Optimal Data-Dependent Hashing for Approximate Near Neighbors (1501.01062v3)

Abstract: We show an optimal data-dependent hashing scheme for the approximate near neighbor problem. For an $n$-point data set in a $d$-dimensional space our data structure achieves query time $O(d n^{\rho+o(1)})$ and space $O(n^{1+\rho+o(1)} + dn)$, where $\rho=\tfrac{1}{2c^2-1}$ for the Euclidean space and approximation $c>1$. For the Hamming space, we obtain an exponent of $\rho=\tfrac{1}{2c-1}$. Our result completes the direction set forth in [AINR14] who gave a proof-of-concept that data-dependent hashing can outperform classical Locality Sensitive Hashing (LSH). In contrast to [AINR14], the new bound is not only optimal, but in fact improves over the best (optimal) LSH data structures [IM98,AI06] for all approximation factors $c>1$. From the technical perspective, we proceed by decomposing an arbitrary dataset into several subsets that are, in a certain sense, pseudo-random.

• The paper presents an optimal data-dependent hashing scheme that outperforms traditional LSH methods for ANN search.
• It adopts a novel decomposition of data into pseudo-random subsets to achieve optimal query time of O(d·n^(ρ+o(1))) and space complexity.
• The approach offers both theoretical advancements and practical benefits for large-scale, high-dimensional data retrieval applications.

Optimal Data-Dependent Hashing for Approximate Near Neighbors

The paper "Optimal Data-Dependent Hashing for Approximate Near Neighbors," authored by Alexandr Andoni and Ilya Razenshteyn, provides significant contributions to the field of data structures for approximate nearest neighbor (ANN) search. By introducing an optimal data-dependent hashing scheme, the research advances the theoretical and practical understanding of this area, particularly when compared to traditional Locality Sensitive Hashing (LSH) methods.

Technical Summary

The authors present a novel data structure for the ANN problem that achieves both optimal query time and space complexity while relying on data-dependent hashing. In this structure, for a dataset of $n$ points in a $d$-dimensional Euclidean space, the query time is $O(d \cdot n^{\rho+o(1)})$ and the space complexity is $O(n^{1+\rho+o(1)} + d \cdot n)$, where $\rho = \tfrac{1}{2c^2-1}$ and $c > 1$ denotes the approximation factor. The paper also extends these results to the Hamming space, achieving an exponent of $\rho = \tfrac{1}{2c-1}$. The proposed approach therefore not only matches optimal data-independent LSH schemes but, crucially, surpasses them for all approximation factors $c > 1$.

A key technical contribution of this paper is the decomposition of a dataset into subsets that exhibit pseudo-random characteristics. This decomposition underpins the effectiveness of the data-dependent hashing scheme, which adapts to the specific input data distribution to optimize performance. While previous work, cited as \cite{ainr-blsh-14}, provided foundational insights into the potential superiority of data-dependent hashing, this paper extends those results to achieve optimality, addressing limitations of the existing methods.

Practical and Theoretical Implications

This research holds substantial implications for both theoretical exploration and practical applications in computational geometry and related fields. The optimality in terms of query time and space complexity makes this hashing scheme highly applicable for large-scale machine learning tasks and databases where fast retrieval of approximate neighbors is crucial. Particularly in high-dimensional data spaces, the improvements over traditional LSH methods could yield considerable efficiency gains.

From a theoretical perspective, the results in this paper open avenues for further exploration into data-dependent methods for other computational problems. The demonstrated ability to refine data-dependent hashing to outperform classical methods across a range of approximation factors suggests potential for broader applications and enhancements in algorithmic performance.

Future Directions

The framework and results presented in this research prompt several speculative avenues for future inquiry. A natural progression would be to incorporate these findings into real-world systems and assess empirical performance across diverse datasets and application domains. Moreover, extending the principles of data-dependent decomposition to other data structures or computational paradigms could bear fruitful advancements.

In summary, the paper by Andoni and Razenshteyn creates an inflection point for the paper of approximate nearest neighbors, highlighting the advantages of data-dependent hashing over conventional LSH approaches and setting a benchmark for future academic and practical endeavors in the domain of efficient data retrieval systems.