# On Lebesgue Integral Quadrature

arXiv: https://arxiv.org/abs/1807.06007 · HTML: https://mlsanity.com/paper/1807.06007/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jul 2018.

## Built on by (most cited first, 2 of 2)

- [Measure Theoretic Weighted Model Integration](https://mlsanity.com/paper/2103.13901/) (2021)
- [On Machine Learning Knowledge Representation In The Form Of Partially Unitary Operator. Knowledge Generalizing Operator](https://mlsanity.com/paper/2212.14810/) (2022)

## Abstract (arXiv)

A new type of quadrature is developed. The Gaussian quadrature, for a given measure, finds optimal values of a function's argument (nodes) and the corresponding weights. In contrast, the Lebesgue quadrature developed in this paper, finds optimal values of function (value-nodes) and the corresponding weights. The Gaussian quadrature groups sums by function argument; it can be viewed as a n-point discrete measure, producing the Riemann integral. The Lebesgue quadrature groups sums by function value; it can be viewed as a n-point discrete distribution, producing the Lebesgue integral. Mathematically, the problem is reduced to a generalized eigenvalue problem: Lebesgue quadrature value-nodes are the eigenvalues and the corresponding weights are the square of the averaged eigenvectors. A numerical estimation of an integral as the Lebesgue integral is especially advantageous when analyzing irregular and stochastic processes. The approach separates the outcome (value-nodes) and the probability of the outcome (weight). For this reason, it is especially well-suited for the study of non-Gaussian processes. The software implementing the theory is available from the authors.