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arxiv: 2506.02276 · v2 · submitted 2025-06-02 · 💻 cs.LG · stat.ML

Latent Stochastic Interpolants

Saurabh Singh , Dmitry Lagun This is my paper

Pith reviewed 2026-05-19 10:38 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords latent variable modelsstochastic interpolantsgenerative modelingevidence lower boundend-to-end optimizationImageNet generation
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The pith

Latent stochastic interpolants enable joint end-to-end optimization of encoder, decoder, and generative model in a learned latent space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops Latent Stochastic Interpolants to move the stochastic interpolant framework into a jointly trained latent variable model. A continuous-time ELBO is derived that supports simultaneous optimization of the encoder's aggregated posterior, the decoder, and the interpolant that maps an arbitrary prior into that posterior. This removes the requirement for direct high-dimensional samples from both distributions while retaining the flexibility of stochastic interpolants. The resulting models avoid the restrictive priors typical of diffusion approaches and lower the computational cost of applying interpolants in observation space.

Core claim

LSI derives a continuous-time Evidence Lower Bound directly in latent space, allowing an arbitrary prior distribution to be transformed into the encoder-defined aggregated posterior through end-to-end optimization of the encoder, decoder, and latent stochastic interpolant models.

What carries the argument

Continuous-time ELBO objective for stochastic interpolants applied to the encoder's aggregated posterior in latent space.

If this is right

  • An arbitrary prior can be used instead of a fixed simple distribution such as a standard normal.
  • Generative sampling occurs by drawing from the prior, applying the learned interpolant in latent space, and then decoding to observations.
  • Computational cost is reduced by performing interpolation entirely in the lower-dimensional latent space.
  • The same framework supports learning both the representation and the generative process without separate pre-training stages.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may allow stochastic interpolants to be applied to modalities where direct high-dimensional sampling is especially expensive, such as video or 3D data.
  • Different choices of prior distribution could be tested to improve sample quality or training stability beyond what is shown on ImageNet.
  • The latent-space formulation might be combined with other latent variable objectives to further relax assumptions on the form of the generative process.

Load-bearing premise

The stochastic interpolant must remain well-defined and sufficiently flexible when applied to samples drawn from the encoder's aggregated posterior rather than from direct high-dimensional observations.

What would settle it

If the derived continuous-time ELBO fails to provide a valid lower bound on the marginal likelihood when the interpolant is optimized jointly with the encoder, the joint learning procedure would not be guaranteed to work as claimed.

Figures

Figures reproduced from arXiv: 2506.02276 by Dmitry Lagun, Saurabh Singh.

Figure 1
Figure 1. Figure 1: Effect of loss trade-off β and encoder noise scale c: In the left panel, we evaluate the effect of loss trade-off weight β for 128 × 128 models and observe that FID improves with β, until the degradation in reconstruction quality (PSNR) starts degrading FID. In the right panel, we evaluate the effect of encoder noise scale on FID. We also plot the FID for a model with learned scale as dashed line. A determ… view at source ↗
Figure 2
Figure 2. Figure 2: LSI supports CFG sampling. Class conditional samples are visualized with increasing guidance weight λ leading to more typical samples for the class. See text for details. InterpFlow parameterization performs better than alternatives: In table 3 we compare differ￾ent parameterizations discussed in section 4 and appendix A. Among the alternatives considered, the InterpFlow parameterization consistently led t… view at source ↗
Figure 3
Figure 3. Figure 3: LSI supports flexible sampling. We demonstrate inversion of an ‘Original’ image, using reverse probability flow ODE (similar to DDIM inversion), followed by forward stochastic sampling to yield samples similar to it, with diversity increasing with γ (eq. (18)). See text for details. Diffusion Models: Diffusion models, originating from foundational work on score matching [Vincent, 2011, Song and Ermon, 2019… view at source ↗
Figure 4
Figure 4. Figure 4: Schedule for t. A visualization of the schedule for t(s) with s ∈ [0, 1] as c is varied. As c increases, larger t values are favored, thereby sampling interpolants closer to t = 1 more frequently. . if z0 is gaussian, we can replace the linear combination of two normal random variables ϵ, z0 with a single random variable zˆ0 ∼ N(ˆµ, Σ) ˆ . Assuming z0 ∼ N(0, I), the mean µˆ = 0 and covariance Σˆ can be com… view at source ↗
Figure 5
Figure 5. Figure 5: An overview of the architecture of various components for [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: LSI supports flexible sampling. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: LSI supports CFG sampling. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
read the original abstract

Stochastic Interpolants (SI) is a powerful framework for generative modeling, capable of flexibly transforming between two probability distributions. However, its use in jointly optimized latent variable models remains unexplored as it requires direct access to the samples from the two distributions. This work presents Latent Stochastic Interpolants (LSI) enabling joint learning in a latent space with end-to-end optimized encoder, decoder and latent SI models. We achieve this by developing a principled Evidence Lower Bound (ELBO) objective derived directly in continuous time. The joint optimization allows LSI to learn effective latent representations along with a generative process that transforms an arbitrary prior distribution into the encoder-defined aggregated posterior. LSI sidesteps the simple priors of the normal diffusion models and mitigates the computational demands of applying SI directly in high-dimensional observation spaces, while preserving the generative flexibility of the SI framework. We demonstrate the efficacy of LSI through comprehensive experiments on the standard large scale ImageNet generation benchmark.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper introduces Latent Stochastic Interpolants (LSI) as an extension of Stochastic Interpolants to latent variable models. It derives a continuous-time Evidence Lower Bound (ELBO) that enables joint end-to-end optimization of an encoder, decoder, and latent SI model, allowing an arbitrary prior to be transformed into the encoder-induced aggregated posterior q(z) while learning effective latent representations. The approach is positioned as sidestepping restrictive priors in diffusion models and avoiding direct high-dimensional SI application, with efficacy shown via experiments on the ImageNet generation benchmark.

Significance. If the ELBO derivation is correct and the latent-space path measure remains well-defined, LSI would offer a principled way to combine flexible generative interpolants with amortized inference, potentially improving upon standard latent diffusion models by supporting non-Gaussian priors and joint optimization. The ImageNet results, if competitive, would provide concrete evidence of practical utility for large-scale generation.

major comments (1)
  1. [§3] §3 (ELBO derivation): The continuous-time objective is claimed to be a valid lower bound on the marginal likelihood for the joint model. However, because the aggregated posterior q(z) depends on the encoder parameters through the data x, the change-of-measure between the interpolant path and the prior/posterior measures may introduce additional Girsanov or entropy correction terms that are not explicitly accounted for in the final ELBO. If these terms are omitted, the claimed joint optimization optimizes a different functional than stated, undermining the central guarantee.
minor comments (2)
  1. [§2] Notation for the stochastic interpolant I_t and its velocity/score fields should be made consistent between the latent-space formulation and the high-dimensional SI background section to avoid reader confusion.
  2. [§4] The ImageNet experimental section would benefit from an ablation isolating the effect of the arbitrary prior choice versus a standard Gaussian prior under the same latent SI architecture.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a potential subtlety in the continuous-time ELBO derivation. We address this point directly below.

read point-by-point responses

  1. Referee: [§3] §3 (ELBO derivation): The continuous-time objective is claimed to be a valid lower bound on the marginal likelihood for the joint model. However, because the aggregated posterior q(z) depends on the encoder parameters through the data x, the change-of-measure between the interpolant path and the prior/posterior measures may introduce additional Girsanov or entropy correction terms that are not explicitly accounted for in the final ELBO. If these terms are omitted, the claimed joint optimization optimizes a different functional than stated, undermining the central guarantee.

    Authors: We respectfully disagree that additional Girsanov or entropy corrections are omitted. The derivation in §3 begins from the marginal likelihood and constructs the interpolant path measure directly between the fixed prior and the aggregated posterior q(z) := ∫ q(z|x)p(x)dx. Because the path is defined entirely in latent space, the Radon–Nikodym derivative (via Girsanov) is taken with respect to the reference Wiener measure on that space; the data dependence enters only through the definition of q(z) itself, which is already marginalized in the expectation that yields the ELBO. Consequently the standard entropy term between the path and the posterior measure already accounts for the variational gap, and no further correction arises from the parametric dependence of the encoder. The resulting functional is therefore the claimed lower bound for any fixed encoder, and joint optimization proceeds by differentiating through this bound. We stand by the derivation as written. revision: no

Circularity Check

0 steps flagged

Continuous-time ELBO derivation for LSI is self-contained and does not reduce to fitted inputs or self-citation chains.

full rationale

The paper derives a principled ELBO directly in continuous time to enable joint optimization of encoder, decoder, and latent stochastic interpolant models, transforming an arbitrary prior into the encoder-defined aggregated posterior. No equations or steps in the abstract or described derivation chain exhibit self-definition, where a claimed prediction equals a fitted parameter by construction, or load-bearing self-citations that import uniqueness or ansatzes without independent verification. The central objective is presented as following from first-principles change-of-measure considerations in latent space, with experiments on ImageNet serving as external validation rather than internal tautology. This qualifies as an honest non-finding of circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard assumptions from variational inference and continuous-time generative modeling; no new free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption A continuous-time ELBO can be derived for the latent stochastic interpolant that supports end-to-end optimization of encoder and decoder.
    Invoked in the abstract as the mechanism enabling joint learning.

pith-pipeline@v0.9.0 · 5679 in / 1217 out tokens · 38139 ms · 2026-05-19T10:38:19.144911+00:00 · methodology

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Reference graph

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