Aspect dilution: several vectors beat one, and why

The claim and the confound

In short: Cards have to beat an equally large set of blind chunks, not just the single vector, or the experiment measures capacity rather than the idea.

Giving a document several cards means giving it several embeddings. Cutting the same document into arbitrary windows also means several embeddings. If cards beat a single pooled vectorPooled (single-vector) embedding. One numeric fingerprint for a whole document. Cheap and standard, but for a document about several things it is an average of all of them, so it matches each of them weakly., that proves nothing about purposes, because chunks would beat it too. Every experiment therefore carries three arms.

The encoder is the same in every arm (all-MiniLM-L6-v2, a small general-purpose sentence encoder), everything is scored at document level, and the control is matchedMatched control. A comparison arm given the same budget as the method under test, here the same number of embeddings per document, so the experiment measures the idea rather than the extra capacity. on units per document.

Aspect dilution on a controlled corpus

In short: As a document grows from one aspect to ten, the pooled index falls from 0.815 to 0.294 nDCG@10; any multi-vector representation holds it above 0.63.

Each synthetic document concatenates k passages drawn from k of ten unrelated topic pools (legal, logistics, finance, engineering, personnel, marketing, facilities, research, security, procurement). Every passage realises one fact, a query asks about one fact in wording that shares no template with the passages, and every document carrying a passage that realises the queried fact is relevant. Relevance therefore comes from the generative process rather than from an assumption that one document is the answer; relevance sets grow from 3.7 documents per query at k=1 to 33 at k=10. Five hundred documents and two hundred queries per setting.

Against the matched chunk control, cards built from hand-written anchors score +0.023 at k=3 (p=0.12, not significant), +0.099 at k=5, +0.098 at k=7, and +0.188 at k=10 (95 percent interval +0.152 to +0.225, p=0.0001, 135 wins to 34 losses), while using 9.8 units per document against the control's 13.0, which is 25 percent fewer embeddings. The earlier template-anchor version of the same experiment gave +0.109 at k=10. The reason the cruder anchors do better is intuitive once seen: a topical description is a better prototype for an aspect than one rendered sentence carrying one specific fact.

The mechanism behaves as the theory says it should. The mean cosine similarity between a query and its target document's best card stays flat across k (0.476 at k=1, 0.466 at k=10), while the pooled vector's similarity decays monotonically from 0.476 to 0.225. Treating the pooled vector as the normalised mean of k near-orthogonal aspect vectors predicts a decay of one over the square root of k; the measured ratio follows k to the power −0.32, above the prediction at every k. Aspects drawn from natural language are not orthogonal, since all the passages share register and syntax, and encoder similarities occupy a narrow cone that compresses differences. The orthogonality argument is an upper bound on dilution, not a prediction the data confirm.

Capacity or alignment? The LIMIT test

In short: On a benchmark built to break single-vector retrieval, purpose cards recover almost everything, but so does a fine enough blind window. Most of the benefit is capacity.

LIMIT is constructed to expose the representational ceiling of single-vector retrieval: for a fixed embedding dimension, some combinations of documents can never be returned together as a top-k result by any single-vector index, a bound that state-of-the-art embedders visibly hit (Weller, Boratko, Naim, Lee, arXiv:2508.21038, Aug 2025). Each document names a person and lists the things they like; each query asks who likes one specific thing. The small variant has 46 documents with 44 attributes each and 1,000 queries. The task looks trivial and defeats dense retrievers, and it hands over an unambiguous aspect decomposition for free: one card per attribute.

The first reading of this table attributed the 0.674 gain to alignment, because 30-word chunks gained only 0.076. An adversarial re-run, described on the method page, varied only the chunk granularity and found that nDCG@10 is a smooth function of units per document.

Two things follow. Capacity is necessary and explains most of the LIMIT result; the decomposition has to match the query distribution for the remainder to appear. And LIMIT supplies a perfect decomposition for free while its queries are effectively exact-attribute lookups, which is why BM25BM25 (keyword search). The classic keyword ranking formula behind most search engines. It matches exact words, so it wins on names, codes, and quotations, where meaning-based search is weak. wins outright and the hybridHybrid retrieval (reciprocal rank fusion). Running keyword search and meaning-based search side by side and merging the two ranked lists by rank position, so each covers the other's blind spots. only matches it. Real corpora arrive undecomposed, the taxonomy must be designed and may be misaligned, so the honest reading of the table is an upper bound on what alignment can buy. The storage cost is also a property of the corpus: 44 cards per document here, far above the small constant the pattern implies elsewhere.

What this establishes

Independently measured The synthetic and LIMIT results reproduced byte-for-byte or to floating-point precision across two machines and two processor architectures.