Semantic Regions

HNSW as Hierarchical Clustering

HNSW builds a multi-level graph where each level has fewer, more “central” nodes. These hubs emerge naturally and act as unsupervised cluster centroids. CVX assigns every node to its nearest hub via greedy descent — $\sim N/M^L$ regions at level $L$.

💡 No training required
Unlike k-means or DBSCAN, HNSW regions emerge from index construction. No hyperparameter tuning, no iterative optimization. O(N) single-pass assignment.

Setup

import chronos_vector as cvx
import numpy as np
np.random.seed(42)

index = cvx.TemporalIndex(m=4, ef_construction=50, ef_search=50)
for cluster in range(3):
    center = np.random.randn(16).astype(np.float32) * 2
    for entity in range(10):
        for t in range(20):
            drift = np.random.randn(16).astype(np.float32) * 0.1 * t
            vec = center + drift + np.random.randn(16).astype(np.float32) * 0.3
            index.insert(cluster * 100 + entity, t * 86400, vec.tolist())
print(f"Index: {len(index)} points")

Index: 600 points

Region Discovery and Purity

assignments = index.region_assignments(level=1)
for hub_id, members in sorted(assignments.items(), key=lambda x: -len(x[1]))[:3]:
    clusters = {}
    for eid, ts in members:
        clusters[eid // 100] = clusters.get(eid // 100, 0) + 1
    dominant = max(clusters.values())
    print(f"  Hub {hub_id}: {len(members)} members, purity={dominant/len(members):.0%}")

  Hub 38: 32 members, purity=100%
  Hub 161: 28 members, purity=100%
  Hub 301: 26 members, purity=100%

Region Trajectory (EMA-Smoothed)

Track membership evolution with exponential smoothing: $\mathbf{s}(t) = \alpha \cdot \mathbf{c}(t) + (1-\alpha) \cdot \mathbf{s}(t-1)$

Distributional Distances

Compare region distributions with geometry-aware metrics:

Metric	Formula	Range
Fisher-Rao	$2\arccos(\sum_i \sqrt{p_i q_i})$	$[0, \pi]$
Hellinger	$\frac{1}{\sqrt{2}}\sqrt{\sum(\sqrt{p_i}-\sqrt{q_i})^2}$	$[0, 1]$
Wasserstein	Optimal transport with region centroids	$[0, \infty)$