Inference via Interpolation: Contrastive Representations Provably Enable
  Planning and Inference
Benjamin Eysenbach Vivek Myers Ruslan Salakhutdinov Sergey Levine
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.