Optimal Hashing-based Time-Space Trade-offs for Approximate Near Neighbors (1608.03580v2)

Abstract: [See the paper for the full abstract.] We show tight upper and lower bounds for time-space trade-offs for the $c$-Approximate Near Neighbor Search problem. For the $d$-dimensional Euclidean space and $n$-point datasets, we develop a data structure with space $n^{1 + \rho_u + o(1)} + O(dn)$ and query time $n^{\rho_q + o(1)} + d n^{o(1)}$ for every $\rho_u, \rho_q \geq 0$ such that: \begin{equation} c^2 \sqrt{\rho_q} + (c^2 - 1) \sqrt{\rho_u} = \sqrt{2c^2 - 1}. \end{equation} This is the first data structure that achieves sublinear query time and near-linear space for every approximation factor $c > 1$, improving upon [Kapralov, PODS 2015]. The data structure is a culmination of a long line of work on the problem for all space regimes; it builds on Spherical Locality-Sensitive Filtering [Becker, Ducas, Gama, Laarhoven, SODA 2016] and data-dependent hashing [Andoni, Indyk, Nguyen, Razenshteyn, SODA 2014] [Andoni, Razenshteyn, STOC 2015]. Our matching lower bounds are of two types: conditional and unconditional. First, we prove tightness of the whole above trade-off in a restricted model of computation, which captures all known hashing-based approaches. We then show unconditional cell-probe lower bounds for one and two probes that match the above trade-off for $\rho_q = 0$, improving upon the best known lower bounds from [Panigrahy, Talwar, Wieder, FOCS 2010]. In particular, this is the first space lower bound (for any static data structure) for two probes which is not polynomially smaller than the one-probe bound. To show the result for two probes, we establish and exploit a connection to locally-decodable codes.

• The paper presents a hashing-based data structure that achieves optimal time-space trade-offs for ANN by enabling sublinear query times with near-linear space complexity.
• It defines a critical trade-off equation relating space and query time exponents, which guides the optimal balancing for different approximation factors.
• Numerical examples, such as for c=2, illustrate practical configurations with space complexities ranging from n^1.14 to n^1.77, significantly improving upon earlier LSH methods.

Optimal Hashing-based Time-Space Trade-offs for Approximate Near Neighbors

This paper by Alexandr Andoni et al. focuses on establishing upper and lower bounds for the time-space trade-offs involved in solving the $c$-Approximate Near Neighbor Search (ANN) problem in high-dimensional Euclidean spaces. It outlines a rigorous approach to create efficient data structures for ANN, which is pivotal for applications in similarity search, optimization, and cryptanalysis.

The authors present a new hashing-based data structure that can achieve sublinear query times while maintaining near-linear space consumption for any approximation factor $c > 1$. This stands as a significant improvement over previous works, such as those by Indyk and Motwani (1998) involving locality-sensitive hashing (LSH). Unlike these earlier methods, which often result in exponential dependencies on the space and time parameters with respect to dimensionality, the solutions proposed here provide a balanced trade-off, making them practically viable for large datasets characteristic of many real-world applications.

Key Insights and Methods

1. Hashing-based Data Structures: The paper introduces a data structure that efficiently processes ANN queries by using hashing methods tuned to the underlying geometry of the dataset spaces. This enhancement is grounded on spherical locality-sensitive filtering and data-dependent hashing techniques.
2. Trade-off Equation: The authors define a critical relation:

$$c^2 \sqrt{\rho_q} + (c^2 - 1) \sqrt{\rho_u} = \sqrt{2c^2 - 1},$$

where $\rho_u$ and $\rho_q$ denote the exponents of space and query time, respectively. This equation characterizes the optimal trade-off achievable with their data structure.

1. Numerical Examples: For $c = 2$, several scenarios are discussed, including:
   • Space complexity $n^{1.77}$ with query time $n^{o(1)}$.
   • Space complexity $n^{1.14}$ and query time $n^{0.14}$.
   • Space usage $n^{1 + o(1)}$ with query time $n^{0.43}$.

These examples illustrate optimal configurations for differing priorities between space and query time, where improvements over previous data structures are evident.

Experimental Design and Results

The authors unified two prior preprints and compared their theoretical findings against existing results. Using both conditional and unconditional lower bounds, it provides robust evidence that matches the theoretical expectations for sublinear query times and near-linear space utilization. The results showcased that for a single cell-probe model, the space bounds were $n^{\left(\frac{c}{c - 1}\right)^2 - o(1)}$, and similar results were demonstrated for two cell probes.

Implications and Speculations

This paper contributes theoretically sound frameworks that can be further explored to influence the future design of AI-driven applications requiring rapid and efficient data retrieval processes. From a practical standpoint, simplifying these sophisticated data structures will be a crucial step toward integrating them into real-world systems. Moreover, there's an open question regarding their extension to other metric spaces and possibly further improvements by utilizing locally-decodable codes (LDC).

Conclusion

Overall, the paper presents substantial advancements in the ANN problem, offering promising strategies for balancing space and query efficiency. The proposed trade-offs are optimal within the hashing-based frameworks, leading to breakthroughs in scaling LSH to large-scale applications. The ongoing question of bridging these algorithmic results with pragmatic, easily implementable solutions remains an exciting avenue for further research.