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The flat space problem: planarization in vector retrieval

How we discovered that our embedding space had collapsed into something nearly one-dimensional — and the three things we did to fix it.

This paper grew out of building Tune Search — a search engine that retrieves, critiques its own coverage, and lets a council of voices argue for what should be returned. The questions it raised are what this piece tries to answer.

We were building a search engine over a corpus of scripture and commentary. The query was supposed to match emotional states — someone types “I can’t sleep” and the system finds passages about peace, or stillness, or the dark night of the soul. To do that, we tagged every passage with a vocabulary of themes: grief, perseverance, compassion, longing, and so on. Then we noticed something strange. Wildly different queries kept returning almost exactly the same cluster of tags — something like {sadness, love, compassion} over and over, regardless of what was actually typed. The system looked confident. It was always confident. And it was nearly always wrong in the same direction.

We called it planarization: the tag space had collapsed into something almost one-dimensional, so every query landed in roughly the same region and every result looked equally relevant. The machine never shrugged because, geometrically, it couldn’t tell the difference between queries that were similar and queries that merely weren’t far enough apart. This piece explains what planarization is, why it happens, and the three things we did to fix it.

How search works, briefly

Keyword search. The oldest idea: match the words. The modern workhorse is BM25 — it scores a document by how many of the query’s words it contains, weighting rare words more heavily. Fast and literal: it cannot match “sorrow” to “grief” because they share no letters. Elastic have a good walkthrough.

Semantic search (embeddings). Each passage is turned into a vector positioned so that things with similar meaning sit near each other. A query becomes a vector too; retrieval returns its nearest neighbours by cosine similarity. Now “sorrow” and “grief” land close together. Pinecone’s What are embeddings? is a reliable primer.

Hybrid + RAG. In practice you run both and merge the rankings. The standard merge is Reciprocal Rank Fusion: score each result as 1/(k + rank) in each list and sum. Then in Retrieval-Augmented Generation (RAG), the top passages go to a language model, which writes a grounded answer. Weaviate explain how hybrid search works. This ladder works remarkably well. And it is exactly this design that answers “I broke my car” with a passage about perseverance.

Planarization: the geometry is flatter than it looks

When you embed text into a high-dimensional vector space, you expect the space to be — well, high-dimensional. Queries about grief should land somewhere different from queries about gratitude, which should land somewhere different from queries about doubt. But in practice, trained embeddings tend to occupy a narrow cone rather than spreading evenly. Gao et al. (ICLR 2019) named the training cause the “Representation Degeneration Problem”: most learned embeddings collapse into a narrow cone, “which largely limits the representation power.” Timkey & van Schijndel (EMNLP 2021) found that in BERT and GPT-2, just 1–3 “rogue” dimensions dominate cosine scores — not the dimensions the model actually uses for meaning. When everything is packed into a cone, even unrelated items share high cosine similarity. Every query has a close neighbour. The machine is always confident. And confidence is meaningless.

This is the phenomenon in the literature under the name anisotropy. Related effects go by different names: dimensional collapse in contrastive learning (Jing et al., ICLR 2022), where embeddings span only a sub-space; hubness (Radovanović et al., JMLR 2010), where a few points become everyone’s nearest neighbour in high dimensions. Different mechanisms, same symptom: the similarity metric stops discriminating.

In our case there was a second, separate cause on top of this. Affect really does have low-dimensional structure — psychology’s circumplex model (Russell, 1980) places most emotions on roughly two axes, valence and arousal. So sadness, love, and compassion genuinely live near the same affective subspace, independently of any embedding pathology. We were being hit by both problems at once: a cone-shaped vector space and a genuinely low-dimensional domain. Tags that we intended to be distinct kept meaning the same geometric thing.

The standard mathematical fix is whitening — redistributing embeddings into a more isotropic space. BERT-whitening (Su et al., 2021) and BERT-flow (Li et al., EMNLP 2020) both take this route. But whitening improves the similarity diagnostics without reliably improving downstream retrieval, as Steck et al. (WWW 2024) showed: cosine may not be measuring what you think it is in the first place. We wanted to attack the problem closer to the source.

What we did: evolve the tags

The root cause was our tag vocabulary. We had designed it by hand — a list of themes that felt distinct to us semantically but turned out to cluster together geometrically. The fix was to let a genetic algorithm redesign it.

The GA treated the tag set as a chromosome and optimised for a single objective: spread in the embedding space. A tag set scored well if, when passages were tagged and their tag vectors compared, the resulting space was isotropic — queries landing in different regions, results discriminating meaningfully between inputs. Tag sets that produced a flat, cone-shaped space scored poorly and died; tag sets that produced spread survived and recombined. After enough generations, the evolved vocabulary was qualitatively different from our hand-authored one: tags that had seemed natural to a human but collapsed onto the same vector were replaced by tags that felt less intuitive but produced genuinely distinct geometry.

This did not fully eliminate planarization — the affective subspace is real and no vocabulary can make “grief” and “sorrow” live far apart. But it made the space meaningfully more isotropic. Queries that previously returned identical tag clusters now produced distinguishable results. The machine could, for the first time, disagree with itself.

Two more layers: the pastoral index and the council

Better tags help. But we added two architectural layers that work alongside them.

The pastoral index. A curated set of hand-mapped life-situations — “diagnosis came back,” “can’t sleep,” “someone died” — injected directly into the top of the retrieval pool, bypassing cosine entirely for known patterns. For a corpus of religious texts, the set of situations a person might bring to it is finite and nameable. Where the geometry still fails to discriminate, the index guarantees the right passage surfaces anyway. This is not a geometric fix; it is an acknowledgment that for some problems, curation beats optimisation.

The council. Where a plain pipeline returns one cosine-ranked list, Tune Search convenes a council of per-author agents — Augustine, Calvin, Lewis, and the rest — each nominating the single best passage from their own writings for the query. A Judge then adjudicates: it vetoes nominations that are off-tone or low quality and keeps a diverse quorum. This bypasses global cosine dominance: even if the vector space is flat and Augustine and Calvin live in the same region, they are constrained to nominate from different parts of the corpus. The per-author nomination structure introduces diversity that geometry alone cannot provide. Multi-agent deliberation as a retrieval fix has literature behind it — multi-agent debate (Du et al., 2023), LLM-as-a-judge (Zheng et al., 2023) — but the motivation here is geometric as much as epistemic.

The not-found problem: secondary, and deliberately so

There is a second, well-known failure mode in retrieval: the system returns a confident answer even when nothing in the corpus actually addresses the query. Self-RAG (Asai et al., 2023) and CRAG (Yan et al., 2024) are built around exactly this. The damage is measurable: irrelevant passages dropped answer accuracy by up to 25% in controlled experiments (Cuconasu et al., SIGIR 2024).

Tune Search does not abstain. Its self-critique loop broadens the query and tries again when coverage is low, but always returns its best match. For a corpus of prayers and commentary, this is actually the right call. Someone who types “I broke my car” into a prayer search is not looking for car-repair manuals. Returning the most relevant passage about brokenness is the correct behaviour. The not-found problem matters acutely when a retriever is wired to a general factual knowledge base; it matters much less when the corpus is a deliberately narrow thing the user already knows they are searching.

Planarization is a geometry problem, and the right response is to attack the geometry — in our case, by evolving a tag vocabulary that actually spreads across the space rather than collapsing into a cone. The pastoral index and the council then operate on top of a space that has real discriminating power. The result is a system that can, for the first time, return different answers to different questions.

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