December 5, 2022 Computer Vision

Regression-based Monte Carlo Integration

  • 50 minutes
  • Corentin Salaün, Adrien Gruson, Binh-Son Hua, Toshiya Hachisuka, Gurprit Singh

  • SIGGRAPH 2022
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Abstract

Monte Carlo integration is typically interpreted as an estimator of the expected value using stochastic samples. There exists an alternative interpretation in calculus where Monte Carlo integration can be seen as estimating a constant function—from the stochastic evaluations of the integrand—that integrates to the original integral. The integral mean value theorem states that this constant function should be the mean (or expectation) of the integrand. Since both interpretations result in the same estimator, little attention has been devoted to the calculus-oriented interpretation. We show that the calculus-oriented interpretation actually implies the possibility of using a more complex function than a constant one to construct a more efficient estimator for Monte Carlo integration. We build a new estimator based on this interpretation and relate our estimator to control variates with least-squares regression on the stochastic samples of the integrand. Unlike prior work, our resulting estimator is provably better than or equal to the conventional Monte Carlo estimator. To demonstrate the strength of our approach, we introduce a practical estimator that can act as a simple drop-in replacement for conventional Monte Carlo integration. We experimentally validate our framework on various light transport integrals. The code is available at https://github.com/iribis/regressionmc.

Bibtex

@article{10.1145/3528223.3530095,
author = {Sala\”{u}n, Corentin and Gruson, Adrien and Hua, Binh-Son and Hachisuka, Toshiya and Singh, Gurprit},
title = {Regression-Based Monte Carlo Integration},
year = {2022},
issue_date = {July 2022},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {41},
number = {4},
issn = {0730-0301},
url = {https://doi.org/10.1145/3528223.3530095},
doi = {10.1145/3528223.3530095},
abstract = {Monte Carlo integration is typically interpreted as an estimator of the expected value using stochastic samples. There exists an alternative interpretation in calculus where Monte Carlo integration can be seen as estimating a constant function—from the stochastic evaluations of the integrand—that integrates to the original integral. The integral mean value theorem states that this constant function should be the mean (or expectation) of the integrand. Since both interpretations result in the same estimator, little attention has been devoted to the calculus-oriented interpretation. We show that the calculus-oriented interpretation actually implies the possibility of using a more complex function than a constant one to construct a more efficient estimator for Monte Carlo integration. We build a new estimator based on this interpretation and relate our estimator to control variates with least-squares regression on the stochastic samples of the integrand. Unlike prior work, our resulting estimator is provably better than or equal to the conventional Monte Carlo estimator. To demonstrate the strength of our approach, we introduce a practical estimator that can act as a simple drop-in replacement for conventional Monte Carlo integration. We experimentally validate our framework on various light transport integrals. The code is available at https://github.com/iribis/regressionmc.},
journal = {ACM Trans. Graph.},
month = {jul},
articleno = {79},
numpages = {14},
keywords = {light transport simulation, Monte Carlo integration, regression, control variates}
}

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  • 50 minutes
  • Corentin Salaün, Adrien Gruson, Binh-Son Hua, Toshiya Hachisuka, Gurprit Singh

  • SIGGRAPH 2022
Share

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