Inner Product Similarity Search using Compositional Codes

This paper addresses the nearest neighbor search problem under inner product similarity and introduces a compact code-based approach. The idea is to approximate a vector using the composition of several elements selected from a source dictionary and to represent this vector by a short code composed of the indices of the selected elements. The inner product between a query vector and a database vector is efficiently estimated from the query vector and the short code of the database vector. We show the superior performance of the proposed group $M$-selection algorithm that selects $M$ elements from $M$ source dictionaries for vector approximation in terms of search accuracy and efficiency for compact codes of the same length via theoretical and empirical analysis. Experimental results on large-scale datasets ($1M$ and $1B$ SIFT features, $1M$ linear models and Netflix) demonstrate the superiority of the proposed approach.

💡 Research Summary

The paper tackles the problem of approximate nearest‑neighbor (ANN) search under inner‑product similarity, which is central to many large‑scale vision and recommendation tasks. Instead of using traditional distance‑based quantization, the authors propose a compact coding scheme that represents each database vector as a sum of M dictionary elements. The indices of the selected elements form a short code of length M·log K bits, where K is the size of each dictionary. Because the inner product distributes over addition, the similarity between a query q and a database vector x can be approximated by summing the pre‑computed inner products ⟨q, c⟩ for the M dictionary atoms indicated by the code, yielding an O(M) evaluation cost per vector after an initial O(M K d) pre‑computation of the query‑dictionary table.

Three increasingly expressive models are introduced. The baseline is a standard k‑means quantizer (1‑combination), where each vector is approximated by a single centroid. The second model, M‑selection, allows the same dictionary to be used M times with repetitions, improving approximation without increasing code length. The third and most novel model, group M‑selection, learns M distinct dictionaries C₁,…,C_M and selects one atom from each, effectively constructing a compositional dictionary of size K^M while keeping the code length unchanged. This “code‑length‑preserving expansion of the dictionary” yields substantially tighter vector approximations.

Mathematically, the learning objective is to minimize the Frobenius norm ‖X −