How do false positives and false negatives manifest in ANN search results, and how do they relate to the concepts of precision and recall respectively in a vector search evaluation?

False positives and false negatives in Approximate Nearest Neighbor (ANN) search occur due to the trade-offs inherent in balancing speed and accuracy. A false positive happens when the ANN system returns an item that is not truly a nearest neighbor to the query vector. For example, in a product recommendation system, a false positive might be a shoe shown to a user searching for “running shoes,” even though the result is a formal dress shoe with no relevance. Conversely, a false negative occurs when a true nearest neighbor exists in the dataset but is not returned in the search results. For instance, a relevant running shoe matching the query might be excluded due to approximations in the ANN algorithm, such as using hierarchical navigable small world (HNSW) graphs or product quantization, which prioritize speed over exhaustive checks.

These errors directly map to the evaluation metrics precision and recall. Precision measures the proportion of retrieved results that are relevant (i.e., true positives). A high number of false positives lowers precision because irrelevant items dominate the results. For example, if an image search returns 10 images and 4 are unrelated, precision drops to 60%. Recall, on the other hand, measures the fraction of all relevant items in the dataset that are successfully retrieved. False negatives reduce recall because they represent missed relevant items. If a dataset contains 50 relevant images for a query but the ANN returns only 30 of them, recall is 60%. In ANN systems, adjusting parameters like the number of probes in IVF indexes or the search radius in locality-sensitive hashing (LSH) can influence this balance: stricter criteria reduce false positives (raising precision) but risk increasing false negatives (lowering recall), while looser settings do the opposite.

When evaluating ANN systems, developers must prioritize based on use-case requirements. For applications like legal document retrieval, high recall is critical—missing a relevant document (false negative) could have serious consequences. Here, techniques like increasing the search scope or using higher-quality embeddings might be prioritized. Conversely, in real-time recommendation engines, high precision is often more important to avoid frustrating users with irrelevant results. Adjusting parameters like the number of neighbors (k) or using post-filtering can help. Understanding this precision-recall trade-off allows developers to tune ANN systems effectively, whether through algorithm selection (e.g., choosing HNSW for better recall or tree-based methods for precision) or by optimizing hyperparameters during index construction and query execution.