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Maximum Inner-Product Search

Find a stored vector whose raw inner product with a query vector is maximal.

Version
v1 · 2026-10-03 · History
Domain-specific #
13422
Aliases
MIPS, Max Inner Product Search

Core Idea

Maximum inner-product search (MIPS) selects a stored vector \(x\) maximizing \(\langle q,x\rangle\) for query \(q\). The necessary roles are a stored vector set, a query vector and this raw dot-product objective. Top-\(k\), approximate answers and indexes are extensions or solution methods, not defining requirements.[ref-49bd71d4e528][ref-df9730bf35d3]

Scope of Application

Matrix-factorization recommendation can rank item embeddings against a user embedding by dot product. Any dense retrieval model using query–item dot product has the same query structure; systems using a different score do not. Exact flat scanning and approximate indexed search both solve versions of MIPS.[ref-49bd71d4e528][ref-df9730bf35d3]

Clarity

Raw inner product is not Euclidean distance or necessarily cosine similarity. Candidate norms can change the ranking; normalizing them may change the target objective. Equal candidate norms or a proved order-preserving transformation can connect MIPS to nearest-neighbor procedures under conditions.[ref-49bd71d4e528][ref-c6aad91194c6]

Manages Complexity

Specifying the score separately from the solver allows an exact scan, approximate hash or transformed nearest-neighbor method to be assessed against the same target. An approximate answer requires its own recall/quality qualification and should not be mislabeled exact.[ref-c6aad91194c6][ref-df9730bf35d3]

Abstract Reasoning

For candidates \(x_i,x_j\), their relative score is the sign of \(\langle q,x_i-x_j\rangle\). Euclidean distance adds candidate-norm terms, explaining why nearest-neighbor and MIPS can diverge. The proposed strict DAG genus is live Optimization; nearest-neighbor search is a conditional reduction target, not the universal parent.[^ref-49bd71d4e528]

Knowledge Transfer

Identify candidates, query and whether magnitude is part of intended relevance. Then choose a search implementation and test whether normalization, quantization or asymmetric mapping preserves the original dot-product ordering under actual bounds.[ref-49bd71d4e528][ref-df9730bf35d3]

[^ref-49bd71d4e528]: Behnam Neyshabur and Nathan Srebro, “On Symmetric and Asymmetric LSHs for Inner Product Search,” PMLR 37 (2015), original full paper §1 Eq. (1) and §2 directly checked. [^ref-c6aad91194c6]: Anshumali Shrivastava and Ping Li, “Asymmetric LSH (ALSH) for Sublinear Time Maximum Inner Product Search (MIPS),” NeurIPS (2014), original abstract/§1 directly checked. [^ref-df9730bf35d3]: Faiss documentation and index summary, project-maintainer descriptions of exact IndexFlatIP and approximate indexes directly checked.

Relationships to Other Abstractions

Local relationship map for Maximum Inner-Product SearchParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.MaximumInner-Product SearchDOMAINPrime abstraction: Optimization — is a kind ofOptimizationPRIME

Current abstraction Maximum Inner-Product Search Domain-specific

Parents (1) — more general patterns this builds on

  • Maximum Inner-Product Search is a kind of Optimization Prime

    MIPS is finite-candidate optimization under a specified inner-product objective.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

Maximum Inner-Product Search sits in a moderately populated region (58th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Codes, Matrices & Combinatorial Problems (30 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08