# Monotonic Calibrated Interpolated Look-Up Tables

arXiv: https://arxiv.org/abs/1505.06378 · HTML: https://mlsanity.com/paper/1505.06378/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 May 2015.

## Built on by (most cited first, 30 of 35)

- [Explanations for Monotonic Classifiers](https://mlsanity.com/paper/2106.00154/) (2021)
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- [Matching Learned Causal Effects of Neural Networks with Domain Priors](https://mlsanity.com/paper/2111.12490/) (2021)
- [Size and depth of monotone neural networks: interpolation and approximation](https://mlsanity.com/paper/2207.05275/) (2022)
- [A Cram'er Distance perspective on Quantile Regression based Distributional Reinforcement Learning](https://mlsanity.com/paper/2110.00535/) (2021)
- [Advancing Constrained Monotonic Neural Networks: Achieving Universal Approximation Beyond Bounded Activations](https://mlsanity.com/paper/2505.02537/) (2025)
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- [A framework for massive scale personalized promotion](https://mlsanity.com/paper/2108.12100/) (2021)
- [DeepSaDe: Learning Neural Networks that Guarantee Domain Constraint Satisfaction](https://mlsanity.com/paper/2303.01141/) (2023)
- [Distribution Embedding Networks for Generalization from a Diverse Set of Classification Tasks](https://mlsanity.com/paper/2202.01940/) (2022)
- [Shape-constrained Symbolic Regression with NSGA-III](https://mlsanity.com/paper/2209.13851/) (2022)
- [Global Optimization Networks](https://mlsanity.com/paper/2202.01277/) (2022)
- [Neural Inverse Transform Sampler](https://mlsanity.com/paper/2206.11172/) (2022)
- [XClusters: Explainability-first Clustering](https://mlsanity.com/paper/2209.10956/) (2022)
- [Regularization Strategies for Quantile Regression](https://mlsanity.com/paper/2102.05135/) (2021)
- [Parsing the Language of Expression: Enhancing Symbolic Regression with Domain-Aware Symbolic Priors](https://mlsanity.com/paper/2503.09592/) (2025)
- [Efficient and Scalable Estimation of Distributional Treatment Effects with Multi-Task Neural Networks](https://mlsanity.com/paper/2507.07738/) (2025)
- [Why are there many equally good models? An Anatomy of the Rashomon Effect](https://mlsanity.com/paper/2601.06730/) (2026)
- [Monotonic Kolmogorov-Arnold Networks: A Theoretical and Empirical Study of Monotonicity as an Inductive Bias](https://mlsanity.com/paper/2606.17886/) (2026)
- [Constrained Machine Learning Through Hyperspherical Representation](https://mlsanity.com/paper/2504.08415/) (2025)
- [Responsiveness Verification: Will Predictions Change? How Much? How Often?](https://mlsanity.com/paper/2507.02169/) (2025)
- [An Interpretable Deep Learning Model for General Insurance Pricing](https://mlsanity.com/paper/2509.08467/) (2025)
- [MonoCon: A general framework for learning ultra-compact high-fidelity representations using monotonicity constraints](https://mlsanity.com/paper/2509.22931/) (2025)
- [Multi-Horizon Time Series Forecasting of non-parametric CDFs with Deep Lattice Networks](https://mlsanity.com/paper/2511.13756/) (2025)

## Abstract (arXiv)

Real-world machine learning applications may require functions that are fast-to-evaluate and interpretable. In particular, guaranteed monotonicity of the learned function can be critical to user trust. We propose meeting these goals for low-dimensional machine learning problems by learning flexible, monotonic functions using calibrated interpolated look-up tables. We extend the structural risk minimization framework of lattice regression to train monotonic look-up tables by solving a convex problem with appropriate linear inequality constraints. In addition, we propose jointly learning interpretable calibrations of each feature to normalize continuous features and handle categorical or missing data, at the cost of making the objective non-convex. We address large-scale learning through parallelization, mini-batching, and propose random sampling of additive regularizer terms. Case studies with real-world problems with five to sixteen features and thousands to millions of training samples demonstrate the proposed monotonic functions can achieve state-of-the-art accuracy on practical problems while providing greater transparency to users.