Stability classification of symmetry-preserving critical points

Determine whether the symmetry-preserving solutions produced by exact gradient descent for the t-SNE energy are stable critical points or saddle points.

Background

The paper constructs and numerically studies embeddings that preserve prescribed pairs of discrete isometries. Exact gradient descent preserves these symmetries, whereas practical approximations such as sparsification, Barnes–Hut repulsion, momentum, and adaptive gains can break them.

Although some symmetry-preserving runs attain higher energy than approximate implementations and appear numerically structured, the paper does not determine whether the corresponding symmetry solutions are dynamically stable critical points or unstable saddles.

References

In many cases the outcomes maintain a high degree of visual symmetry, and still a relatively low value of $\mathcal{SE}$, and understanding whether the symmetry solutions are stable critical points or saddle points is an interesting question that we do not resolve here.

— On the Abundance of Critical Points of the t-SNE Energy  (2609.04379 - Haridas et al., 3 Sep 2026) in Example 5, Section 3.2, subsection “Numerical Illustrations”