations (higher-order structures) to encode ordered relationships. By introducing hyperedges that can connect multiple nodes in a directed manner, the invention adds a dimensional layer that captures sequence and ordering information that cannot be represented in standard vector spaces
2Difficulty of detecting and measuring
If data is simplified by assuming it lies on an embedded non-linear manifold within higher-dimensional space, then data can be visualized in low dimensional space, but all current techniques assume homogenous spaces and only one manifold per space
Solution Approach 1:
The patent segments the data space into multiple distinct manifolds, each representing a different semantic class or category. Instead of forcing all data into a single homogeneous manifold, the invention divides the higher-dimensional space into multiple manifolds (M1, M2, ..., Mn) where each can have its own geometry and properties, allowing heterogeneous data types to be represented appropriately in their own manifolds
Solution Approach 2:
The patent creates a universal framework that can handle multiple types of manifolds and data representations within a single system. The hypergraph structure serves as a universal container that can represent points, vectors, and manifolds of various types, allowing the system to adapt to different data types and manifold geometries while maintaining a unified processing approach
Data Source
AI summary
Described herein is a method and system of geometrically encoding data including partitioning data into a plurality of semantic classes based on a dissimilarity metric, generating a subspace formed by first and second data elements, the first and second data elements being included in first and second numbers of partitioned semantic classes, encoding the first data element with respect to the second data element such that the generated subspace formed by the first data element and the second data element is orthogonal, computing a weight distribution of the first data element with respect to the second data element, the weight distribution being performed for each of the first number of semantic classes and the second number of semantic classes, and determining a dominant semantic class corresponding to an ordered sequence of the first data element and the second data element, the dominant semantic class having a maximum weight distribution.