Distribution Aggregation Calculator for Floating-Point Underflow
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Solution Overview
Problem
The multiplication of floating-point numbers in the range of 0 to 1, such as 10^-10, 10^-20, or less, often results in arithmetic underflow, leading to erroneous results due to the inability of machines to represent these values, which are approximated to zero, especially when aggregating a large number of distributions.
Innovation Solution
A calculator that iteratively groups distributions into subsets, performs term-by-term product operations, and normalizes the results to ensure values remain above the minimum representable value, avoiding arithmetic underflow by using a reduction tree structure.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If term-by-term product operations are performed on a large number of probability distributions, then aggregation accuracy is improved, but arithmetic underflow occurs causing values to be rounded to zero
Solution Approach 1:
The patent divides the aggregation process into multiple stages by grouping distributions into subsets. Instead of computing the product of all N distributions directly, the method segments them into groups of K distributions each, computes products within each group, and then combines results. This segmentation prevents the intermediate product from becoming too small and underflowing, while still achieving accurate aggregation through the multi-stage process.
2Device complexity
If all distributions are aggregated in a single product operation, then computational complexity is reduced, but machine representation limits cause information loss
Solution Approach 1:
The patent segments the aggregation into multiple computational stages with intermediate normalization steps. Each stage processes a manageable subset of distributions, keeping intermediate values within representable ranges. This multi-stage segmented approach prevents information loss from underflow while maintaining reasonable computational complexity through efficient subset processing.
Solution Approach 2:
The patent applies preliminary normalization to each subset product before combining with other subsets. By normalizing intermediate results early in the process, the method prevents subsequent underflow issues and maintains numerical stability throughout the aggregation, avoiding information loss that would occur with a single large product operation.
3Measurement precision
If floating-point numbers below minimum representable value are computed, then complete probability space is captured, but values are rounded to zero causing erroneous results
Solution Approach 1:
The patent segments the probability space aggregation into multiple stages, each handling a subset of distributions. By computing products in smaller groups and normalizing intermediately, the method keeps all probability values within the representable range, preventing both underflow to zero and loss of precision, thus capturing the complete probability space accurately.
Solution Approach 2:
The patent changes the computational parameters by introducing intermediate normalization steps that rescale probability values. This parameter change ensures that intermediate products remain within the representable range of floating-point numbers, preventing both underflow and precision loss while maintaining the complete probability distribution.
Data Source
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AI summary
A calculator aggregating initial distributions into an aggregated distribution, said initial distributions being associated with vectors of real values representable according to a predefined representation format in which the values representable according to this format are between a minimum value and a maximum representable value, wherein, for the aggregation of the initial distributions, the calculator is configured to implement, iteratively: - a grouping of distributions to form a plurality of subsets of distributions, and - product operations, term by term, on the subsets of distributions, said products resulting in at least one intermediate distribution, followed by a normalization operation of each intermediate distribution, until the aggregated distribution is obtained,in which each grouping of distributions is implemented so that product operations on the subsets of distributions lead to resulting values strictly greater than the minimum representable value.,