Moonshot guess: Harnessing the Universal TOPOLOGY of embeddings.

Preserving Topology in Text Embedding Spaces: A Topological Perspective on the vec2vec Method

Edit: a lot more needs to be thought about in respect of the “order” defined below. as it stands, the ordering leads to “trivial” topologies. More to follow.

In recent research on unsupervised text embedding translation (https://arxiv.org/pdf/2505.12540), the vec2vec method has shown how we can translate embeddings between different models without needing paired data, encoders, or predefined matches. While this method was initially celebrated for aligning embeddings from models with different architectures, it also has a potential deeper connection to topology that I find particularly fascinating.

The vec2vec approach preserves the relative geometry of embeddings, but what does that mean from a topological perspective? It’s not made entirely clear in the paper, but I would *guess” that it would imply that the method preserves the partial order of embeddings across spaces, which is a fundamental concept in order theory. The partial order induced by the distances between embeddings translates naturally into topological structures, where the embeddings form a simplicial complex (a space built from simplices like points, edges, and triangles) based on their relative closeness.

Partial Order and Topology: What’s the Connection?

If I’m right, by maintaining the partial order between embeddings, vec2vec preserves the topological structure of the associated partially ordered sets of the embedding spaces. This is where things get interesting. In topology, a poset topology (or partial order topology) can be mapped to a simplicial complex. When the vec2vec method preserves the partial order between embeddings, it also preserves the underlying topological structure, creating a natural mapping between simplicial complexes.

The Functor and Structure Preservation: A Deeper Topological Insight

Here’s where the fun begins: when applying the Classifying Space functor (a tool from algebraic topology and category theory), this partial order preservation translates into a structure-preserving mapping between the associated simplicial complexes. While the mapping may not be bijective (injective and surjective), it still ensures that the topological structure — the way embeddings are connected and related — is preserved in the translated space.

The mapping respects the relative geometry and nearness relationships between embeddings, which can be understood as preserving some elements of the topological structure in a weaker sense. This implies that the translated embeddings share the same topological properties as the original embeddings, though they are not exactly the same in a bijective sense.

It would be very interesting to understand if this mapping is actually a homological or homotopical equivalence. Might also be interesting to understand how universality of embeddings maps to universality of associated topological spaces via the classifying space functor.

Implications: A New Lens on Embedding Spaces

This topological perspective adds a new layer of depth to our understanding of text embeddings. It shows that the vec2vec method doesn’t just align vectors or preserve semantics; it preserves the fundamental topological structure of the embeddings in terms of their relative ordering and closeness.

By ensuring that the mapping between these spaces respects the partial order, it guarantees that the embeddings’ nearness relations and semantic structure are maintained — even if the mapping isn’t strictly one-to-one or onto. This provides a powerful tool for aligning embeddings across models and spaces while retaining their topological essence.

Conclusion: A Powerful Intersection of Geometry, Topology, and AI

At its core, vec2vec provides more than just a way to translate embeddings across different models. It offers insights into how geometry and the topology of an associated partially ordered set relate to preserving semantic relationships. This intersection between embedding spaces, partial order, and topology is an exciting frontier in AI and natural language processing.