# SPARK: A General Goodness-of-Fit Assessment via Residual Projection

arXiv: https://arxiv.org/abs/2609.37705 · HTML: https://mlsanity.com/paper/2609.37705/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 50 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 10 of 10)

- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Cauchy combination test: a powerful test with analytic p-value calculation under arbitrary dependency structures](https://mlsanity.com/paper/1808.09011/) (2018)
- [Is a Classification Procedure Good Enough? A Goodness-of-Fit Assessment Tool for Classification Learning](https://mlsanity.com/paper/1911.03063/) (2019)
- [Goodness-of-fit testing in high-dimensional generalized linear models](https://mlsanity.com/paper/1908.03606/) (2019)
- [Sup-Norm Convergence of Deep Neural Network Estimator for Nonparametric Regression by Adversarial Training](https://mlsanity.com/paper/2307.04042/) (2023)
- [Goodness of fit tests for high-dimensional linear models](https://mlsanity.com/paper/1511.03334/) (2015)
- [Test and Measure for Partial Mean Dependence Based on Machine Learning Methods](https://mlsanity.com/paper/2212.12874/) (2022)
- [GRASP: A Goodness-of-Fit Test for Classification Learning](https://mlsanity.com/paper/2209.02064/) (2022)
- [Effective Positive Cauchy Combination Test](https://mlsanity.com/paper/2410.10345/) (2024)

## Abstract (arXiv)

Goodness-of-fit testing is a basic tool for assessing whether a fitted procedure has captured the systematic information contained in the covariates. While traditional theory has largely focused on parametric regression models, modern data analysis increasingly relies on flexible black-box learners, whose predictive success alone is insufficient to assess model accuracy. In this paper, we propose SPARK, a general framework for goodness-of-fit testing that applies to traditional statistical models and general black-box learning procedures, continuous and binary responses, and low- and high-dimensional predictors. Based on a debiasing strategy, the residuals from an initial fit of a learning procedure are projected onto nearly orthogonal directions to extract any remaining signal. To capture information across all projection directions, we propose a kernel-based projection method and establish both its asymptotic properties and the consistency of a bootstrap procedure. Comprehensive simulations and real data analyses illustrate the effectiveness and flexibility of our proposed method.