Delaunay Triangulation Interpolation for Sparse Data Classification

Solution Overview

Problem

Traditional isotonic regression methods in binary classification systems often fail to provide optimal results, especially when training data is sparse, as they either violate monotonicity or continuity during interpolation.

Innovation Solution

A computer-implemented method that uses Delaunay triangulation to determine an interpolation function satisfying monotonicity constraints, which maximizes the likelihood of observations and is continuous and piecewise linear on Delaunay simplices, ensuring accurate probability estimation.

Engineering Contradictions & Design Principles

VSEngineering Contradiction Analysis

1Reliability

If traditional isotonic regression methods are used for binary classification, then the classification process can be performed, but the results are suboptimal especially when training data is sparse due to violations of monotonicity or continuity during interpolation

Engineering Contradiction:

Improveclassification accuracyVSAvoidprobability estimation accuracy

Core Design Contradiction:

ReliabilityVSMeasurement precision

Solution Approach 1:

The feature space is segmented into Delaunay simplices (triangles in 2D, tetrahedra in 3D, etc.) formed by the training data points. Each simplex becomes an independent region where linear interpolation is performed, allowing the method to handle sparse data by dividing the space into manageable geometric units that collectively cover the entire feature space.

Inventive Principle:

Principle #1Segmentation

Solution Approach 2:

The patent employs Delaunay triangulation which creates a geometric structure that naturally adapts to the distribution of training points. The triangulation creates curved boundaries in the probability space that follow the data distribution, providing smoother and more accurate probability estimates compared to traditional linear isotonic regression methods.

Inventive Principle:

Principle #14Spheroidality (Curvature)

2Adaptability or versatility

If interpolation is performed in traditional isotonic regression, then probabilities can be estimated for unseen data points, but monotonicity constraints are violated during the interpolation process

Engineering Contradiction:

Improveinterpolation capabilityVSAvoidmonotonicity constraint satisfaction

Core Design Contradiction:

Adaptability or versatilityVSStability of the object's composition

Solution Approach 1:

The patent employs Delaunay triangulation which creates a geometric structure that naturally adapts to the distribution of training points. The triangulation creates curved boundaries in the probability space that follow the data distribution, providing smoother and more accurate probability estimates compared to traditional linear isotonic regression methods.

Inventive Principle:

Principle #14Spheroidality (Curvature)

Solution Approach 2:

The patent transforms the probability estimation problem by using barycentric coordinates within each Delaunay simplex as intermediate parameters. This parameter transformation allows the method to maintain monotonicity by ensuring that probability increases monotonically with respect to the input features while still providing smooth interpolation across the feature space.

Inventive Principle:

Principle #35Parameter changes

3Adaptability or versatility

If interpolation is performed in traditional isotonic regression, then probabilities can be estimated for unseen data points, but continuity is violated during the interpolation process

Engineering Contradiction:

Improveinterpolation capabilityVSAvoidcontinuity maintenance

Core Design Contradiction:

Adaptability or versatilityVSStability of the object's composition

Solution Approach 1:

The feature space is segmented into Delaunay simplices (triangles in 2D, tetrahedra in 3D, etc.) formed by the training data points. Each simplex becomes an independent region where linear interpolation is performed, allowing the method to handle sparse data by dividing the space into manageable geometric units that collectively cover the entire feature space.

Inventive Principle:

Principle #1Segmentation

Solution Approach 2:

The patent ensures continuous probability estimation across the entire feature space by using Delaunay triangulation to create adjacent simplices that share common boundaries. The linear interpolation within each simplex and the continuous transition between adjacent simplices maintain continuity of the probability surface, eliminating discontinuities that plague traditional methods.

Inventive Principle:

Principle #20Continuity of useful action

Data Source

PatentUS9189752B2Interpolating isotonic regression for binary classification of spam, explicit material or malware using interpolation based on a Delaunay triangulation

Publication Date: 2015.11.17 GOOGLE LLC

AI summary

Described is a machine learning system for binary classifications. The system improves interpolation schemes used in isotonic regressions by providing a continuous function that also is monotonic. The system outputs a probability estimating function on a signal space that is both monotonic and varies continuously with the input signals. More specifically, described is an interpolation function that is continuous and piecewise linear on Delaunay simplices. Accordingly, the resulting probability estimation function may more accurately match actual probabilities especially when training data is sparse.