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Fiber-Navigable Search: A Geometric Approach to Filtered ANN

Thuong Dang

Abstract

We present a geometric framework for filtered approximate nearest neighbor (ANN) search. Filtering a proximity graph by a metadata predicate produces a subgraph, a fiber, whose connectivity and geometry can differ sharply from the full graph. Using local signals, we propose a two-phase search algorithm that combines full-graph exploration with filtered-neighbor descent when the local geometry is favorable. These signals also classify search failures into three regimes: topological cuts, geometric folds, and genuine basins. A key observation is that all three share a common resolution: restarting the search in a fiber-present cluster near the query. To support this, we introduce a lightweight anchor structure that identifies such regions and restarts the search accordingly. We show empirically that the method outperforms FAISS HNSW on filtered search and the three failure regimes separate cleanly and shift predictably with filter selectivity.

Fiber-Navigable Search: A Geometric Approach to Filtered ANN

Abstract

We present a geometric framework for filtered approximate nearest neighbor (ANN) search. Filtering a proximity graph by a metadata predicate produces a subgraph, a fiber, whose connectivity and geometry can differ sharply from the full graph. Using local signals, we propose a two-phase search algorithm that combines full-graph exploration with filtered-neighbor descent when the local geometry is favorable. These signals also classify search failures into three regimes: topological cuts, geometric folds, and genuine basins. A key observation is that all three share a common resolution: restarting the search in a fiber-present cluster near the query. To support this, we introduce a lightweight anchor structure that identifies such regions and restarts the search accordingly. We show empirically that the method outperforms FAISS HNSW on filtered search and the three failure regimes separate cleanly and shift predictably with filter selectivity.

Paper Structure

This paper contains 63 sections, 1 theorem, 5 equations, 6 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Paper outline.
  4. Related Work
  5. Geometric Framework
  6. Problem Setting
  7. Data.
  8. Proximity graph.
  9. Fibers over Metadata
  10. Local Signals
  11. Fiber density.
  12. Drift.
  13. Algorithm Overview
  14. Beam search (cluster-level navigation).
  15. Drift-guided search (cluster- and fiber-level navigation).
  16. ...and 48 more sections

Key Result

Lemma 4.1

Let each of the $n$ points be assigned to exactly one of $K$ clusters, and let each point carry metadata across $F$ fields. The total storage for members, and cluster_index is $O(n \cdot F)$, independent of the vocabulary sizes $|V_f|$ and the number of clusters $K$. $\blacktriangleleft$$\blacktrian

Figures (6)

  • Figure 1: Fibers over metadata.
  • Figure 2: Drift and potential best descent direction (bold lines) as filtered neighborhood diagnostics.
  • Figure 3: Local descent.
  • Figure 4: Transition function to glue charts in differential geometry.
  • Figure 5: Discrete and continuous fiber structures.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Lemma 4.1: Atlas storage
  • proof