Orthoplex Quantization for LSH Hash Computation
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Solution Overview
Problem
Existing locality-sensitive hashing (LSH) techniques face challenges in efficiently computing hash values and achieving balanced distribution of items into buckets, leading to high computational complexity and reduced effectiveness in nearest-neighbor searches, especially when dealing with large datasets.
Innovation Solution
The use of Orthoplex Quantization, which involves inscribing an orthoplex within a multi-dimensional space and employing pseudo-random rotations using the Fast Hadamard Transform to map embedding vectors into buckets, reducing computational complexity and improving distribution balance.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If traditional LSH hashing is used to map embedding vectors to buckets, then the technique can aggregate similar items into partitions, but the computational complexity becomes O(D·2B) which is too high for efficient processing
Solution Approach 1:
The patent segments the high-dimensional embedding space by inscribing an orthoplex (cross-polytope) within the multi-dimensional space. This geometric segmentation divides the space into distinct regions defined by the orthoplex vertices, allowing efficient mapping of embedding vectors to buckets based on their proximity to vertices rather than exhaustive computation across all possible hash combinations.
Solution Approach 2:
The patent changes the parameter representation by using pseudo-random rotations applied to the embedding vectors before mapping. This transformation modifies the coordinate system and distribution characteristics of the data, enabling the use of simpler distance calculations to achieve balanced bucket distribution while reducing computational complexity from O(D·2B) to O(B·2B).
2Reliability
If LSH buckets are created to reduce candidate comparisons, then similar items are aggregated into partitions, but poor distribution balance causes all items to map to the same bucket, rendering hashing ineffective
Solution Approach 1:
The patent introduces asymmetry through pseudo-random rotations that are uniquely applied to different embedding vectors or batches of vectors. This asymmetric transformation ensures that identical embedding vectors do not consistently map to the same bucket due to numerical precision issues, while still maintaining the locality-sensitive property that similar vectors are likely to hash to the same or adjacent buckets. The asymmetry breaks ties and distributes items more evenly across buckets.
3Speed
If simple comparisons are used instead of hashing, then computation is faster, but the ability to efficiently identify nearest neighbors in large datasets is lost
Solution Approach 1:
The patent performs preliminary action by pre-defining the orthoplex geometry and its vertices in the multi-dimensional space before actual hashing occurs. The orthoplex structure, with its symmetric vertices positioned at specific coordinates, is established in advance, allowing embedding vectors to be quickly mapped to nearest vertices using simple distance calculations rather than complex hash function evaluations during query processing.
Data Source
AI summary
Systems and methods are presented for efficient and effective computation of LSH hashing functions for use in various environments, including nearest-neighbor search. A corpus of items is maintained, with each item of the corpus being associated with information identifying one of a plurality of LSH partitions to which the item has been mapped or hashed. In response to an input item, the input item is projected one or more times into an orthoplex inscribed within a multi-dimensional space. For each mapping, a determination is made as to the nearest vertex of the orthoplex to the projected input item, and the LSH partitions associated with the nearest vertex are aggregated and associated with the input item as the input item's LSH partition.