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arxiv: 2512.19373 · v3 · pith:JWSPYY3Enew · submitted 2025-12-22 · 📊 stat.ML · cs.LG

Cluster-Based Generalized Additive Models Informed by Random Fourier Features

Xin Huang , Jia Li , Jun Yu This is my paper

Pith reviewed 2026-05-21 17:02 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords generalized additive modelsrandom Fourier featuresheterogeneous regressioninterpretable modelsspectral featuresclusteringprincipal component analysis
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The pith

Cluster-based generalized additive models using random Fourier features improve regression on heterogeneous data

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces a regression approach for data with varying relationships by learning a spectral representation from random Fourier features that captures predictive structure. The representation is reduced with principal component analysis and then softly clustered with a Gaussian mixture model to find regimes. Within each regime, a separate generalized additive model is fit using smooth functions for each covariate. The final output mixes these local models, allowing the method to adapt to heterogeneity. Readers interested in interpretable machine learning would care because the approach aims to match black-box performance while keeping the transparency of additive models that reveal individual variable contributions.

Core claim

By constructing a response-informed spectral feature map from a fitted random Fourier feature regression, compressing it via principal component analysis, and applying a Gaussian mixture model for soft regime discovery, the method enables the fitting of cluster-specific generalized additive models with spline smooths. The predictor is then a weighted combination of these local models, which the authors show improves upon global interpretable methods and competes with black-box models on benchmark datasets.

What carries the argument

Response-informed spectral feature map from random Fourier features compressed by PCA and partitioned by Gaussian mixture model to enable localized generalized additive models

Load-bearing premise

The low-dimensional embedding from the spectral features contains separable structure that a Gaussian mixture model can use to define regimes where local additive models provide meaningful improvements.

What would settle it

Demonstrating that on the benchmark regression datasets, a single global generalized additive model achieves similar or better performance than the proposed clustered version.

Figures

Figures reproduced from arXiv: 2512.19373 by Jia Li, Jun Yu, Xin Huang.

Figure 1
Figure 1. Figure 1: The overall pipeline of the RFF-informed mixture-of-GAMs. Stage 1 learns RFF model coefficients and [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Diagram with graphical representations of the workflow of the mixture-of-GAMs method informed with [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Root mean square error of the trained random Fourier feature model [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Test root mean square error on California housing dataset evaluated over a grid of hyperparameter config [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Partial dependence plots for selected features of the California Housing dataset. [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Spatial distributions of training data for each GMM cluster in the California housing dataset, using [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Spatial distributions of training data for each GMM cluster in the California housing dataset, based on [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Illustration of the inverse relation between the spatial and spectral scales of Gaussian kernels. Left: [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Plot of California housing price data (left) and 2D empirical histogram of frequency samples [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Histograms of the five covariates in the Airfoil Self-Noise dataset. [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Partial dependence plots on the five covariates of the Airfoil Self-noise dataset. [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Test root mean square error on the airfoil self-noise dataset evaluated over a grid of hyperparameter [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Average posterior responsibilities γ¯ℓ(h) of the eight mixture components as functions of the hour of day, computed over the training dataset. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
read the original abstract

In developing data-driven modeling methodologies, there is an ongoing need to reconcile the strong predictive performance of opaque black-box models with the transparency required for critical applications. This work introduces an interpretable and computationally tractable regression framework for heterogeneous data by combining response-informed spectral representation learning with localized additive modeling. The method first fits a random Fourier feature regression model and constructs a spectral feature map from the learned amplitudes and adaptively resampled frequencies, so that the representation reflects predictive variation in the data. This representation is then compressed by principal component analysis to obtain a low-dimensional latent embedding, in which a Gaussian mixture model performs soft regime discovery. Within each regime, a cluster-specific generalized additive model captures nonlinear covariate effects through interpretable spline-based univariate smooth functions. The final predictor is formed as a soft mixture of these local additive models, enabling flexible modeling of a nonlinear, heterogeneous structure while preserving interpretability. Numerical experiments across several benchmark regression datasets show that the proposed method consistently improves upon classical globally interpretable baselines while remaining competitive with more flexible black-box models. Overall, the framework provides a unified approach to heterogeneous regression that combines predictive adaptivity with interpretable local covariate effects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a regression framework that integrates response-informed random Fourier features with principal component analysis and Gaussian mixture modeling to discover regimes, followed by fitting cluster-specific generalized additive models whose predictions are combined via soft weighting. The central claim is that this approach yields improved predictive performance over global interpretable baselines on benchmark regression datasets while remaining competitive with black-box models and preserving interpretability through local spline-based effects.

Significance. If the empirical results hold after addressing the noted gaps, the work provides a constructive pipeline for interpretable modeling of heterogeneous regression data, bridging global GAMs and flexible mixtures. The use of response-informed spectral features to guide the latent embedding is a positive design choice that merits further validation.

major comments (2)
  1. [Numerical Experiments] Numerical Experiments section: the reported benchmark improvements lack an ablation that isolates the contribution of the PCA-GMM regime discovery from the baseline capacity of a soft mixture of GAMs. Without this comparison, it remains unclear whether gains derive from meaningful structure in the response-informed embedding or simply from the added flexibility of multiple local models, directly impacting the central claim.
  2. [Method] Method section (around the GMM and PCA steps): the free parameters (number of mixture components, retained principal components, spline degrees and knots) are acknowledged but the manuscript provides no systematic selection rule or sensitivity analysis tied to the performance tables. This weakens the assertion of consistent improvements across datasets.
minor comments (2)
  1. [Abstract] Abstract and introduction: the phrase 'adaptively resampled frequencies' is introduced without a precise algorithmic description; a short pseudocode or equation reference would improve clarity.
  2. [Notation] Notation: ensure consistent use of symbols for the spectral feature map and the soft weights across equations and text to avoid minor reader confusion.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 0 invented entities

The framework rests on standard machine-learning components but introduces a custom pipeline whose success depends on several modeling choices and domain assumptions about data heterogeneity.

free parameters (3)
  • number of Gaussian mixture components
    Determines the number of soft regimes discovered in the latent space.
  • number of principal components retained
    Controls the dimensionality of the embedding used for clustering.
  • spline basis degrees and knot placements
    Control the flexibility of the univariate smooth functions inside each cluster-specific GAM.
axioms (1)
  • domain assumption The data-generating process exhibits heterogeneous structure that is recoverable as soft clusters in the response-informed spectral feature space.
    This premise justifies the GMM regime-discovery step and the subsequent local modeling.

pith-pipeline@v0.9.0 · 5728 in / 1369 out tokens · 67839 ms · 2026-05-21T17:02:16.902880+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The method first fits a random Fourier feature regression model and constructs a spectral feature map from the learned amplitudes and adaptively resampled frequencies... This representation is then compressed by principal component analysis... a Gaussian mixture model performs soft regime discovery. Within each regime, a cluster-specific generalized additive model captures nonlinear covariate effects through interpretable spline-based univariate smooth functions.

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Numerical experiments across several benchmark regression datasets show that the proposed method consistently improves upon classical globally interpretable baselines

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · 1 internal anchor

  1. [1]

    Avron, H., Kapralov, M., Musco, C., Musco, C., Velingker, A., Zandieh, A.,Random Fourier Features for Kernel Ridge Regression: Approximation Bounds and Statistical Guarantees, Pro- ceedings of the 34th International Conference on Machine Learning, PMLR, 70, 253–262, 2017

    work page 2017

  2. [2]

    Bach, F.,On the Equivalence between Kernel Quadrature Rules and Random Feature Expan- sions, Journal of Machine Learning Research, 18, 1–38, 2017

    work page 2017

  3. [3]

    Bach, F.,Learning Theory from First Principles, Adaptive Computation and Machine Learning series, The MIT Press, 2024

    work page 2024

  4. [4]

    Springer-Verlag, 2006

    Bishop, C.M.,Pattern Recognition and Machine Learning (Information Science and Statistics). Springer-Verlag, 2006

    work page 2006

  5. [5]

    Springer-Verlag New York, 1978

    de Boor, C.,A Practical Guide to Splines. Springer-Verlag New York, 1978

    work page 1978

  6. [6]

    Machine Learning, 45, 5–32, 2001

    Breiman, L.,Random Forests. Machine Learning, 45, 5–32, 2001

    work page 2001

  7. [7]

    NASA Reference Publication 1218, 1989

    Brooks, T.F., Pope, D.S., and Marcolini, M.A.,Airfoil Self-Noise and Prediction. NASA Reference Publication 1218, 1989

    work page 1989

  8. [8]

    Proceedings of the 21st ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2015

    Caruana, R., Lou, Y., Gehrke, J., Koch, P., Sturm, M., and Elhadad, N.,Intelligible models for healthcare: Predicting pneumonia risk and hospital 30-day readmission. Proceedings of the 21st ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2015

    work page 2015

  9. [9]

    and Guestrin, C.,XGBoost: A scalable tree boosting system

    Chen, T. and Guestrin, C.,XGBoost: A scalable tree boosting system. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2016

    work page 2016

  10. [10]

    Proceedings of the 2012 IEEE 12th International Conference on Data Mining, 161–170, 2012

    Chitta, R., Jin, R., and Jain, A.K.Efficient Kernel Clustering Using Random Fourier Features. Proceedings of the 2012 IEEE 12th International Conference on Data Mining, 161–170, 2012

    work page 2012

  11. [11]

    Cambridge University Press, 2020

    Deisenroth, M.P., Faisal, A.A., and Ong, C.S.,Mathematics for Machine Learning. Cambridge University Press, 2020

    work page 2020

  12. [12]

    Proceedings of the International Congress of Mathematicians, 2, 914–954, 2022

    E, W.,A Mathematical Perspective of Machine Learning. Proceedings of the International Congress of Mathematicians, 2, 914–954, 2022

    work page 2022

  13. [13]

    Progress in Artificial Intelligence, 2, 113–127, 2014

    Fanaee-T,H., Gama, J.Event Labeling Combining Ensemble Detectors and Background Knowl- edge. Progress in Artificial Intelligence, 2, 113–127, 2014

    work page 2014

  14. [14]

    Pattern Recognition, 134:109057, 2023

    Fang, K., Liu, F., Huang, X., and Yang, Y.,End-to-End Kernel Learning via Generative Random Fourier Features. Pattern Recognition, 134:109057, 2023. 28

    work page 2023

  15. [15]

    The Annals of Statistics, 19, 1–67, 1991

    Friedman, J.H.,Multivariate Adaptive Regression Splines. The Annals of Statistics, 19, 1–67, 1991

    work page 1991

  16. [16]

    The Annals of Statistics, 29, 1189–1232, 2001

    Friedman, J.H.Greedy Function Approximation: A Gradient Boosting Machine. The Annals of Statistics, 29, 1189–1232, 2001

    work page 2001

  17. [17]

    The MIT Press, 2016

    Goodfellow, I., Bengio, Y., and Courville, A.Deep Learning. The MIT Press, 2016

    work page 2016

  18. [18]

    and Tibshirani, R.,Generalized Additive Models

    Hastie, T. and Tibshirani, R.,Generalized Additive Models. Chapman and Hall, New York, 1990

    work page 1990

  19. [19]

    Springer New York, NY, 2009

    Hastie, T., Tibshirani, R., and Friedman, J.H.,The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer New York, NY, 2009

    work page 2009

  20. [20]

    Hodges, J.,Richly Parametrized Linear Models: Additive, Time Series, and Spatial Models Using Random Effects, Boca Raton: Chapman & Hall/CRC Texts in Statistical Science, 2014

    work page 2014

  21. [21]

    Convergence for adaptive resampling of random Fourier features

    Huang, X., Kammonen, A., Pandey, A., Sandberg, M., von Schwerin, E., Szepessy, A., and Tempone, R.,Convergence for Adaptive Resampling of Random Fourier Features. Preprint, https://doi.org/10.48550/arXiv.2509.03151, 2025

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2509.03151 2025

  22. [22]

    and Johnson, K.,Applied Predictive Modeling

    Kuhn, M. and Johnson, K.,Applied Predictive Modeling. Springer New York, NY, 2013

    work page 2013

  23. [23]

    The MIT Press, 2022

    Murphy, K.P.,Probabilistic Machine Learning: An introduction. The MIT Press, 2022

    work page 2022

  24. [24]

    Proceedings of Machine Learning Research, 95, 129–144, 2018

    Nguyen, K., Dam, N., Le, T., Nguyen, T.D., and Phung, D.,Clustering Induced Kernel Learning. Proceedings of Machine Learning Research, 95, 129–144, 2018

    work page 2018

  25. [25]

    Preprint, https://doi.org/10.48550/arXiv.1909.09223, 2019

    Nori, H., Jenkins, S., Koch, P., and Caruana, R.,InterpretML: A Unified Framework for Machine Learning Interpretability. Preprint, https://doi.org/10.48550/arXiv.1909.09223, 2019

    work page doi:10.48550/arxiv.1909.09223 1909

  26. [26]

    and Barry, R.,Sparse Spatial Autoregressions

    Pace, R.K. and Barry, R.,Sparse Spatial Autoregressions. Statistics & Probability Letters, 33(3), 291–297, 1997

    work page 1997

  27. [27]

    The Annals of Math- ematical Statistics, 33, 1065–1076, 1962

    Parzen, E.,On Estimation of a Probability Density Function and Mode. The Annals of Math- ematical Statistics, 33, 1065–1076, 1962

    work page 1962

  28. [28]

    Journal of Machine Learning Research, 12, 2825–2830, 2011

    Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., and Duchesnay, E.,Scikit-learn: Machine Learning in Python. Journal of Machine Learning Research, 12, 2825–2830, 2011

    work page 2011

  29. [29]

    and Recht, B.,Random Features for Large-Scale Kernel Machines

    Rahimi, A. and Recht, B.,Random Features for Large-Scale Kernel Machines. Advances in Neural Information Processing Systems, 2007

    work page 2007

  30. [30]

    and Williams, C.K.I.,Gaussian Processes for Machine Learning, The MIT Press, 2006

    Rasmussen, C.E. and Williams, C.K.I.,Gaussian Processes for Machine Learning, The MIT Press, 2006

    work page 2006

  31. [31]

    and Khajavi, H.,A Novel Study on Forecasting the Airfoil Self-Noise, Using a Hybrid Model Based on the Combination of CatBoost and Arithmetic Optimization Algorithm

    Rastgoo, A. and Khajavi, H.,A Novel Study on Forecasting the Airfoil Self-Noise, Using a Hybrid Model Based on the Combination of CatBoost and Arithmetic Optimization Algorithm. Expert Systems with Applications, 229, 120576, 2023. 29

    work page 2023

  32. [32]

    Proceedings of the 39th Conference on Neural Information Processing Systems (NeurIPS 25), 2025

    Reddy, T.S., Saketh, V.N.S., and Chandran, M.,Interpretable Graph Neural Networks with Random Fourier Features. Proceedings of the 39th Conference on Neural Information Processing Systems (NeurIPS 25), 2025

    work page 2025

  33. [33]

    Rosenblatt, M.,Remarks on Some Nonparametric Estimates of a Density Function.TheAnnals of Mathematical Statistics, 27, 832–837, 1956

    work page 1956

  34. [34]

    Rudi, A. and Rosasco, L.,Generalization Properties of Learning with Random Features, Proceedings of the 31st International Conference on Neural Information Processing Systems (NeurIPS 17), 3218–3228, 2017

    work page 2017

  35. [35]

    Rumelhart, D.E., Hinton, G.E., and Williams, R.J.,Learning representations by back- propagating errors, Nature, 323, 533–536, 1986

    work page 1986

  36. [36]

    and Smola, A.J.,Learning with Kernels: Support Vector Machines, Regulariza- tion, Optimization, and Beyond

    Schölkopf, B. and Smola, A.J.,Learning with Kernels: Support Vector Machines, Regulariza- tion, Optimization, and Beyond. The MIT Press, 2001

    work page 2001

  37. [37]

    and Li, J.,Explainable Machine Learning by SEE-Net: Closing the Gap between Interpretable Models and DNNs

    Seo, B. and Li, J.,Explainable Machine Learning by SEE-Net: Closing the Gap between Interpretable Models and DNNs. Scientific Reports, 14, 26302, 2024

    work page 2024

  38. [38]

    Journal of Computational and Graphical Statistics, 31(4), 1303–1317, 2022

    Seo, B., Lin, L., and Li, J.,Mixture of Linear Models Co-supervised by Deep Neural Networks. Journal of Computational and Graphical Statistics, 31(4), 1303–1317, 2022

    work page 2022

  39. [39]

    and Brummitt, C.,pyGAM: Generalized Additive Models in Python(software), Version 0.4.1, Zenodo, 2018

    Servén, D. and Brummitt, C.,pyGAM: Generalized Additive Models in Python(software), Version 0.4.1, Zenodo, 2018. doi:10.5281/zenodo.1208724

    work page doi:10.5281/zenodo.1208724 2018

  40. [40]

    Journal of the Royal Statis- tical Society, Series B, 58, 267–288, 1996

    Tibshirani, R.,Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statis- tical Society, Series B, 58, 267–288, 1996

    work page 1996

  41. [41]

    Cambridge University Press, 2004

    Wendland, H.,Scattered Data Approximation. Cambridge University Press, 2004

    work page 2004

  42. [42]

    Chapman & Hall/CRC Press, 2017

    Wood, S.N.,Generalized Additive Models: An Introduction with R (2nd ed.). Chapman & Hall/CRC Press, 2017. Appendix A. Implementation and Hyperparameter Details Appendix A.1. Hyperparameters for the Mixture-of-GAMs Framework This subsection summarizes the hyperparameter choices used for training the proposed Mixture- of-GAMs framework, including the resamp...

    work page 2017