Organizations: Department of Statistics and Data Science, Southern University of Science and Technology, 1088 Xueyuan Avenue, Nanshan District, Shenzhen, Guangdong 518055, P.R. China · Department of Statistics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, United Kingdom
Visualizing and interpreting the dependence structure in relational data often benefits from geometric representations via low-dimensional Euclidean embeddings. However, existing approaches are typically restricted to specific response types, rely on burdensome high-dimensional nonconvex optimization, or lack uniform theoretical guarantees for simultaneous coordinate recovery. To address these limitations, we introduce a unified, distance-based latent space framework for both unipartite and bipartite relational data---accommodating continuous, binary, ordinal, and count-valued observations---via a modular two-stage procedure that decouples signal estimation from geometric recovery. First, we estimate the low-rank signal matrix using spectral denoising. For models with nonlinear link functions, we further apply one-step row- and column-wise refinement to achieve uniform entrywise accuracy. Second, we recover the latent configurations through direct spectral analysis for unipartite data and a low-dimensional alignment optimization for bipartite data. We establish uniform recovery guarantees that characterize the propagation of error from signal estimation to the embedding space. To illustrate the scalability and interpretability of our approach, we apply the proposed method to two real-world datasets: a statistician citation network and European Parliament voting records.
Figures & tables
Figure 1: Schematic illustrations of latent Euclidean configurations for unipartite (left) and bipartite (right) relational data. In the unipartite setting, relations are observed among objects from a single set; in the bipartite setting, objects from two distinct sets are jointly represented in a common Euclidean space while only cross-set relations are observed.
Figure 2: Max-norm errors of MSVD and MRSD for the unipartite model with K=3 .
Figure 3: Configuration recovery error dc(Θ,Θ∗) for the unipartite model with K=3 .
Figure 4: Max-norm errors of MSVD and MRSD for the bipartite model with K=3 .
Figure 5: ∣τ−τ0∣ and dc(Θ,Θ∗) for the bipartite model with K=3 .
K
Observation model
J=300
J=500
J=1000
J=2000
Unipartite
K=3
Homoskedastic Gaussian
0.007 (0.002)
0.018 (0.015)
0.118 (0.082)
1.102 (0.800)
Heteroskedastic Gaussian
0.007 (0.005)
0.014 (0.014)
0.065 (0.048)
0.715 (0.638)
Binary
0.186 (0.230)
0.291 (0.069)
0.947 (0.253)
7.016 (3.703)
Poisson
0.219 (0.186)
0.457 (0.350)
1.100 (0.272)
6.318 (3.337)
K=5
Homoskedastic Gaussian
0.008 (0.006)
0.013 (0.004)
0.090 (0.052)
0.803 (0.627)
Table 1: Mean runtime (seconds), with standard errors in parentheses, over 50 replications.
J
Method
dc(Θ,Θ∗)
∣τ−τ0∣
Runtime (seconds)
300
RSD + CR
0.668 (0.064)
0.145 (0.011)
0.277 (0.247)
Low-rank MLE + CR
0.643 (0.053)
0.143 (0.011)
7.884 (5.356)
Joint MLE
0.609 (0.061)
0.088 (0.010)
8.037 (1.304)
500
RSD + CR
0.505 (0.037)
0.085 (0.007)
0.620 (0.649)
Low-rank MLE + CR
0.492 (0.036)
0.084 (0.007)
14.353 (3.104)
Joint MLE
0.465 (0.037)
0.051 (0.006)
21.046 (5.960)
Table 2: Comparison of RSD + CR, low-rank MLE + CR, and joint MLE in the nonlinear bipartite binary setting with K=3 and N=2J . Each entry reports the mean, with standard errors in parentheses, over 50 replications.
Figure 6: Two-dimensional visualization of the estimated embedding vectors for 142 statisticians.
Figure 7: Two views of the recovered European Parliament latent configuration. Left: estimated embedding vectors of 867 MEPs, colored by official political group affiliation. Right: joint representation of MEPs and legislative procedures in the recovered bipartite configuration.
Group
Dim 1 Mean
Dim 1 Median
Dim 2 Mean
Dim 2 Median
ID
-1.626
-1.690
0.365
0.306
GUE_NGL
-0.616
-0.614
-0.921
-1.038
ECR
-0.599
-0.466
0.776
0.848
NI
-0.556
-0.788
-0.030
0.047
GREEN_EFA
0.023
0.118
-1.210
-1.300
SD
0.181
0.336
-0.378
-0.396
Table 3: Coordinate statistics of political groups on the first and second latent dimensions.
Appendix figures & tables5 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 8: Max-norm errors of MSVD and MRSD for the unipartite model with K=5 .
Figure 9: Configuration recovery error dc(Θ,Θ∗) for the unipartite model with K=5 .
Figure 10: Max-norm errors of MSVD and MRSD for the bipartite model with K=5 .
Figure 11: ∣τ−τ0∣ and dc(Θ,Θ∗) for the bipartite model with K=5 .
Political Group
Nearest Legislative Procedure
Second-Nearest Legislative Procedure
ECR
Flexibility Instrument: adjustments to the amounts mobilised for 2019 to be used for migration, refugee inflows and security threats
Additional technical screening criteria for climate change mitigation and adaptation activities
EPP
The EU Trade Policy Review
Organic production: date of application and certain other dates
GREEN_EFA
Role of tax policy in times of crisis
The establishment of the EU ethics body
GUE_NGL
Renewal of the authorisation for products containing genetically modified soybean A5547-127
Renewal of the authorisation for products containing genetically modified cotton 281-24-236 × 3006-210-23
ID
Acceptance period of vaccination certificates issued in the EU Digital COVID Certificate format
Children’s rights on the occasion of the 30th anniversary of the UN Convention on the Rights of the Child
NI
The deteriorating situation of human rights in Egypt, including the case of EIPR activists
Objection concerning maximum residue levels for tricyclazole
Appendix
Table 4: Top-2 closest legislative procedures to each political group center.
Instituto Federal de Educação, Ciência e Tecnologia do Ceará, Canindé, Ceará, Brazil · Federal University of Ceara, Fortaleza, Ceará, Brazil · Sigma Nova, Paris, France
Department of Statistics and Data Science, Carnegie Mellon University · Department of Statistics, Columbia University · Department of Neurology, University of Pittsburgh +1