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arxiv: 2410.16698 · v2 · submitted 2024-10-22 · 💻 cs.LG

Hyperboloid GPLVM for Discovering Continuous Hierarchies via Nonparametric Estimation

Koshi Watanabe , Keisuke Maeda , Takahiro Ogawa , Miki Haseyama This is my paper

Pith reviewed 2026-05-23 19:14 UTC · model grok-4.3

classification 💻 cs.LG
keywords hyperbolic geometrygaussian process latent variable modeldimensionality reductionhierarchical datanonparametric estimationhyperboloid embeddinggenerative modeling
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The pith

Hyperboloid Gaussian process latent variable models embed high-dimensional hierarchical data continuously using nonparametric estimation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes hyperboloid GP latent variable models to embed high-dimensional hierarchical data in low-dimensional hyperbolic spaces while maintaining implicit continuity in the hierarchies. Traditional neighbor embedding methods often break these continuous relations. By employing generative modeling with Gaussian processes, the approach enables effective hierarchical embedding and addresses hyperparameter tuning issues through nonparametric methods. Three variants are developed using different estimation strategies, and learning algorithms incorporate Riemannian optimization and approximation schemes.

Core claim

hGP-LVMs embed high-dimensional hierarchical data with implicit continuity via nonparametric estimation using generative GP modeling in hyperboloid space.

What carries the argument

Hyperboloid Gaussian process latent variable models (hGP-LVMs) that use generative GP modeling for embedding in hyperbolic geometry.

Load-bearing premise

Adopting generative modeling with the GP brings effective hierarchical embedding and resolves ill-posed hyperparameter tuning.

What would settle it

A comparison where hGP-LVMs fail to maintain continuous hierarchical relations better than existing neighbor embedding methods on standard hierarchical datasets.

Figures

Figures reproduced from arXiv: 2410.16698 by Keisuke Maeda, Koshi Watanabe, Miki Haseyama, Takahiro Ogawa.

Figure 1
Figure 1. Figure 1: An illustration of hyperboloid Gaussian process latent variable models (hGP-LVMs). We learn the latent variables on the Lorentz model and visualize them on the Poincaré ball. Gaussian process (GP) latent variable models (GP￾LVMs) (Lawrence, 2005; Titsias and Lawrence, 2010) are one of the representatives in the nonparametric data-embedding methods. GP-LVMs assume the GP decoder of the observed variables (R… view at source ↗
Figure 2
Figure 2. Figure 2: GP prior comparison between the hyper￾boloid exponential kernel (upper, κ = 5) and Euclidean exponential kernel (bottom). The color gives the value of the sampled GP. We input the latent variables on the Poincaré ball model when M = L Q (left) and those on the unit circle when M = E (right). if dM(x, x ′ ) is a conditionally negative definite (CND) metric (Feragen et al., 2015). Since the hyperbolic metric… view at source ↗
Figure 3
Figure 3. Figure 3: An illustration of the synthetic binary tree [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Qualitative results on the SBT dataset (d = 6). (a) embedding comparison between generative models, (b) embedding of hGP-LVM with different length scales, and (c) color code of embedding. validated the effectiveness of hGP-LVM toward gen￾erative models with the experiment typically used for evaluating hyperbolic generative models. Embedding Comparison with Different Length￾scale In Section 3.1, we state th… view at source ↗
Figure 5
Figure 5. Figure 5: Experimental results on the scRNA-seq dataset. (a) The canonical hematopoietic cell lineage tree. (b) Two-dimensional embedding of UMAP, PoincaréMap, Sparse hGP-LVM, and Bayesian hGP-LVM. The colors correspond to those of the lineage tree. (c) The error bar plot of comparative methods. We ran the same experiment 30 times and computed the mean error with standard deviation. (k = 5) as the local metric to us… view at source ↗
read the original abstract

Dimensionality reduction (DR) offers a useful representation of complex high-dimensional data. Recent DR methods focus on hyperbolic geometry to derive a faithful low-dimensional representation of hierarchical data. However, existing methods are based on neighbor embedding, frequently ruining the continual relation of the hierarchies. This paper presents hyperboloid Gaussian process (GP) latent variable models (hGP-LVMs) to embed high-dimensional hierarchical data with implicit continuity via nonparametric estimation. We adopt generative modeling using the GP, which brings effective hierarchical embedding and executes ill-posed hyperparameter tuning. This paper presents three variants that employ original point, sparse point, and Bayesian estimations. We establish their learning algorithms by incorporating the Riemannian optimization and active approximation scheme of GP-LVM. For Bayesian inference, we further introduce the reparameterization trick to realize Bayesian latent variable learning. In the last part of this paper, we apply hGP-LVMs to several datasets and show their ability to represent high-dimensional hierarchies in low-dimensional spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper proposes hyperboloid Gaussian process latent variable models (hGP-LVMs) to perform dimensionality reduction on high-dimensional hierarchical data. It uses generative GP modeling in hyperbolic (hyperboloid) space to embed data while preserving implicit continuity in hierarchies via nonparametric estimation, addressing limitations of neighbor-embedding approaches. Three variants are developed—point estimation, sparse point estimation, and Bayesian estimation—with learning procedures that incorporate Riemannian optimization, active approximation, and the reparameterization trick for the Bayesian case. Experiments on several datasets are used to illustrate the embeddings.

Significance. If the continuity-preserving property holds under the GP construction and the variants scale without introducing new instabilities, the work could provide a useful nonparametric alternative to existing hyperbolic DR methods for tree-structured data. The explicit use of generative modeling and Riemannian tools is a clear strength, as is the provision of multiple estimation regimes.

major comments (2)
  1. [§3.2] §3.2, the sparse variant: the active approximation scheme is stated to inherit from standard GP-LVM, but no derivation or bound is given showing that the inducing-point selection preserves the hyperboloid geometry or the continuity property claimed in the abstract; this is load-bearing for the scalability claim.
  2. [§4] §4, experimental section: the reported visualizations and qualitative hierarchy recovery are not accompanied by a quantitative continuity metric (e.g., a hierarchy violation score or geodesic-distance correlation with ground-truth tree depth); without it the central claim that neighbor-embedding methods “ruin the continual relation” cannot be directly compared.
minor comments (3)
  1. The notation for the hyperboloid manifold and its tangent-space operations is introduced without a short self-contained definition or reference to a standard source; this affects readability for readers outside hyperbolic geometry.
  2. [Algorithm 1] Algorithm 1 (point estimation) and Algorithm 2 (Bayesian) list the same Riemannian optimizer step; a brief note on any differences in step-size scheduling or retraction would clarify the distinction.
  3. The abstract states that GP modeling “executes ill-posed hyperparameter tuning,” yet the text does not quantify how many hyperparameters are actually optimized versus fixed; a short table of free vs. fixed parameters per variant would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive summary and recommendation for minor revision. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3.2] §3.2, the sparse variant: the active approximation scheme is stated to inherit from standard GP-LVM, but no derivation or bound is given showing that the inducing-point selection preserves the hyperboloid geometry or the continuity property claimed in the abstract; this is load-bearing for the scalability claim.

    Authors: We acknowledge that §3.2 states the active approximation inherits from standard GP-LVM without providing an explicit derivation or bound confirming preservation of hyperboloid geometry and the continuity property under inducing-point selection. The construction places inducing points on the manifold and applies the same Riemannian optimization as the point-estimate variant, but the manuscript does not derive a formal guarantee. In the revision we will add a short paragraph in §3.2 supplying a first-order argument that the inducing-point scheme preserves the nonparametric GP prior on the hyperboloid and therefore the continuity property. revision: yes

  2. Referee: [§4] §4, experimental section: the reported visualizations and qualitative hierarchy recovery are not accompanied by a quantitative continuity metric (e.g., a hierarchy violation score or geodesic-distance correlation with ground-truth tree depth); without it the central claim that neighbor-embedding methods “ruin the continual relation” cannot be directly compared.

    Authors: We agree that the absence of a quantitative continuity metric limits direct comparison with neighbor-embedding baselines. The current experiments rely on qualitative visualization of hierarchy recovery. In the revised manuscript we will augment §4 with at least one quantitative measure (e.g., Spearman correlation between embedding geodesic distances and ground-truth tree depths, or a hierarchy-violation count) on the synthetic and real tree-structured datasets. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents hGP-LVMs as an adaptation of existing GP-LVM machinery to hyperboloid geometry, introducing three estimation variants (point, sparse, Bayesian) along with Riemannian optimization, active approximation, and the reparameterization trick. No load-bearing derivation step is shown reducing to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain. The motivation for using generative GP modeling is stated as a methodological choice that addresses hierarchical continuity and hyperparameter issues, without equations or claims that collapse by construction to the inputs. The construction remains self-contained against external GP-LVM benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Abstract-only; ledger populated from stated assumptions in the summary. Relies on GP generative modeling for hierarchies and hyperbolic suitability without independent verification visible.

free parameters (1)
  • GP hyperparameters
    Tuned via the proposed method; abstract notes ill-posed tuning as motivation for GP approach.
axioms (2)
  • domain assumption Hyperbolic geometry faithfully represents hierarchical data structures
    Core motivation stated in abstract for using hyperboloid model.
  • domain assumption GP generative modeling resolves ill-posed hyperparameter tuning for hierarchical embedding
    Directly stated as benefit of adopting GP in the abstract.

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Reference graph

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