arXiv:2610.33224v1 Announce Type: new Abstract: In the age of AI, privacy-preserving approximate nearest neighbor (ANN) search over high-dimensional embeddings has emerged as a critical yet frustratingly challenging domain, dominated by tradeoffs between speed, privacy, and accuracy. Traditional tree and graph-based indexing structures (e.g., HNSW, ScaNN) rely fundamentally on non-linear operations that are computationally intractable under Fully Homomorphic Encryption (FHE) or blind server-side constraints. In this paper, we introduce Holo-NN (Holographic Nearest Neighbor Search), an algebraic framework that replaces data-dependent branching with a single linear projection. Our key observation is that while distances across a database are dense, neighborhood membership is inherently sparse: only a handful of points ever lie close to a query. By pairing locality-preserving feature transforms with Compressive Sensing, Holo-NN folds the entire database offline into a compact holographic sensing matrix. At query time, encrypted search reduces to a single, non-interactive matrix-vector multiplication---requiring no FHE bootstrapping, no multi-round interaction, and no client-side index state. This turns private retrieval into an embarrassingly parallel, accuracy-tunable computation that runs out of the box on both CPUs and GPUs, scaling efficiently to production databases. We offer a comparison against other state-of-the-art Private ANN algorithms to help situate the method in the broader context of production-grade private retrieval.