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Discontinuities can be fairly arbitrary but also cause a significant impact on outcomes in social systems. Indeed, their arbitrariness is why they have been used to infer causal relationships among variables in numerous settings. Regression discontinuity from econometrics assumes the existence of a discontinuous variable that splits the population into distinct partitions to estimate causal effects. Here we consider the design of partitions for a given discontinuous variable to optimize a certain effect. To do so, we propose a quantization-theoretic approach to optimize the effect of interest, first learning the causal effect size of a given discontinuous variable and then applying dynamic programming for optimal quantization design of discontinuities that balance the gain and loss in the effect size. We also develop a computationally-efficient reinforcement learning algorithm for the dynamic programming formulation of optimal quantization. We demonstrate our approach by designing optimal time zone borders for counterfactuals of social capital.
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This content will become publicly available on June 8, 2026
Dyadic Partition-Based Training Schemes for TV/TGV Denoising
Due to their ability to handle discontinuous images while having a well-understood behavior, regularizations with total variation (TV) and total generalized variation (TGV) are some of the best-known methods in image denoising. However, like other variational models including a fidelity term, they crucially depend on the choice of their tuning parameters. A remedy is to choose these automatically through multilevel approaches, for example by optimizing performance on noisy/clean image pairs. In this work, we consider such methods with space-dependent parameters which are piecewise constant on dyadic grids, with the grid itself being part of the minimization. We prove existence of minimizers for fixed discontinuous parameters under mild assumptions on the data, which lead to existence of finite optimal partitions. We further establish that these assumptions are equivalent to the commonly used box constraints on the parameters. On the numerical side, we consider a simple subdivision scheme for optimal partitions built on top of any other bilevel optimization method for scalar parameters, and demonstrate its improved performance on some representative test images when compared with constant optimized parameters.
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- Award ID(s):
- 2205627
- PAR ID:
- 10598764
- Editor(s):
- Morel, Jean Michel
- Publisher / Repository:
- Springer Nature
- Date Published:
- Journal Name:
- Journal of mathematical imaging and vision
- Volume:
- 66
- ISSN:
- 1573-7683
- Page Range / eLocation ID:
- 1070-1108
- Subject(s) / Keyword(s):
- Total variation Total generalized variation Discontinuous weights Spatially_dependent regularization parameters Box constraint· Bilevel optimization
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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