# Improved Representation of Asymmetrical Distances with Interval Quasimetric Embeddings

arXiv: https://arxiv.org/abs/2211.15120 · HTML: https://mlsanity.com/paper/2211.15120/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 36 references that are in the map.
- First posted on arXiv 28 Nov 2022.

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [Conservative Q-Learning for Offline Reinforcement Learning](https://mlsanity.com/paper/2006.04779/) (2020)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Learning Invariant Representations for Reinforcement Learning without Reconstruction](https://mlsanity.com/paper/2006.10742/) (2020)
- [Understanding disentangling in β-VAE](https://mlsanity.com/paper/1804.03599/) (2018)
- [Visual Reinforcement Learning with Imagined Goals](https://mlsanity.com/paper/1807.04742/) (2018)
- [Hyperbolic Entailment Cones for Learning Hierarchical Embeddings](https://mlsanity.com/paper/1804.01882/) (2018)
- [On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes](https://mlsanity.com/paper/1805.08114/) (2018)
- [Order-Embeddings of Images and Language](https://mlsanity.com/paper/1511.06361/) (2015)
- [Denoised MDPs: Learning World Models Better Than the World Itself](https://mlsanity.com/paper/2206.15477/) (2022)
- [Model-Based Visual Planning with Self-Supervised Functional Distances](https://mlsanity.com/paper/2012.15373/) (2020)
- [An Inductive Bias for Distances: Neural Nets that Respect the Triangle Inequality](https://mlsanity.com/paper/2002.05825/) (2020)
- [Hyperbolic Disk Embeddings for Directed Acyclic Graphs](https://mlsanity.com/paper/1902.04335/) (2019)
- [Self-supervised Learning of Distance Functions for Goal-Conditioned Reinforcement Learning](https://mlsanity.com/paper/1907.02998/) (2019)
- [On the Learning and Learnability of Quasimetrics](https://mlsanity.com/paper/2206.15478/) (2022)
- [Metric Residual Networks for Sample Efficient Goal-Conditioned Reinforcement Learning](https://mlsanity.com/paper/2208.08133/) (2022)
- [Shortest path distance approximation using deep learning techniques](https://mlsanity.com/paper/2002.05257/) (2020)
- [Quasimetric embeddings and their applications](https://mlsanity.com/paper/1608.01396/) (2016)

## Built on by (most cited first, 24 of 24)

- [OGBench: Benchmarking Offline Goal-Conditioned RL](https://mlsanity.com/paper/2410.20092/) (2024)
- [Optimal Goal-Reaching Reinforcement Learning via Quasimetric Learning](https://mlsanity.com/paper/2304.01203/) (2023)
- [Offline Goal-conditioned Reinforcement Learning with Quasimetric Representations](https://mlsanity.com/paper/2509.20478/) (2025)
- [Option-aware Temporally Abstracted Value for Offline Goal-Conditioned Reinforcement Learning](https://mlsanity.com/paper/2505.12737/) (2025)
- [Value-guided action planning with JEPA world models](https://mlsanity.com/paper/2601.00844/) (2025)
- [Horizon Generalization in Reinforcement Learning](https://mlsanity.com/paper/2501.02709/) (2025)
- [Self-Predictive Representations for Combinatorial Generalization in Behavioral Cloning](https://mlsanity.com/paper/2506.10137/) (2025)
- [Flattening Hierarchies with Policy Bootstrapping](https://mlsanity.com/paper/2505.14975/) (2025)
- [Physics-informed Temporal Difference Metric Learning for Robot Motion Planning](https://mlsanity.com/paper/2505.05691/) (2025)
- [Goal Reaching with Eikonal-Constrained Hierarchical Quasimetric Reinforcement Learning](https://mlsanity.com/paper/2512.12046/) (2025)
- [Offline Goal-Conditioned Reinforcement Learning with Projective Quasimetric Planning](https://mlsanity.com/paper/2506.18847/) (2025)
- [State Chrono Representation for Enhancing Generalization in Reinforcement Learning](https://mlsanity.com/paper/2411.06174/) (2024)
- [VLD: Visual Language Goal Distance for Reinforcement Learning Navigation](https://mlsanity.com/paper/2512.07976/) (2025)
- [Learning The Minimum Action Distance](https://mlsanity.com/paper/2506.09276/) (2025)
- [Compositional Transduction with Latent Analogies for Offline Goal-Conditioned Reinforcement Learning](https://mlsanity.com/paper/2605.20609/) (2026)
- [Latent Representation Alignment for Offline Goal-Conditioned Reinforcement Learning](https://mlsanity.com/paper/2605.25740/) (2026)
- [QuasiNav: Asymmetric Cost-Aware Navigation Planning with Constrained Quasimetric Reinforcement Learning](https://mlsanity.com/paper/2410.16666/) (2024)
- [Intrinsic-Energy Joint Embedding Predictive Architectures Induce Quasimetric Spaces](https://mlsanity.com/paper/2602.12245/) (2026)
- [ChronoForest: Closed-Loop Multi-Tree Diffusion Planning for Efficient Bridge Search and Route Composition](https://mlsanity.com/paper/2606.06618/) (2026)
- [GTRL: Grounding Divide-and-Conquer Value Learning with Temporal Differences](https://mlsanity.com/paper/2609.33259/) (2026)
- [Learning Transferable Skills using Goal-Conditioned Bisimulation](https://mlsanity.com/paper/2610.00676/) (2026)
- [Learning Goal-Reaching Quasimetric Geometry From Finite-Time Reachability](https://mlsanity.com/paper/2610.00778/) (2026)
- [World Modeling through Spectral Alignment](https://mlsanity.com/paper/2610.15429/) (2026)
- [Outperforming 1000 Euclidean Layers With Hyperbolic Goal-Conditioned RL](https://mlsanity.com/paper/2610.20009/) (2026)

## Abstract (arXiv)

Asymmetrical distance structures (quasimetrics) are ubiquitous in our lives and are gaining more attention in machine learning applications. Imposing such quasimetric structures in model representations has been shown to improve many tasks, including reinforcement learning (RL) and causal relation learning. In this work, we present four desirable properties in such quasimetric models, and show how prior works fail at them. We propose Interval Quasimetric Embedding (IQE), which is designed to satisfy all four criteria. On three quasimetric learning experiments, IQEs show strong approximation and generalization abilities, leading to better performance and improved efficiency over prior methods. Project Page: https://www.tongzhouwang.info/interval_quasimetric_embedding Quasimetric Learning Code Package: https://www.github.com/quasimetric-learning/torch-quasimetric