# Inference via Interpolation: Contrastive Representations Provably Enable Planning and Inference

arXiv: https://arxiv.org/abs/2403.04082 · HTML: https://mlsanity.com/paper/2403.04082/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 69 of its 105 references that are in the map.
- First posted on arXiv 6 Mar 2024.

## Builds on (most cited first, 30 of 69)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Exploring Simple Siamese Representation Learning](https://mlsanity.com/paper/2011.10566/) (2020)
- [Improved Baselines with Momentum Contrastive Learning](https://mlsanity.com/paper/2003.04297/) (2020)
- [Unsupervised Feature Learning via Non-Parametric Instance-level Discrimination](https://mlsanity.com/paper/1805.01978/) (2018)
- [Decision Transformer: Reinforcement Learning via Sequence Modeling](https://mlsanity.com/paper/2106.01345/) (2021)
- [D4RL: Datasets for Deep Data-Driven Reinforcement Learning](https://mlsanity.com/paper/2004.07219/) (2020)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [Generating Diverse High-Fidelity Images with VQ-VAE-2](https://mlsanity.com/paper/1906.00446/) (2019)
- [Contrastive Multiview Coding](https://mlsanity.com/paper/1906.05849/) (2019)
- [R3M: A Universal Visual Representation for Robot Manipulation](https://mlsanity.com/paper/2203.12601/) (2022)
- [Offline Reinforcement Learning as One Big Sequence Modeling Problem](https://mlsanity.com/paper/2106.02039/) (2021)
- [On Variational Bounds of Mutual Information](https://mlsanity.com/paper/1905.06922/) (2019)
- [A Theoretical Analysis of Contrastive Unsupervised Representation Learning](https://mlsanity.com/paper/1902.09229/) (2019)
- [Time-Contrastive Networks: Self-Supervised Learning from Video](https://mlsanity.com/paper/1704.06888/) (2017)
- [VIP: Towards Universal Visual Reward and Representation via Value-Implicit Pre-Training](https://mlsanity.com/paper/2210.00030/) (2022)
- [Spatiotemporal Contrastive Video Representation Learning](https://mlsanity.com/paper/2008.03800/) (2020)
- [Adversarial Autoencoders](https://mlsanity.com/paper/1511.05644/) (2015)
- [A Recurrent Latent Variable Model for Sequential Data](https://mlsanity.com/paper/1506.02216/) (2015)
- [Successor Features for Transfer in Reinforcement Learning](https://mlsanity.com/paper/1606.05312/) (2016)
- [Data-Efficient Reinforcement Learning with Self-Predictive Representations](https://mlsanity.com/paper/2007.05929/) (2020)
- [Embed to Control: A Locally Linear Latent Dynamics Model for Control from Raw Images](https://mlsanity.com/paper/1506.07365/) (2015)
- [Decoupling Representation Learning from Reinforcement Learning](https://mlsanity.com/paper/2009.08319/) (2020)
- [Contrastive Learning as Goal-Conditioned Reinforcement Learning](https://mlsanity.com/paper/2206.07568/) (2022)
- [Adversarially Learned Inference](https://mlsanity.com/paper/1606.00704/) (2016)

## Built on by (most cited first, 17 of 17)

- [Temporal Straightening for Latent Planning](https://mlsanity.com/paper/2603.12231/) (2026)
- [A Single Goal is All You Need: Skills and Exploration Emerge from Contrastive RL without Rewards, Demonstrations, or Subgoals](https://mlsanity.com/paper/2408.05804/) (2024)
- [Contrastive Representation for Data Filtering in Cross-Domain Offline Reinforcement Learning](https://mlsanity.com/paper/2405.06192/) (2024)
- [Horizon Generalization in Reinforcement Learning](https://mlsanity.com/paper/2501.02709/) (2025)
- [Temporal Representation Alignment: Successor Features Enable Emergent Compositionality in Robot Instruction Following](https://mlsanity.com/paper/2502.05454/) (2025)
- [Learning to Assist Humans without Inferring Rewards](https://mlsanity.com/paper/2411.02623/) (2024)
- [UORA: Uniform Orthogonal Reinitialization Adaptation in Parameter-Efficient Fine-Tuning of Large Models](https://mlsanity.com/paper/2505.20154/) (2025)
- [Multistep Quasimetric Learning for Scalable Goal-conditioned Reinforcement Learning](https://mlsanity.com/paper/2511.07730/) (2025)
- [Context Structure Reshapes the Representational Geometry of Language Models](https://mlsanity.com/paper/2601.22364/) (2026)
- [Contrastive Representations for Temporal Reasoning](https://mlsanity.com/paper/2508.13113/) (2025)
- [NFTR: From Provable Mode-Averaging to Geodesic Subgoal Selection in Offline Goal-Conditioned RL](https://mlsanity.com/paper/2607.07855/) (2026)
- [Implicit Contact Diffuser: Sequential Contact Reasoning with Latent Point Cloud Diffusion](https://mlsanity.com/paper/2410.16571/) (2024)
- [The "Law" of the Unconscious Contrastive Learner: Probabilistic Alignment of Unpaired Modalities](https://mlsanity.com/paper/2501.11326/) (2025)
- [Learning Abstract World Models with a Group-Structured Latent Space](https://mlsanity.com/paper/2506.01529/) (2025)
- [Learning to Plan from Random Exploration](https://mlsanity.com/paper/2609.38383/) (2026)
- [Direction-Conditioned Policies for Online Goal-Conditioned Reinforcement Learning](https://mlsanity.com/paper/2610.05087/) (2026)
- [Hyperspherical Contrastive Representations Unlock Planning and Exploration](https://mlsanity.com/paper/2610.20409/) (2026)

## Abstract (arXiv)

Given time series data, how can we answer questions like "what will happen in the future?" and "how did we get here?" These sorts of probabilistic inference questions are challenging when observations are high-dimensional. In this paper, we show how these questions can have compact, closed form solutions in terms of learned representations. The key idea is to apply a variant of contrastive learning to time series data. Prior work already shows that the representations learned by contrastive learning encode a probability ratio. By extending prior work to show that the marginal distribution over representations is Gaussian, we can then prove that joint distribution of representations is also Gaussian. Taken together, these results show that representations learned via temporal contrastive learning follow a Gauss-Markov chain, a graphical model where inference (e.g., prediction, planning) over representations corresponds to inverting a low-dimensional matrix. In one special case, inferring intermediate representations will be equivalent to interpolating between the learned representations. We validate our theory using numerical simulations on tasks up to 46-dimensions.