Introduction \(\beta\) — CFM and ELBO

Clarification (what this note is—and is not—claiming)

This note puts Conditional Flow Matching (CFM) and variational objectives (ELBO) side by side for a very specific reason: when designing a computable loss, both frameworks face a similar-looking obstacle—some marginal object is intractable, and we introduce conditional structure (auxiliary variables, conditional distributions, conditional expectations) to turn the optimization into an expectation we can sample or estimate.

However, this is a methodological similarity, not a semantic or “physical” equivalence. The objectives live on different mathematical objects:

• ELBO is a variational lower bound on a log-marginal likelihood \(\log p_\theta(x)=\log\int p_\theta(x,z)\,dz\). Its construction is inseparable from Jensen’s inequality and an explicit inference distribution \(q_\phi(z\mid x)\).
• Flow Matching / CFM is a path-matching / generator-regression viewpoint: we prescribe a probability path \((p_t)\) and learn a velocity field along that path. The intractable marginal target (e.g., a marginal velocity defined via posterior averaging) is replaced by a tractable conditional supervision signal, and the correctness of this replacement is justified by precise statements such as marginalization identities and gradient equivalence under Bregman losses.

So the purpose of this note is not to “derive Flow Matching from ELBO” (or vice versa), but to clarify:

1. where the analogy does hold (turning intractable marginals into computable conditional objectives), and
2. where it breaks (what is being optimized, what assumptions are required, and what guarantees we get).

With that scope fixed, we now recap the two optimization stories.