Piece-wise Polynomial Data Representation via Recursive Segmentation
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Solution Overview
Problem
Representing large volumes of data with multi-dimensional input vectors and corresponding output elements using polynomials is challenging due to the need for extensive memory and computations, especially when curve fitting techniques fail to find a suitable curve for all data points, leading to inefficient resource utilization.
Innovation Solution
The method involves determining piece-wise polynomials by splitting the data set along dimensions, using recursion to simplify programming logic, and minimizing the number of combinations of orders tried by ruling out combinations that do not lead to acceptable accuracy, ensuring that the polynomials accurately model the data set with reduced computational requirements.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Quantity of substance
If a polynomial is used to model large volumes of multi-dimensional data, then memory usage is reduced, but computational complexity increases
Solution Approach 1:
The patent divides the multi-dimensional data space into multiple hyper-rectangular regions using axis-aligned splits. Each region is independently modeled by a simpler polynomial, avoiding the need for a single complex high-degree polynomial to capture all data variations. This segmentation reduces the overall computational complexity while maintaining modeling accuracy.
Solution Approach 2:
The patent introduces the dimension of spatial partitioning by dividing the data space into multiple regions. Instead of using a single polynomial in the original N-dimensional space, the approach creates multiple lower-complexity polynomials distributed across different spatial regions, effectively adding a partitioning dimension to the problem structure.
2Reliability
If traditional curve fitting techniques are used, then a single polynomial model is obtained, but accuracy is insufficient when no suitable curve exists for all data points
Solution Approach 1:
The patent segments the data space into multiple hyper-rectangular regions where each region can be independently fitted with a simpler polynomial. This allows accurate local modeling in each region without requiring a single complex polynomial to fit all data points globally, thereby improving overall modeling accuracy while keeping individual polynomial models simple.
Solution Approach 2:
The patent applies different polynomial models to different regions of the data space based on local characteristics. Each hyper-rectangular region receives a polynomial tailored to its specific data patterns, allowing the model to adapt to local variations in the data rather than forcing a single global polynomial structure.
3Measurement precision
If the data set is partitioned into multiple subsets for piece-wise polynomial representation, then modeling accuracy improves, but the number of polynomials and computational requirements increase
Solution Approach 1:
The patent partitions the data space into hyper-rectangular regions and fits simpler polynomials to each region rather than using a single high-degree polynomial. This segmentation strategy achieves high modeling precision through multiple low-degree polynomials, reducing the computational burden of evaluating and determining polynomial coefficients compared to a single complex polynomial.
Solution Approach 2:
The patent applies piece-wise polynomial approximation where each region receives a polynomial model of appropriate complexity for that specific region. Rather than uniformly applying high-degree polynomials everywhere, the method uses lower-degree polynomials in regions where they suffice, only increasing complexity where necessary to achieve the desired precision.
Data Source
AI summary
Determining piece-wise polynomials which together would represent large data sets having multi-dimensional input vectors and corresponding output element. In an embodiment, a function/procedure/routine is recursively called/invoked to determine piece-wise polynomial is a data set cannot be entirely modeled by one polynomial. Another aspect of the present invention reduces the number of combinations (of orders for sub-polynomials forming the polynomials) to be tried in determining polynomials, meeting various accuracy requirements. Such a reduction is obtained based on a recognition that when the order in one dimension alone is increased and the result does not lead to acceptable accuracy of the polynomial, the combinations with a lesser number for the order (of the dimension) can be ruled out.