Graph-based approaches to nearest neighbor search are popular and powerful tools for handling large datasets in practice, but they have limited theoretical guarantees. We study the worst-case performance of recent graph-based approximate nearest neighbor search algorithms, such as HNSW, NSG and DiskANN. For DiskANN, we show that its "slow preprocessing" version provably supports approximate nearest neighbor search query with constant approximation ratio and poly-logarithmic query time, on data sets with bounded "intrinsic" dimension. For the other data structure variants studied, including DiskANN with "fast preprocessing", HNSW and NSG, we present a family of instances on which the empirical query time required to achieve a "reasonable" accuracy is linear in instance size. For example, for DiskANN, we show that the query procedure can take at least 0.1n steps on instances of size nbefore it encounters any of the 5nearest neighbors of the query.

We present a new algorithm for the approximate near neighbor problem that combines classical ideas from group testing with locality-sensitive hashing (LSH). We reduce the near neighbor search problem to a group testing problem by designating neighbors as "positives," non-neighbors as "negatives," and approximate membership queries as group tests.

Graph search is one of the most successful algorithmic trends in near neighbor search. Severalofthemostpopular andempirically successful algorithms are,at their core, a greedy walk along a pruned near neighbor graph.

For large-scale databases the sequential scan with theO(nd) complexity is not feasible, and the efficient approximate methods arerequired.

Ifthese partitions rank very well, we can setk0 very low,and search through fewpartitions while still achieving great MIPS recall.

Graph search is one of the most successful algorithmic trends in near neighbor search. Several of the most popular and empirically successful algorithms are, at their core, a greedy walk along a pruned near neighbor graph. However, graph traversal applications often suffer from poor memory access patterns, and near neighbor search is no exception to this rule. Our measurements show that popular search indices such as the hierarchical navigable small-world graph (HNSW) can have poor cache miss performance. To address this issue, we formulate the graph traversal problem as a cache hit maximization task and propose multiple graph reordering as a solution. Graph reordering is a memory layout optimization that groups commonly-accessed nodes together in memory.

Geometric methods rely on tensors that can be encoded using a symbolic formula and data arrays, such as kernel and distance matrices. We present an extension for standard machine learning frameworks that provides comprehensive support for this abstraction on CPUs and GPUs: our toolbox combines a versatile, transparent user interface with fast runtimes and low memory usage. Unlike general purpose acceleration frameworks such as XLA, our library turns generic Python code into binaries whose performances are competitive with state-of-the-art geometric libraries - such as FAISS for nearest neighbor search - with the added benefit of flexibility. We perform an extensive evaluation on a broad class of problems: Gaussian modelling, K-nearest neighbors search, geometric deep learning, non-Euclidean embeddings and optimal transport theory. In practice, for geometric problems that involve 1k to 1M samples in dimension 1 to 100, our library speeds up baseline GPU implementations by up to two orders of magnitude.

We present a new algorithm for the approximate near neighbor problem that combines classical ideas from group testing with locality-sensitive hashing (LSH). We reduce the near neighbor search problem to a group testing problem by designating neighbors as positives, non-neighbors as negatives, and approximate membership queries as group tests.