# Conditional Selective Inference for Robust Regression and Outlier Detection using Piecewise-Linear Homotopy Continuation

arXiv: https://arxiv.org/abs/2104.10840 · HTML: https://mlsanity.com/paper/2104.10840/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 82 references that are in the map.
- First posted on arXiv 22 Apr 2021.

## Builds on (most cited first, 20 of 20)

- [Exact post-selection inference, with application to the lasso](https://mlsanity.com/paper/1311.6238/) (2013)
- [Exact Post-Selection Inference for Sequential Regression Procedures](https://mlsanity.com/paper/1401.3889/) (2014)
- [Valid Inference Corrected for Outlier Removal](https://mlsanity.com/paper/1711.10635/) (2017)
- [Computing Valid p-value for Optimal Changepoint by Selective Inference using Dynamic Programming](https://mlsanity.com/paper/2002.09132/) (2020)
- [Computing Valid p-values for Image Segmentation by Selective Inference](https://mlsanity.com/paper/1906.00629/) (2019)
- [Selective Inference for Group-Sparse Linear Models](https://mlsanity.com/paper/1607.08211/) (2016)
- [A significance test for the lasso](https://mlsanity.com/paper/1301.7161/) (2013)
- [Post-Selection Inference for Changepoint Detection Algorithms with Application to Copy Number Variation Data](https://mlsanity.com/paper/1812.03644/) (2018)
- [Computing Full Conformal Prediction Set with Approximate Homotopy](https://mlsanity.com/paper/1909.09365/) (2019)
- [Selecting the number of principal components: estimation of the true rank of a noisy matrix](https://mlsanity.com/paper/1410.8260/) (2014)
- [A significance test for forward stepwise model selection](https://mlsanity.com/paper/1405.3920/) (2014)
- [Selective Sequential Model Selection](https://mlsanity.com/paper/1512.02565/) (2015)
- [Post Selection Inference with Kernels](https://mlsanity.com/paper/1610.03725/) (2016)
- [Post Selection Inference with Incomplete Maximum Mean Discrepancy Estimator](https://mlsanity.com/paper/1802.06226/) (2018)
- [Selective inference in regression models with groups of variables](https://mlsanity.com/paper/1511.01478/) (2015)
- [Safe Grid Search with Optimal Complexity](https://mlsanity.com/paper/1810.05471/) (2018)
- [Integrative Methods for Post-Selection Inference Under Convex Constraints](https://mlsanity.com/paper/1605.08824/) (2016)
- [More Powerful and General Selective Inference for Stepwise Feature Selection using the Homotopy Continuation Approach](https://mlsanity.com/paper/2012.13545/) (2020)
- [Selective inference for the problem of regions via multiscale bootstrap](https://mlsanity.com/paper/1711.00949/) (2017)
- [Regularization Path of Cross-Validation Error Lower Bounds](https://mlsanity.com/paper/1502.02344/) (2015)

## Built on by (most cited first, 16 of 16)

- [More Powerful Conditional Selective Inference for Generalized Lasso by Parametric Programming](https://mlsanity.com/paper/2105.04920/) (2021)
- [Valid P-Value for Deep Learning-Driven Salient Region](https://mlsanity.com/paper/2301.02437/) (2023)
- [Valid and Exact Statistical Inference for Multi-dimensional Multiple Change-Points by Selective Inference](https://mlsanity.com/paper/2110.08989/) (2021)
- [CAD-DA: Controllable Anomaly Detection after Domain Adaptation by Statistical Inference](https://mlsanity.com/paper/2310.14608/) (2023)
- [Exact Statistical Inference for the Wasserstein Distance by Selective Inference](https://mlsanity.com/paper/2109.14206/) (2021)
- [Statistical Test for Feature Selection Pipelines by Selective Inference](https://mlsanity.com/paper/2406.18902/) (2024)
- [Statistical Test for Auto Feature Engineering by Selective Inference](https://mlsanity.com/paper/2410.19768/) (2024)
- [Quantifying Statistical Significance of Deep Nearest Neighbor Anomaly Detection via Selective Inference](https://mlsanity.com/paper/2502.12978/) (2025)
- [Controllable RANSAC-based Anomaly Detection via Hypothesis Testing](https://mlsanity.com/paper/2410.15133/) (2024)
- [Change Point Detection in the Frequency Domain with Statistical Reliability](https://mlsanity.com/paper/2502.03062/) (2025)
- [Bounded P-values in Parametric Programming-based Selective Inference](https://mlsanity.com/paper/2307.11351/) (2023)
- [Statistical Test for Diffusion-Based Anomaly Localization via Selective Inference](https://mlsanity.com/paper/2402.11789/) (2024)
- [Statistical Inference for the Dynamic Time Warping Distance, with Application to Abnormal Time-Series Detection](https://mlsanity.com/paper/2202.06593/) (2022)
- [Statistical Inference for Clustering-based Anomaly Detection](https://mlsanity.com/paper/2504.18633/) (2025)
- [Automatic Statistical Test for Rationally Expressible Algorithms by Selective Inference, with Applications to Feature Selection](https://mlsanity.com/paper/2608.04667/) (2026)
- [Post-Anomaly Detection Inference for Deep SVDD](https://mlsanity.com/paper/2609.37935/) (2026)

## Abstract (arXiv)

In practical data analysis under noisy environment, it is common to first use robust methods to identify outliers, and then to conduct further analysis after removing the outliers. In this paper, we consider statistical inference of the model estimated after outliers are removed, which can be interpreted as a selective inference (SI) problem. To use conditional SI framework, it is necessary to characterize the events of how the robust method identifies outliers. Unfortunately, the existing methods cannot be directly used here because they are applicable to the case where the selection events can be represented by linear/quadratic constraints. In this paper, we propose a conditional SI method for popular robust regressions by using homotopy method. We show that the proposed conditional SI method is applicable to a wide class of robust regression and outlier detection methods and has good empirical performance on both synthetic data and real data experiments.