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Quotient-Based Posterior Analysis for Euclidean Latent Space Models

Kisung You, Mauro Giuffrè

Abstract

Latent space models are widely used in statistical network analysis and are often fit by Markov chain Monte Carlo. However, posterior summaries of latent coordinates are not canonical because the likelihood depends only on pairwise distances and is invariant under rigid motions of the latent space. Standard post hoc alignment can aid visualization, but the resulting summaries depend on an arbitrary reference configuration. We propose a quotient-based posterior analysis for Euclidean latent space models using the centered Gram map, which represents identifiable latent structure while removing nonidentifiability. This yields intrinsic posterior summaries of mean structure and uncertainty that can be computed directly from posterior samples, together with basic theoretical guarantees including canonicality, existence, and stability. Through simulations and analyses of the Florentine marriage network and a statisticians' coauthorship network, the proposed framework clarifies when alignment-based summaries are stable, when they become reference-sensitive, and which nodes or relationships are weakly identified. These results show how coherent posterior analysis can reveal latent relational structure beyond a single embedding.

Quotient-Based Posterior Analysis for Euclidean Latent Space Models

Abstract

Latent space models are widely used in statistical network analysis and are often fit by Markov chain Monte Carlo. However, posterior summaries of latent coordinates are not canonical because the likelihood depends only on pairwise distances and is invariant under rigid motions of the latent space. Standard post hoc alignment can aid visualization, but the resulting summaries depend on an arbitrary reference configuration. We propose a quotient-based posterior analysis for Euclidean latent space models using the centered Gram map, which represents identifiable latent structure while removing nonidentifiability. This yields intrinsic posterior summaries of mean structure and uncertainty that can be computed directly from posterior samples, together with basic theoretical guarantees including canonicality, existence, and stability. Through simulations and analyses of the Florentine marriage network and a statisticians' coauthorship network, the proposed framework clarifies when alignment-based summaries are stable, when they become reference-sensitive, and which nodes or relationships are weakly identified. These results show how coherent posterior analysis can reveal latent relational structure beyond a single embedding.

Paper Structure

This paper contains 30 sections, 7 theorems, 123 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation and Background
  3. Identifiability in latent space models
  4. Posterior sampling versus posterior summarization
  5. Limitations of alignment-based summaries
  6. A Quotient Representation for Latent Space Models
  7. Centered Gram map and invariance
  8. Connection to squared distance matrices and classical MDS
  9. Identifiability and the quotient state space
  10. Quotient distance and computation
  11. Geometric operations on the rank-$r$ stratum
  12. Posterior Summaries
  13. Posterior mean
  14. Global and local posterior variability
  15. Pairwise distance inference
  16. ...and 15 more sections

Key Result

Lemma 1

Let $\boldsymbol{X},\boldsymbol{W}\in\mathbb{R}^{n\times r}$ and write $\tilde{\boldsymbol{X}}=\boldsymbol{H} \boldsymbol{X}$, $\tilde{\boldsymbol{W}}=\boldsymbol{H} \boldsymbol{W}$. Suppose $\mathrm{rank}(\tilde{\boldsymbol{X}})=\mathrm{rank}(\tilde{\boldsymbol{W}})=r$. Then Equivalently, $\blacktriangleleft$$\blacktriangleleft$

Figures (7)

  • Figure 1: Representative dataset from the weakly identified regime. (a) True centered latent positions. (b) Quotient Fréchet mean. (c) Procrustes mean using the first posterior draw as reference. (d) Procrustes mean using the quotient medoid draw as reference. Panels (b)--(d) are aligned to the true centered configuration for display only.
  • Figure 2: Simulation summaries. (a) Reference-sensitivity index $S_{\mathrm{ref}}$ by regime. (b) Node-level uncertainty index $U_i$ by true group for the representative weak-regime dataset. (c) Relationship between $U_i$ and the invariant node-wise loss $L_i$.
  • Figure 3: Posterior summaries for the Florentine marriage network. (a) Quotient Fréchet mean embedding with observed marriage edges overlaid; point size and shade are proportional to $U_i$. (b) Node-level uncertainty $U_i$ versus degree. (c) Node-level uncertainty $U_i$ versus betweenness. (d) Families with the largest posterior uncertainty.
  • Figure 4: Dyad-level posterior summaries for the Florentine marriage network. (a) Posterior mean distance by dyad type. (b) Posterior variance of distance by dyad type. (c) Business-only dyads with the smallest posterior mean distances. The business network was not used in fitting and is used here only for interpretation.
  • Figure 5: Quotient Fréchet mean embedding for the coauthorship network, with observed coauthorship edges overlaid. Colors indicate the three biostatistics communities, and node size is proportional to the node-level uncertainty index $U_i$.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Lemma 1: Gram characterization
  • Theorem 2: Canonicality
  • Theorem 3: Existence
  • Theorem 4: Stability under Wasserstein perturbations
  • Theorem 5: Consistency of pairwise distances
  • Corollary 6: Consistency of edge probabilities
  • Theorem 7: Vanishing node-level uncertainty