The curse of dimensionality—the challenge of analyzing data in high-dimensional spaces—forces indexing techniques for vector search to prioritize efficiency and approximation over exact results. In high dimensions, data becomes sparse, and distance metrics lose discriminative power, making traditional tree-based structures (e.g., k-d trees) ineffective. These methods rely on partitioning space, which fails when dimensions are too high because partitions become too numerous or overlapping. To address this, modern indexing techniques adopt strategies like approximate nearest neighbor (ANN) algorithms, dimensionality reduction, and specialized data structures that trade precision for speed.
One key adaptation is the use of probabilistic or hashing-based methods. Locality-Sensitive Hashing (LSH), for example, maps similar vectors into the same "buckets" with high probability using random projections. This avoids exhaustive comparisons by narrowing the search to a subset of candidates. Similarly, graph-based indexes like Hierarchical Navigable Small World (HNSW) create layered proximity graphs, enabling fast traversal by exploiting the small-world property (few hops between nodes). These methods sidestep the inefficiency of exact partitioning by focusing on local proximity, which remains meaningful even in high dimensions. For instance, FAISS, a popular vector search library, combines inverted file indices (clustering vectors into cells) with product quantization (compressing vectors into smaller codes) to reduce computational and memory overhead.
Another design consideration is dimensionality reduction or metric learning. Techniques like Principal Component Analysis (PCA) or random projections reduce the effective dimensionality of data, making indexing more manageable. However, this risks losing information, so some systems integrate learned embeddings (e.g., via neural networks) to preserve semantic relationships. Additionally, distance computation optimizations—such as using cosine similarity instead of Euclidean distance for normalized vectors—help mitigate the "uniformity" of distances in high dimensions. Overall, the curse of dimensionality pushes indexing techniques to balance resource constraints, accuracy, and scalability, favoring approximate, compressed, or learned representations over exact geometric partitioning.