Florian Gunsilius, David Van Dijcke
arXiv 26 Sep 2023 · Econometrics · 1 citations (OpenAlex)
arXiv:2309.14630 · PDF · DOI · OpenAlex · Extracted main text
Sharp, multidimensional changepoints-abrupt shifts in a regression surface whose locations and magnitudes are unknown-arise in settings as varied as gene-expression profiling, financial covariance breaks, climate-regime detection, and urban socioeconomic mapping. Despite their prevalence, there are no current approaches that jointly estimate the location and size of the discontinuity set in a one-shot approach with statistical guarantees. We therefore introduce Free Discontinuity Regression (FDR), a fully nonparametric estimator that simultaneously (i) smooths a regression surface, (ii) segments it into contiguous regions, and (iii) provably recovers the precise locations and sizes of its jumps. By extending a convex relaxation of the Mumford-Shah functional to random spatial sampling and correlated noise, FDR overcomes the fixed-grid and i.i.d. noise assumptions of classical image-segmentation approaches, thus enabling its application to real-world data of any dimension. This yields the first identification and uniform consistency results for multivariate jump surfaces: under mild SBV regularity, the estimated function, its discontinuity set, and all jump sizes converge to their true population counterparts. Hyperparameters are selected automatically from the data using Stein's Unbiased Risk Estimate, and large-scale simulations up to three dimensions validate the theoretical results and demonstrate good finite-sample performance. Applying FDR to an internet shutdown in India reveals a 25-35% reduction in economic activity around the estimated shutdown boundaries-much larger than previous estimates. By unifying smoothing, segmentation, and effect-size recovery in a general statistical setting, FDR turns free-discontinuity ideas into a practical tool with formal guarantees for modern multivariate data.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Caroccia, M., Chambolle, A., and Slepcev, D (2020) Mumford–Shah functionals on graphs and their asymptotics | 0.950 | 7 | 4 | 86% |
| 2 | Pock, T., Cremers, D., Bischof, H., and Chambolle, A (2009) An algorithm for minimizing the Mumford–Shah functional | 0.950 | 7 | 3 | 86% |
| 3 | Politis, D. N., Romano, J. P., and Wolf, M (1999) Subsampling | 0.941 | 6 | 4 | 83% |
| 4 | Chambolle, A. and Pock, T (2021) Learning consistent discretizations of the total variation | 0.874 | 6 | 3 | 67% |
| 5 | Strekalovskiy, E., Chambolle, A., and Cremers, D (2012) A convex representation for the vectorial Mumford–Shah functional | 0.737 | 4 | 3 | 50% |
| 6 | Lucas, C.-G., Pascal, B., Pustelnik, N., and Abry, P (2022) Hyperparameter selection for discrete Mumford–Shah | 0.737 | 3 | 2 | 100% |
| 7 | Richardson, T. J (1992) Limit theorems for a variational problem arising in computer vision | 0.737 | 3 | 2 | 100% |
| 8 | Irle, A (1997) On consistency in nonparametric estimation under mixing conditions | 0.659 | 7 | 2 | 43% |
| 9 | Alberti, G., Bouchitté, G., and Dal Maso, G (2003) The calibration method for the Mumford–Shah functional and free-discontinuity problems | 0.644 | 2 | 2 | 100% |
| 10 | Chan, T. F. and Shen, J (2005) Image processing and analysis: variational, PDE, wavelet, and stochastic methods | 0.644 | 2 | 2 | 100% |
Showing the top 10 of 132 scored citations.