# Deep Lattice Networks and Partial Monotonic Functions

arXiv: https://arxiv.org/abs/1709.06680 · HTML: https://mlsanity.com/paper/1709.06680/

## Key facts

- Cited by 63 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Sep 2017.

## Built on by (most cited first, 30 of 63)

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- [Explanations for Monotonic Classifiers](https://mlsanity.com/paper/2106.00154/) (2021)
- [Constrained Monotonic Neural Networks](https://mlsanity.com/paper/2205.11775/) (2022)
- [Feature Necessity & Relevancy in ML Classifier Explanations](https://mlsanity.com/paper/2210.15675/) (2022)
- [Deep Non-Crossing Quantiles through the Partial Derivative](https://mlsanity.com/paper/2201.12848/) (2022)
- [Monotonic Neural Additive Models: Pursuing Regulated Machine Learning Models for Credit Scoring](https://mlsanity.com/paper/2209.10070/) (2022)
- [Neural Clustering based Visual Representation Learning](https://mlsanity.com/paper/2403.17409/) (2024)
- [A Probabilistic Neuro-symbolic Layer for Algebraic Constraint Satisfaction](https://mlsanity.com/paper/2503.19466/) (2025)
- [Matching Learned Causal Effects of Neural Networks with Domain Priors](https://mlsanity.com/paper/2111.12490/) (2021)
- [Advancing Constrained Monotonic Neural Networks: Achieving Universal Approximation Beyond Bounded Activations](https://mlsanity.com/paper/2505.02537/) (2025)
- [Expressive Monotonic Neural Networks](https://mlsanity.com/paper/2307.07512/) (2023)
- [Robust and Provably Monotonic Networks](https://mlsanity.com/paper/2112.00038/) (2021)
- [Contrastive Learning with Consistent Representations](https://mlsanity.com/paper/2302.01541/) (2023)
- [Monotonicity Regularization: Improved Penalties and Novel Applications to Disentangled Representation Learning and Robust Classification](https://mlsanity.com/paper/2205.08247/) (2022)
- [Smooth Min-Max Monotonic Networks](https://mlsanity.com/paper/2306.01147/) (2023)
- [Deep neural networks for choice analysis: Enhancing behavioral regularity with gradient regularization](https://mlsanity.com/paper/2404.14701/) (2024)
- [A framework for massive scale personalized promotion](https://mlsanity.com/paper/2108.12100/) (2021)
- [Monotonicity for AI ethics and society: An empirical study of the monotonic neural additive model in criminology, education, health care, and finance](https://mlsanity.com/paper/2301.07060/) (2023)
- [DeepSaDe: Learning Neural Networks that Guarantee Domain Constraint Satisfaction](https://mlsanity.com/paper/2303.01141/) (2023)
- [How to address monotonicity for model risk management?](https://mlsanity.com/paper/2305.00799/) (2023)
- [MonoKAN: Certified Monotonic Kolmogorov-Arnold Network](https://mlsanity.com/paper/2409.11078/) (2024)
- [Antithetic Noise in Diffusion Models](https://mlsanity.com/paper/2506.06185/) (2025)
- [A Complement to Neural Networks for Anisotropic Inelasticity at Finite Strains](https://mlsanity.com/paper/2510.04187/) (2025)
- [Global Optimization Networks](https://mlsanity.com/paper/2202.01277/) (2022)
- [XClusters: Explainability-first Clustering](https://mlsanity.com/paper/2209.10956/) (2022)
- [Smoothness and monotonicity constraints for neural networks using ICEnet](https://mlsanity.com/paper/2305.08807/) (2023)
- [Regularization Strategies for Quantile Regression](https://mlsanity.com/paper/2102.05135/) (2021)
- [Learned Accelerator Framework for Angular-Distance-Based High-Dimensional DBSCAN](https://mlsanity.com/paper/2302.03136/) (2023)

## Abstract (arXiv)

We propose learning deep models that are monotonic with respect to a user-specified set of inputs by alternating layers of linear embeddings, ensembles of lattices, and calibrators (piecewise linear functions), with appropriate constraints for monotonicity, and jointly training the resulting network. We implement the layers and projections with new computational graph nodes in TensorFlow and use the ADAM optimizer and batched stochastic gradients. Experiments on benchmark and real-world datasets show that six-layer monotonic deep lattice networks achieve state-of-the art performance for classification and regression with monotonicity guarantees.