Connecting vector spaces through shared references

You already have a collection of text embeddings and want to add vectors from another team. Both sides use cosine similarity. The vectors even have the same length. Putting them in one index seems to require little more than importing the data.

First, though, there is a question to settle: do the numbers describe the same space?

An embedding model produces learned coordinates. Dimension 17 in one model has no inherent correspondence with dimension 17 in another. Two models can understand a collection of texts in similar ways while writing their coordinates differently.

NAC—Nested Anchor Coordinates—in ToWow’s research starts from this problem. Each participant uses its own model to observe a shared set of references, then exchanges the observations. We will first work through an example that can be calculated by hand, then consider how it could help find collaborators.

Imagine a two-dimensional model A that places a text at x = (1, 0). Model B represents the same text as x′ = (0, 1). The cosine similarity between these two raw vectors is 0.

In this teaching example, however, B has simply rotated A’s entire space 90 degrees counterclockwise. All angles between texts are preserved. Retrieval within each model could be identical, while the cross-model comparison treats the same text as an orthogonal direction.

Now introduce two shared reference objects, a and b. In A they lie at (1, 0) and (0, 1) respectively. After rotation, their positions in B are (0, 1) and (-1, 0).

Ask each model two questions: how similar is the text to a, and how similar is it to b?

The raw coordinates differ, but both reference results are [1, 0]. Comparing the reference results reveals the agreement in this example.

We have not learned a matrix that maps B back into A. Each participant computes valid similarities within its own space and writes the results in a shared order. What crosses the boundary is a description of relationships to the same objects.

In NAC, a reference can be a fixed text or a commonly accessible example. The protocol first specifies the reference set and its order. Each participant uses its own encoder E to encode local content x and all references a₁…aₘ, producing:

c_E(x) = [cos(E(x), E(a₁)), …, cos(E(x), E(aₘ))]

The first position now means “relationship to the first reference”; the second means “relationship to the second reference.” Model A’s native vectors might have 384 dimensions and model B’s 1,024. If both compute relationships to the same m references, both can output m relationship values.

Relative representations have a clear research lineage. In Relative Representations, presented at ICLR 2023, Moschella and colleagues construct representations from similarities between samples and anchors, investigating latent-space comparison and model-component stitching. NAC adopts this family of coordinate ideas and investigates progressive transmission, relationship direction, and version evolution for open agent networks.

Shared references remove ambiguity about what each position means. How much semantics they preserve depends on the actual models and references. The rotation example works particularly well because rotation preserves angles. Real models may learn different distinctions.

For example, one model may distinguish carefully between “equipment available to borrow” and “equipment being purchased,” while another groups both near the general topic of equipment. Shared references expose this disagreement on a comparable surface. We still need to determine whether it harms retrieval.

Consider a design example involving an equipment trial. A small team is developing thermal-imaging quality inspection and needs two hours of equipment access with on-site guidance. A nearby laboratory has spare time slots. Another engineer is also looking for a similar camera.

The two seekers may have the most similar text. Equipment names, trial objectives, and technical vocabulary all overlap. The laboratory’s description might instead say “instrument platform,” “short-term access,” and “operational support,” placing it further away in wording.

The signal therefore also needs to express the content’s role in a relationship. NAC’s design allows a local description to be divided into directional fragments: seeking, offering, and background. The team’s “seeking equipment” is compared first with the laboratory’s “offering equipment.” The other seeker could still contribute experience or an introduction; that would be a different reason to collaborate.

Generating one signal can be quite concrete:

1. The local agent selects content relevant to the request and permitted to participate in discovery.
2. Its local model computes that content’s relationship values against the shared references.
3. It attaches the “seeking” direction, reference version, and expiry.
4. The receiver uses corresponding “offering” fragments to compare candidates.

The network receives a relationship description with a semantic direction. Whether the timing and equipment fit can be established gradually in subsequent conversation.

A discovery network may need an inexpensive initial filter before spending more computation on a few candidates. NAC therefore orders its references consistently, allowing a signal to begin with a short prefix and expand when needed.

For example, an initial exchange might contain the first 32 relationship values, expanding to 128 for a promising candidate. These lengths are illustrative. A useful configuration needs task-level measurement: how many useful candidates the short prefix misses, how many the extension recovers, and the computation and transmission it costs.

Ordering matters. If the first 32 references all concern similar industrial procurement situations, they may not discriminate well between artistic or teaching collaborations. A long vector can work well while its beginning works poorly. The ability to truncate and the usefulness of a truncated representation are separate requirements.

This also clarifies the relationship to Matryoshka Representation Learning. That work uses training to make representations useful at different lengths. NAC puts the progressive ordering in external references, allowing participants to retain their own encoders. The extra freedom creates an experimental question: how should references be selected and ordered so that each transmission budget buys a useful signal?

A floating-point array alone is insufficient. A receiver needs to know which reference version was used, how long a prefix it received, and when the information was generated. Otherwise, two equally long arrays could answer different questions.

NAC therefore carries version and direction alongside coordinates. When a local model is upgraded, the agent can return to the original materials and recompute their reference relationships. Updating the reference set requires a transition: generating both versions for a period, for example, or restricting comparison to nodes that explicitly support the same version.

The interesting design choice is to move the shared dependency from “everyone installs the same model” to “everyone can obtain and interpret the same references.” Each party maintains its model choice; reference quality and evolution become shared network concerns.

If all content can be encoded by one service, a shared encoder is often a straightforward starting point. Its vectors already inhabit the same space, simplifying ingestion and retrieval. Where existing embeddings must be reused across spaces, learning a transformation is another option. vec2vec, for example, investigates learning translations from embedding distributions.

NAC is worth investigating where participants want to retain local models while exchanging a commonly interpretable representation whose detail can grow progressively. The engineering choice must account for retrieval quality, reference computation, transmission volume, and maintenance after model changes.

ToWow has produced a signal design and reference implementation; cross-model discovery performance remains to be established. Earlier local diagnostics included anchor coordinates and other representations, but did not complete the evaluation needed to settle this cross-provider approach. This article explains the mechanism and its design rationale.

A shared encoder, learned mappings, and shared references place maintenance costs in different locations. NAC leaves model maintenance with each participant and makes reference quality and ordering a shared responsibility. Its suitability as an exchange representation depends on how much useful cross-model comparison that cost buys.

Research materials: the NAC mechanism profile and current Discovery Front status records; local source locations are in the publication package’s SOURCE-NOTES.md. The two-dimensional coordinates, equipment scenario, and 32/128 prefixes are explanatory examples, not model experiment results.