SparseGeometricRAG

CPU-first sparse geometric retrieval for practical top-10 RAG

No transformer inference at retrieval time. No retrieval GPU requirement. No dense document-vector dot products. No external API.

SparseGeometricRAG is a retrieval system built around one systems objective: make the retrieval layer cheap enough to run on ordinary multicore CPU hardware without turning the corpus into a dense embedding database. It uses sparse TF-IDF geometry, fuzzy branch localization, and a tiny signed local residual code. The richer chunk-level evidence is delayed until after routing and shortlist reduction.

The project is not positioned as an accuracy-at-any-cost replacement for the strongest neural retrievers. Its selling point is the quality / latency / hardware tradeoff: useful top-10 retrieval with small structured state, bounded local computation, no retrieval-time transformer stack, and no requirement for a GPU or hosted inference service.

At a glance

Property	Frozen design
Query representation	sparse TF-IDF
Fuzzy memberships per chunk	F = 4
Sparse branch-center support	B = 64 coordinates
Signed residual support	S = 16 coordinates per membership
Weak routing expansion	bounded sparse neighborhood
Large-route shortlist	P = 100 for the frozen six-dataset row
Final RAG output	top 10 chunks
Retrieval-time transformer	none
Retrieval-time GPU	not required
Dense vector per document	not required

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1. Why this design exists

Most modern retrieval systems optimize a learned representation and then optimize the search engine around that representation. SparseGeometricRAG changes the question: can the representation itself be made sufficiently small and local that the retrieval engine no longer needs heavyweight dense-vector machinery?

Figure 1. SparseGeometricRAG changes the cost structure of retrieval. Dense and learned-sparse stacks retain a neural representation stage; the proposed stack remains sparse and CPU-native at retrieval time.

The key design choice is to store a coarse sparse location plus a tiny local directional code, rather than a dense vector for every chunk. The query remains sparse and real-valued, so it supplies fine amplitude information at runtime while the database stores only coarse branch position and signed local deviations.

This gives three practical consequences:

1. the stored geometric state per chunk is controlled by small fixed handles;
2. the decisive local comparison is bounded by only 16 residual coordinates per routed membership; and
3. detailed lexical, semantic-support, and diversity calculations are postponed until the candidate set has already collapsed.

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2. Architecture

2.1 Offline indexing

Figure 2. Offline indexing converts each chunk into sparse lexical support, four fuzzy branch memberships, and a 16-sign local residual code. Branch centers and sparse term graphs are shared structures.

The index is constructed from sparse normalized TF-IDF. A chunk is assigned to its strongest fuzzy branches, each branch is represented by a sparse center, and the chunk's deviation from that center is compressed to a small signed residual code. A bounded sparse term graph provides weak second-order routing support. The result is a compact index consisting of branch postings, membership weights, local sign codes, binary term support, and shared sparse structures.

The frozen structural handles are:

Handle	Frozen value	Role
F	4	fuzzy branch memberships per chunk
B	64	sparse coordinates retained in each branch center
S	16	signed residual coordinates per chunk-branch membership
L	12	sparse chunk terms retained in the frozen geometry path
P	100	large-route shortlist for the final six-dataset row

2.2 Query-time retrieval

Figure 3. Query-time computation is staged. Sparse routing finds candidate branch memberships; the local geometric comparison touches only 16 coordinates; detailed chunk evidence is evaluated only after shortlist reduction.

A query is converted to sparse TF-IDF amplitudes. Weak second-order expansion is used only for routing; it is not a dense semantic representation. Routed branch postings produce candidate memberships. Each candidate is compared locally using the 16 retained signed residual coordinates, then aggregated at the document level. Cheap whole-chunk support provides an early lexical rescue/pre-score. Only a small shortlist proceeds to the richer final evidence calculation.

The final relevance score combines the geometric tail with whole-chunk lexical evidence, sparse semantic support, rare-term coverage, and coordination. Branch quality is estimated from the strongest branch-specific evidences; only high-quality branches are eligible for the small diversity bonus used for ranks 2-10. Rank 1 remains pure relevance.

2.3 What one chunk actually stores

Figure 4. The chunk-local geometric state is deliberately tiny: four fuzzy memberships and sixteen signed residual positions per membership. Branch centers and term-neighbor graphs are shared across chunks.

The asymmetry between document and query representations is intentional. Document residual amplitudes are discarded after their signs and reliability structure have been retained; query amplitudes remain real-valued. The database therefore carries direction, while the query supplies magnitude at runtime.

This differs from dense retrieval, where every chunk generally contributes a full dense vector to the search object. Here, the local geometry is explicitly bounded by F and S, with sparse lexical support retained separately for the rescue and final evidence stages.

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3. Complexity and why the method is fast

3.1 Query-time computation

Let:

• Q be the number of nonzero query terms;
• K_r be the retained routing neighbors per query term;
• C be the number of routed branch-membership hits;
• U be the number of unique routed documents;
• S = 16 be the residual support;
• P be the final shortlist size; and
• L_d be the average binary-support length of a shortlisted chunk.

The main query-time stages are:

Stage	Work
Sparse query construction	O(query tokens)
Weak routing expansion	O(Q K_r)
Posting traversal	O(C)
Local geometric scoring	O(C S) = O(16 C)
Candidate aggregation	O(C log C) in the frozen reference path; O(C) in the preserved stamp-aggregation optimization
Cheap lexical rescue	proportional to routed/gated binary support
Shortlist selection	approximately linear partial selection in U
Final evidence extraction	performed only on the shortlist P
Top-10 construction	small overhead after shortlist features are available

Figure 5. Corpus scale does not imply corpus-wide expensive scoring. Each stage reduces the active set before the next, richer computation is permitted to run.

The decisive bounded term is the local geometric score: only 16 coordinates are consulted for each routed membership. Rich chunk-level evidence is deliberately positioned after routing and shortlist reduction. This is the main reason the system can stay CPU-native without replacing one expensive dense-search primitive by another.

3.2 Storage complexity

Ignoring implementation dtypes and small metadata, the structured state scales conceptually as

O(N F S)
+ O(M B)
+ O(M (K_assoc + K_route))
+ O(total binary term support)

where N is the number of chunks and M is the vocabulary size. The first term is the chunk-local geometric state; the second and third are shared sparse structures; the final term is the whole-chunk binary lexical support.

With the frozen F = 4 and S = 16, the local residual layer retains only 64 residual positions across the four memberships of a chunk. This is the structural reason the method does not require an N x d dense document matrix.

3.3 Preserved post-benchmark optimizations

The repository also preserves a later optimization branch that was developed after the frozen six-dataset benchmark. It introduces:

• stamp-based O(C) candidate aggregation instead of sort-based deduplication; and
• a gate before whole-chunk lexical scanning.

These optimizations are kept separate from the canonical benchmark implementation so that the reported frozen results are not silently changed after the fact.

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4. Hardware and deployment requirements

Figure 6. Retrieval needs only CPU, system RAM, and local corpus/index storage. A GPU may still be used by the generator, but it is not a dependency of the retriever.

The low-cost hardware story is central, not incidental. SparseGeometricRAG is designed for environments where a dedicated retrieval GPU is undesirable or unavailable: inexpensive servers, lab workstations, teaching machines, air-gapped systems, and deployments where accelerator memory is reserved for generation.

Deployment	Retrieval requirement	Typical reason to use it
Laptop / teaching machine	ordinary CPU + modest RAM	development, instruction, small corpora
Commodity workstation/server	multicore CPU + more RAM	larger corpora and batch evaluation
Air-gapped / cost-constrained	CPU + local storage	no hosted model/API dependency
GPU-equipped RAG system	GPU optional for generator	retrieval does not compete for accelerator memory

The claim is not that GPUs are undesirable. The claim is that the retriever is designed so they are optional rather than mandatory.

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5. Why the objective is top-10 RAG

A practical generator usually consumes only a small number of retrieved chunks. For that reason, this repository treats shortlist size as a RAG operating parameter rather than assuming that the setting that maximizes deep recall must also be best for top-10 context selection.

Figure 7. TREC-COVID and SciFact expose opposite regimes. On the large TREC-COVID route, `P = 100` is a useful denoising operating point. On the tiny SciFact route, quality continues improving as aggressive pruning is relaxed.

This is why the final benchmark emphasizes nDCG@10, MRR@10, P@10, R@10, Hit@10, and query latency. Deep recall remains useful as a diagnostic, but it is not allowed to determine the final RAG shortlist by itself.

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6. Frozen six-dataset CPU results

Figure 8. Frozen `100 -> 10` effectiveness across six datasets, with representative median latencies. The result should be read as a quality/cost tradeoff rather than an accuracy-at-any-cost claim.

6.1 OURS: concise benchmark row

Dataset	nDCG@10	MRR@10	P@10	R@10	Hit@10	Median latency (ms)	p95 latency (ms)
SciFact	0.5685	0.5452	0.0737	0.6663	0.6833	0.947	1.031
TREC-COVID	0.5990	0.8252	0.6520	0.0163	1.0000	1.051	1.193
Quora	0.7366	0.7287	0.1124	0.8407	0.8904	120.301	175.494
MS MARCO / DL19	0.3400	0.5189	0.2674	0.0916	0.7209	65.195	142.647
HotpotQA	0.4670	0.6255	0.0964	0.4820	0.7507	28.944	42.568
NQ	0.2579	0.2262	0.0462	0.3983	0.4342	33.686	44.161

The strongest neural systems remain ahead in pure effectiveness on many datasets. SparseGeometricRAG instead targets the low-cost corner of the design space: CPU-first retrieval with bounded sparse computation and no retrieval-time neural inference.

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7. Full benchmark suite

The full suite is intentionally retained. Missing entries are shown as NR; baselines are not removed merely because a compatible value is unavailable.

7.2.1 nDCG@10

Method	SciFact	TREC-COVID	Quora	MS MARCO / DL19	HotpotQA	NQ
Exact TF-IDF	0.5780	0.3738	NR	NR	NR	NR
BM25	0.6650	0.6560	0.7890	0.2280	0.6030	0.3290
MiniLM + FAISS Flat	0.6451	0.4725	0.8756	0.3654	0.4651	0.4387
BGE-base + FAISS Flat	0.7404	0.7807	0.8890	0.4135	0.7260	0.5415
BGE-base + FAISS HNSW	0.7404	0.7807†	0.8890†	0.4135†	0.7260†	0.5415†
BGE-base + FAISS IVF-Flat	0.7255	0.7807†	0.8890†	0.4135†	0.7260†	0.5415†
BGE-base + FAISS IVF-PQ	0.6979	0.7807†	0.8890†	0.4135†	0.7260†	0.5415†
BGE-base + hnswlib HNSW	0.7404	0.7807†	0.8890†	0.4135†	0.7260†	0.5415†
BGE-base + ScaNN	0.6783	0.7807†	0.8890†	0.4135†	0.7260†	0.5415†
Contriever-MS MARCO + FAISS	0.6770	0.5960	0.8650	0.4070	0.6380	0.4980
SPLADE++	0.7040	0.7270	0.8340	0.4330	0.6870	0.5370
Modern ColBERT	0.7645	0.8341	0.8754	0.4499	0.7667	0.6169
OURS — CPU, 100→10	0.5685	0.5990	0.7366	0.3400	0.4670	0.2579

7.2.2 MRR@10

Method	SciFact	TREC-COVID	Quora	MS MARCO / DL19	HotpotQA	NQ
Exact TF-IDF	0.5437	0.5915	NR	NR	NR	NR
BM25	0.6460	0.8530	0.7790	0.1800	0.8030	0.2630
MiniLM + FAISS Flat	0.6110	0.7244	NR	NR	0.4446	NR
BGE-base + FAISS Flat	0.7034	0.9180	0.8823	0.3502	0.8611	0.4924
BGE-base + FAISS HNSW	0.7034	0.9180†	0.8823†	0.3502†	0.8611†	0.4924†
BGE-base + FAISS IVF-Flat	0.6879	0.9180†	0.8823†	0.3502†	0.8611†	0.4924†
BGE-base + FAISS IVF-PQ	0.6615	0.9180†	0.8823†	0.3502†	0.8611†	0.4924†
BGE-base + hnswlib HNSW	0.7034	0.9180†	0.8823†	0.3502†	0.8611†	0.4924†
BGE-base + ScaNN	0.6494	0.9180†	0.8823†	0.3502†	0.8611†	0.4924†
Contriever-MS MARCO + FAISS	0.6207	NR	NR	NR	NR	NR
SPLADE++	0.6699	NR	NR	0.3830	NR	NR
Modern ColBERT	0.7390	0.9533	0.8671	0.3849	0.9188	0.5655
OURS — CPU, 100→10	0.5452	0.8252	0.7287	0.5189	0.6255	0.2262

7.2.3 Precision@10

Method	SciFact	TREC-COVID	Quora	MS MARCO / DL19	HotpotQA	NQ
Exact TF-IDF	0.0797	0.4020	NR	NR	NR	NR
BM25	0.0863	0.6360	~0.1200	NR	NR	NR
MiniLM + FAISS Flat	0.0883	0.5040	0.1337	0.0591	0.0974	0.0770
BGE-base + FAISS Flat	0.0987	0.8300	0.1346	0.0656	0.1515	0.0884
BGE-base + FAISS HNSW	0.0987	0.8300†	0.1346†	0.0656†	0.1515†	0.0884†
BGE-base + FAISS IVF-Flat	0.0973	0.8300†	0.1346†	0.0656†	0.1515†	0.0884†
BGE-base + FAISS IVF-PQ	0.0950	0.8300†	0.1346†	0.0656†	0.1515†	0.0884†
BGE-base + hnswlib HNSW	0.0987	0.8300†	0.1346†	0.0656†	0.1515†	0.0884†
BGE-base + ScaNN	0.0887	0.8300†	0.1346†	0.0656†	0.1515†	0.0884†
Contriever-MS MARCO + FAISS	0.0883	NR	NR	NR	NR	NR
SPLADE++	0.0937	NR	NR	NR	NR	NR
Modern ColBERT	0.0977	0.8820	0.1327	0.0701	0.1548	0.0979
OURS — CPU, 100→10	0.0737	0.6520	0.1124	0.2674	0.0964	0.0462

7.2.4 Recall@10

Method	SciFact	TREC-COVID	Quora	MS MARCO / DL19	HotpotQA	NQ
Exact TF-IDF	0.7135	0.0105	NR	NR	NR	NR
BM25	0.7809	0.0158	0.8854	NR	0.6531	NR
MiniLM + FAISS Flat	0.7833	0.0128	0.9503	0.5676	0.4870	0.6471
BGE-base + FAISS Flat	0.8742	0.0221	0.9574	0.6277	0.7574	0.7469
BGE-base + FAISS HNSW	0.8742	0.0221†	0.9574†	0.6277†	0.7574†	0.7469†
BGE-base + FAISS IVF-Flat	0.8609	0.0221†	0.9574†	0.6277†	0.7574†	0.7469†
BGE-base + FAISS IVF-PQ	0.8441	0.0221†	0.9574†	0.6277†	0.7574†	0.7469†
BGE-base + hnswlib HNSW	0.8742	0.0221†	0.9574†	0.6277†	0.7574†	0.7469†
BGE-base + ScaNN	0.7814	0.0221†	0.9574†	0.6277†	0.7574†	0.7469†
Contriever-MS MARCO + FAISS	0.7868	NR	NR	NR	NR	NR
SPLADE++	0.8230	NR	NR	NR	NR	NR
Modern ColBERT	0.8647	0.0230	0.9516	0.6710	0.7739	0.8239
OURS — CPU, 100→10	0.6663	0.0163	0.8407	0.0916	0.4820	0.3983

7.2.5 Hit@10

Method	SciFact	TREC-COVID	Quora	MS MARCO / DL19	HotpotQA	NQ
Exact TF-IDF	0.7333	0.8600	NR	NR	NR	NR
BM25	0.8033	1.0000	0.9286	NR	NR	NR
MiniLM + FAISS Flat	NR	NR	NR	NR	NR	NR
BGE-base + FAISS Flat	0.8833	NR	NR	NR	NR	NR
BGE-base + FAISS HNSW	0.8833	NR	NR	NR	NR	NR
BGE-base + FAISS IVF-Flat	0.8700	NR	NR	NR	NR	NR
BGE-base + FAISS IVF-PQ	0.8567	NR	NR	NR	NR	NR
BGE-base + hnswlib HNSW	0.8833	NR	NR	NR	NR	NR
BGE-base + ScaNN	0.7900	NR	NR	NR	NR	NR
Contriever-MS MARCO + FAISS	0.7967	NR	NR	NR	NR	NR
SPLADE++	0.8333	NR	NR	NR	NR	NR
Modern ColBERT	0.8100	NR	NR	NR	NR	NR
OURS — CPU, 100→10	0.6833	1.0000	0.8904	0.7209	0.7507	0.4342

7.2.6 Median query latency (ms)

Method	SciFact	TREC-COVID	Quora	MS MARCO / DL19	HotpotQA	NQ
Exact TF-IDF	6.197	50.185	NR	NR	NR	NR
BM25	0.561	4.912	NR	NR	NR	NR
MiniLM + FAISS Flat	NR	NR	NR	NR	NR	NR
BGE-base + FAISS Flat	10.013	NR	NR	NR	NR	NR
BGE-base + FAISS HNSW	10.369	NR	NR	NR	NR	NR
BGE-base + FAISS IVF-Flat	10.050	NR	NR	NR	NR	NR
BGE-base + FAISS IVF-PQ	13.910	NR	NR	NR	NR	NR
BGE-base + hnswlib HNSW	10.594	NR	NR	NR	NR	NR
BGE-base + ScaNN	9.947	NR	NR	NR	NR	NR
Contriever-MS MARCO + FAISS	7.472	NR	NR	NR	NR	NR
SPLADE++	13.493	NR	NR	NR	NR	NR
Modern ColBERT	62.671	NR	NR	NR	NR	NR
OURS — CPU, 100→10	0.947	1.051	120.301	65.195	28.944	33.686

7.2.7 p95 query latency (ms)

Method	SciFact	TREC-COVID	Quora	MS MARCO / DL19	HotpotQA	NQ
Exact TF-IDF	14.605	56.716	NR	NR	NR	NR
BM25	5.645	7.336	NR	NR	NR	NR
MiniLM + FAISS Flat	NR	NR	NR	NR	NR	NR
BGE-base + FAISS Flat	15.398	NR	NR	NR	NR	NR
BGE-base + FAISS HNSW	12.925	NR	NR	NR	NR	NR
BGE-base + FAISS IVF-Flat	16.303	NR	NR	NR	NR	NR
BGE-base + FAISS IVF-PQ	18.496	NR	NR	NR	NR	NR
BGE-base + hnswlib HNSW	13.293	NR	NR	NR	NR	NR
BGE-base + ScaNN	11.858	NR	NR	NR	NR	NR
Contriever-MS MARCO + FAISS	9.628	NR	NR	NR	NR	NR
SPLADE++	19.156	NR	NR	NR	NR	NR
Modern ColBERT	70.994	NR	NR	NR	NR	NR
OURS — CPU, 100→10	1.031	1.193	175.494	142.647	42.568	44.161

Notes. NR means “not reported under a compatible metric / protocol in the current ledger.” † indicates that, outside SciFact, the BGE ANN-backend rows mirror the BGE-base representation-level effectiveness reference rather than a separately rerun backend-specific effectiveness experiment.

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8. Interpreting the latency numbers

Speed comparisons in retrieval are easy to misstate. ANN papers frequently report search-only latency after a dense query embedding already exists, whereas a deployed RAG request pays for query representation, retrieval, shortlist scoring, and final selection.

This repository therefore follows two rules:

1. do not silently compare ANN-only latency with end-to-end retrieval latency; and
2. do not fill missing latency cells using measurements from incompatible hardware or protocols.

The full provenance policy is documented in docs/BASELINE_SUITE.md and docs/LITERATURE_AND_SPEED.md. The frozen OURS timings include the retrieval path used by the reported experiment. For the MS MARCO column, OURS is the 43-query TREC-DL19 run on the full 8.84M-passage corpus; published model-reference values in that column may use MS MARCO dev where applicable.

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9. Repository organization

The repository is deliberately split between reusable retrieval code, clean benchmark runners, the full-scale experimental campaign, and frozen outputs.

Path	Purpose
geomretrieval/	reusable sparse geometric retriever
experiments/beir/	BEIR runners and pool-sweep experiments
experiments/msmarco_scale/	full MS MARCO scale campaign
experiments/postbenchmark_optimizations/	preserved later optimization branch
results/	frozen JSON outputs and benchmark artifacts
configs/	reproduction handles
baselines/	baseline utilities
scripts/	runnable helpers
docs/	method, RAG protocol, baseline, speed, and reproducibility notes
tests/	smoke tests
manifests/	corpus / run manifests

The final six-dataset artifacts are under results/final_100_to_10/. The later stamp-aggregation and lexical-gating optimization results are preserved separately and are not used to rewrite the frozen benchmark row.

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10. Reproducibility

The repository contains the exact frozen result JSONs, benchmark scripts, configuration handles, and tests used to reconstruct the final evaluation. The reference package was checked with the project smoke tests before release.

For a clean reproduction path, start with:

1. docs/METHOD.md - algorithmic description;
2. docs/RAG_PROTOCOL.md - top-10 evaluation protocol;
3. docs/REPRODUCIBILITY.md - environment and run guidance;
4. docs/BASELINE_SUITE.md - baseline/provenance policy; and
5. results/final_100_to_10/ - frozen final outputs.

The code is intentionally CPU-first. Baseline packages that depend on neural encoders or ANN libraries are listed separately from the core requirements.

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11. Scope of the claim

What this repository claims

• a CPU-first retrieval architecture with no retrieval-time transformer inference;
• no requirement for a dense vector per document or a GPU-based retrieval service;
• fixed small structural handles (F = 4, B = 64, S = 16) controlling local geometry;
• local geometric scoring bounded by O(CS) with S = 16 fixed;
• deliberate postponement of richer chunk-level computation until after routing and shortlist reduction;
• a practical top-10 RAG operating point validated across six datasets;
• complete benchmark tables rather than selective reporting of only OURS.

What it does not claim

• dominance over the strongest neural retrievers in pure effectiveness;
• that every latency cell in the literature is directly comparable across hardware and protocol;
• that one shortlist size is mathematically optimal for every dataset or route size;
• that deep recall is irrelevant. It is retained as a diagnostic, but it is not the sole deployment objective.

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12. License

This repository is released under the Fair Noncommercial Research License selected on the Hugging Face repository. Check the repository license metadata and license text before redistribution or commercial use.