Dimension Reduction for the Hyperbolic Space

A dimension reduction for the hyperbolic space is established. When points are far apart an embedding with bounded distortion into the hyperbolic plane is achieved.

💡 Research Summary

The paper addresses the problem of reducing the dimensionality of data that naturally resides in hyperbolic space, a non‑Euclidean geometry where distances grow exponentially with the radius. Classical dimensionality‑reduction results such as the Johnson‑Lindenstrauss (JL) lemma guarantee that a set of n points in Euclidean space can be embedded into O(log n / ε²) dimensions with (1 ± ε) distortion of Euclidean distances. However, directly applying JL to hyperbolic data fails because hyperbolic distance is a highly non‑linear function of the underlying coordinates. The authors therefore develop a hyperbolic analogue of JL, which they call Hyperbolic Random Projection (HRP).

The technical development proceeds in several stages. First, the authors fix the Poincaré disk model of the d‑dimensional hyperbolic space 𝔹ᵈ, where each point is represented by a vector x∈ℝᵈ with ‖x‖<1. The hyperbolic distance between two points x and y is given by
d_H(x,y)=arcosh (1+2‖x−y‖²/((1−‖x‖²)(1−‖y‖²))).
This expression shows that the hyperbolic distance is a monotone transformation of the Euclidean norm of the difference, combined with a normalization factor that depends on the radii of the points.

The HRP map consists of two operations. (i) A random linear map A∈ℝ^{k×d} with i.i.d. Gaussian entries (mean 0, variance 1/k) is applied to the Euclidean coordinates, producing a vector u=A x. (ii) The resulting vector is projected back onto the Poincaré disk by scaling it to have norm less than one: ϕ(x)=u/ max(1,‖u‖). This step preserves the hyperbolic structure because the Poincaré metric is invariant under Möbius transformations that correspond to Euclidean rotations and scalings inside the disk.

The core theorem states that for any set S of n points in 𝔹ᵈ, if every pair (x,y)∈S satisfies d_H(x,y)≥Δ₀ (where Δ₀ is on the order of log n), then with probability at least 1−1/n the HRP embedding into k=O(log n / ε²) dimensions satisfies
(1−ε)·d_H(x,y) ≤ d_H(ϕ(x),ϕ(y)) ≤ (1+ε)·d_H(x,y)
for all pairs in S. The proof leverages the classical JL lemma to control the Euclidean norm ‖x−y‖ after projection, and then uses the monotonicity of the arcosh function together with the lower bound on the radii to translate Euclidean distortion into hyperbolic distortion. The requirement that distances be “large enough” eliminates the pathological case where the denominator (1−‖x‖²)(1−‖y‖²) becomes very small, which would amplify small Euclidean errors into large hyperbolic errors.

To validate the theory, the authors conduct experiments on two types of data. The first is a synthetic hyperbolic graph generated by the Popularity‑Similarity Optimization (PSO) model, containing 10 000 nodes and 50 000 edges. The second consists of real‑world hierarchical data: the Internet Autonomous System (AS) topology and the Wikipedia category hierarchy, both previously embedded into high‑dimensional Poincaré space using the method of Nickel & Kiela (2017). For each dataset they apply HRP to reduce the dimension to 2 and 3, then measure (a) average relative distortion of hyperbolic distances, (b) preservation of clustering structure (using K‑means on the low‑dimensional embeddings), and (c) impact on greedy routing performance (average path‑length stretch).

The empirical results are striking. With ε=0.1 the average distance distortion stays below 8 %, clustering accuracy remains above 95 % of the original high‑dimensional baseline, and routing stretch is less than 1.05×. Visualizations of the 2‑dimensional embeddings reveal that the hierarchical organization of the original data is clearly visible, confirming that HRP retains the essential geometry despite the drastic reduction in dimensionality.

The paper concludes by discussing limitations and future work. The current analysis assumes a lower bound on pairwise distances; extending the guarantee to arbitrary point sets (including points near the origin) remains open. Moreover, while the Poincaré disk was used for exposition, the authors conjecture that analogous results hold for other models of hyperbolic space such as the Klein model or the hyperboloid model. Finally, integrating HRP directly into deep learning pipelines—e.g., as a differentiable layer that jointly learns hyperbolic embeddings and reduces dimensionality—could enable scalable training on massive hierarchical datasets.

In summary, this work provides the first rigorous Johnson‑Lindenstrauss‑type dimensionality‑reduction theorem for hyperbolic space, supplies a simple and efficient algorithm (HRP), and demonstrates both theoretically and experimentally that high‑dimensional hyperbolic data can be compressed to the hyperbolic plane with provably bounded distortion when the points are sufficiently far apart. This opens the door to practical low‑dimensional visualizations, efficient storage, and fast downstream learning on data with intrinsic hierarchical structure.