Uniform first-moment bound for Euclidean t-SNE gradient flows

Establish that, for the t-SNE gradient flow on $X=\mathbb{R}^d$ and $Y=\mathbb{R}^m$ starting from a coupling with finite first moment, the first moment of the embedding marginal remains uniformly bounded for all times.

Background

A uniform moment bound is needed to obtain tightness of the evolving couplings when the embedding space is Euclidean. The paper notes that such a bound is known in some special settings, including finitely many point masses with two-dimensional embeddings, but not in the general continuum setting.

The conjecture is used conditionally to obtain weakly convergent subsequences of the gradient flow and to construct symmetry-invariant critical points in Euclidean embedding spaces.

References

For $X = Rd$, and $Y=Rm$, with an initial coupling $\pi_0$ that has finite first moment, then the first moment remains uniformly bounded for all times.

— On the Abundance of Critical Points of the t-SNE Energy  (2609.04379 - Haridas et al., 3 Sep 2026) in Conjecture 9, Section 3, subsection “Existence of descent paths and their limits”