Dimension Reduction Using Sigmoid Similarity and Alpha Parameter
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Solution Overview
Problem
Current dimension reduction methods fail to preserve the local structure of high-dimensional data in low-dimensional space, leading to issues like the 'crowding problem where small distances in high-dimensional space become excessively large in low-dimensional visualization, causing points to overlap.
Innovation Solution
A computer-readable medium with instructions that transform high-dimensional data into low-dimensional data by selecting observation vectors, computing distances, applying a sigmoid function to determine similarity, and using an optimization method to create an optimized matrix for visualization, allowing for a finer cluster structure and improved data visualization.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If manifold learning methods are used to preserve local structure, then local distance preservation is improved, but the crowding problem worsens
Solution Approach 1:
The patent introduces a new parameter 'alpha' that controls the balance between preserving local structure and avoiding crowding. By adjusting this parameter, the system can transform high-dimensional data to low-dimensional space while maintaining both local distance preservation and reasonable point distribution, resolving the contradiction between these two requirements
Solution Approach 2:
The patent employs an iterative optimization process where the transformation matrix is dynamically adjusted through gradient descent. This dynamic approach allows the system to adaptively find optimal low-dimensional representations that balance local structure preservation with crowding avoidance, rather than using static transformation methods
2Stability of the object's composition
If linear methods are used to preserve global structure, then global structure preservation is improved, but local structure preservation deteriorates
Solution Approach 1:
The patent merges the advantages of linear methods (global structure preservation) and nonlinear methods (local structure preservation) by combining them in a unified optimization framework. The loss function simultaneously includes terms for both global and local structure preservation, allowing the system to achieve both objectives that were previously mutually exclusive
Solution Approach 2:
The patent creates a universal dimensionality reduction framework that can handle both global and local structure preservation requirements through a single optimization process. This multi-functional approach eliminates the need to choose between linear and nonlinear methods, allowing the system to adapt to different data characteristics and preservation requirements
Data Source
AI summary
A computer transforms high-dimensional data into low-dimensional data. A distance is computed between a selected observation vector and each observation vector of a plurality of observation vectors, a nearest neighbors are selected using the computed distances, and a first sigmoid function is applied to compute a distance similarity value between the selected observation vector and each of the selected nearest neighbors where each of the computed distance similarity values is added to a first matrix. The process is repeated with each observation vector of the plurality of observation vectors as the selected observation vector. An optimization method is executed with an initial matrix, the first matrix, and a gradient of a second sigmoid function that computes a second distance similarity value between the selected observation vector and each of the nearest neighbors to transform each observation vector of the plurality of observation vectors into the low-dimensional space.