# Multi-Label Classification Neural Networks with Hard Logical Constraints

arXiv: https://arxiv.org/abs/2103.13427 · HTML: https://mlsanity.com/paper/2103.13427/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 86 references that are in the map.
- First posted on arXiv 24 Mar 2021.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Bidirectional LSTM-CRF Models for Sequence Tagging](https://mlsanity.com/paper/1508.01991/) (2015)
- [DeepXplore: Automated Whitebox Testing of Deep Learning Systems](https://mlsanity.com/paper/1705.06640/) (2017)
- [A Semantic Loss Function for Deep Learning with Symbolic Knowledge](https://mlsanity.com/paper/1711.11157/) (2017)
- [DeepGauge: Multi-Granularity Testing Criteria for Deep Learning Systems](https://mlsanity.com/paper/1803.07519/) (2018)
- [Harnessing Deep Neural Networks with Logic Rules](https://mlsanity.com/paper/1603.06318/) (2016)
- [An approach to reachability analysis for feed-forward ReLU neural networks](https://mlsanity.com/paper/1706.07351/) (2017)
- [Should we really use post-hoc tests based on mean-ranks?](https://mlsanity.com/paper/1505.02288/) (2015)
- [Logic Tensor Networks for Semantic Image Interpretation](https://mlsanity.com/paper/1705.08968/) (2017)
- [Label-Free Supervision of Neural Networks with Physics and Domain Knowledge](https://mlsanity.com/paper/1609.05566/) (2016)
- [Coherent Hierarchical Multi-Label Classification Networks](https://mlsanity.com/paper/2010.10151/) (2020)
- [Augmenting Neural Networks with First-order Logic](https://mlsanity.com/paper/1906.06298/) (2019)
- [A scikit-based Python environment for performing multi-label classification](https://mlsanity.com/paper/1702.01460/) (2017)
- [Learning Deep Structured Models](https://mlsanity.com/paper/1407.2538/) (2014)
- [DeepGO: Predicting protein functions from sequence and interactions using a deep ontology-aware classifier](https://mlsanity.com/paper/1705.05919/) (2017)
- [Collaboration based Multi-Label Learning](https://mlsanity.com/paper/1902.03047/) (2019)
- [Asynchronous Distributed Learning from Constraints](https://mlsanity.com/paper/1911.05473/) (2019)

## Built on by (most cited first, 30 of 36)

- [Semantic Probabilistic Layers for Neuro-Symbolic Learning](https://mlsanity.com/paper/2206.00426/) (2022)
- [A-NeSI: A Scalable Approximate Method for Probabilistic Neurosymbolic Inference](https://mlsanity.com/paper/2212.12393/) (2022)
- [Deep Learning with Logical Constraints](https://mlsanity.com/paper/2205.00523/) (2022)
- [Towards Data-and Knowledge-Driven Artificial Intelligence: A Survey on Neuro-Symbolic Computing](https://mlsanity.com/paper/2210.15889/) (2022)
- [ROAD-R: The Autonomous Driving Dataset with Logical Requirements](https://mlsanity.com/paper/2210.01597/) (2022)
- [Logic-induced Diagnostic Reasoning for Semi-supervised Semantic Segmentation](https://mlsanity.com/paper/2308.12595/) (2023)
- [How Realistic Is Your Synthetic Data? Constraining Deep Generative Models for Tabular Data](https://mlsanity.com/paper/2402.04823/) (2024)
- [A Pseudo-Semantic Loss for Autoregressive Models with Logical Constraints](https://mlsanity.com/paper/2312.03905/) (2023)
- [Beyond the convexity assumption: Realistic tabular data generation under quantifier-free real linear constraints](https://mlsanity.com/paper/2502.18237/) (2025)
- [LOGICSEG: Parsing Visual Semantics with Neural Logic Learning and Reasoning](https://mlsanity.com/paper/2309.13556/) (2023)
- [ENFORCE: Nonlinear Constrained Learning with Adaptive-depth Neural Projection](https://mlsanity.com/paper/2502.06774/) (2025)
- [Semantic Strengthening of Neuro-Symbolic Learning](https://mlsanity.com/paper/2302.14207/) (2023)
- [Refining neural network predictions using background knowledge](https://mlsanity.com/paper/2206.04976/) (2022)
- [Expert Knowledge-Guided Length-Variant Hierarchical Label Generation for Proposal Classification](https://mlsanity.com/paper/2109.06661/) (2021)
- [Hierarchical Interdisciplinary Topic Detection Model for Research Proposal Classification](https://mlsanity.com/paper/2209.13519/) (2022)
- [Machine Learning with Requirements: a Manifesto](https://mlsanity.com/paper/2304.03674/) (2023)
- [EXPLAIN, AGREE, LEARN: Scaling Learning for Neural Probabilistic Logic](https://mlsanity.com/paper/2408.08133/) (2024)
- [Who Should Review Your Proposal? Interdisciplinary Topic Path Detection for Research Proposals](https://mlsanity.com/paper/2203.10922/) (2022)
- [Exploiting T-norms for Deep Learning in Autonomous Driving](https://mlsanity.com/paper/2402.11362/) (2024)
- [LogicMP: A Neuro-symbolic Approach for Encoding First-order Logic Constraints](https://mlsanity.com/paper/2309.15458/) (2023)
- [ConSequence: Synthesizing Logically Constrained Sequences for Electronic Health Record Generation](https://mlsanity.com/paper/2312.05964/) (2023)
- [Semantic Loss Functions for Neuro-Symbolic Structured Prediction](https://mlsanity.com/paper/2405.07387/) (2024)
- [ABS: Enforcing Constraint Satisfaction On Generated Sequences Via Automata-Guided Beam Search](https://mlsanity.com/paper/2506.09701/) (2025)
- [T-ILR: a Neurosymbolic Integration for LTLf](https://mlsanity.com/paper/2508.15943/) (2025)
- [Right for the Right Reasons: Avoiding Reasoning Shortcuts via Prototypical Neurosymbolic AI](https://mlsanity.com/paper/2510.25497/) (2025)
- [The Integer Linear Programming Inference Cookbook](https://mlsanity.com/paper/2307.00171/) (2023)
- [PiShield: A PyTorch Package for Learning with Requirements](https://mlsanity.com/paper/2402.18285/) (2024)
- [Informed deep hierarchical classification: a non-standard analysis inspired approach](https://mlsanity.com/paper/2409.16956/) (2024)
- [Hierarchy-Consistent Learning and Adaptive Loss Balancing for Hierarchical Multi-Label Classification](https://mlsanity.com/paper/2508.13452/) (2025)
- [ROAD-Waymo: A Large-Scale Action Awareness Dataset for Autonomous Driving](https://mlsanity.com/paper/2411.01683/) (2024)

## Abstract (arXiv)

Multi-label classification (MC) is a standard machine learning problem in which a data point can be associated with a set of classes. A more challenging scenario is given by hierarchical multi-label classification (HMC) problems, in which every prediction must satisfy a given set of hard constraints expressing subclass relationships between classes. In this paper, we propose C-HMCNN(h), a novel approach for solving HMC problems, which, given a network h for the underlying MC problem, exploits the hierarchy information in order to produce predictions coherent with the constraints and to improve performance. Furthermore, we extend the logic used to express HMC constraints in order to be able to specify more complex relations among the classes and propose a new model CCN(h), which extends C-HMCNN(h) and is again able to satisfy and exploit the constraints to improve performance. We conduct an extensive experimental analysis showing the superior performance of both C-HMCNN(h) and CCN(h) when compared to state-of-the-art models in both the HMC and the general MC setting with hard logical constraints.