Pith. sign in

REVIEW 2 major objections 4 minor 285 references

On typical inhomogeneous networks, landmark embeddings need only dimension Ω(n^{1-ε} log n) to keep (1±ε) shortest-path distortion, a polynomial saving over classical worst-case bounds.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-14 00:36 UTC pith:46QYKHTD

load-bearing objection Solid average-case improvement on landmark distortion for IHGs, with a clean sandwiching lift to L^{2} kernels and usable GNN transfer experiments. the 2 major comments →

arxiv 2607.10074 v1 pith:46QYKHTD submitted 2026-07-11 cs.LG

Distance-Preserving Embeddings in Inhomogeneous Random Graphs

My Le , Luana Ruiz , Souvik Dhara This is my paper
classification cs.LG
keywords shortest pathdistance-preserving embeddingslandmarksgraph spannersinhomogeneous random graphsheterogeneitygraph neural networkstransferability
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Worst-case theory says that preserving all pairwise shortest-path distances up to a (1±ε) factor forces embedding dimension that grows like a high power of n. This paper shows that the story changes once the graph is drawn from a broad family of inhomogeneous random graphs that capture community structure and heavy-tailed degrees. In the supercritical regime the local neighborhoods expand exponentially at a rate fixed by the spectral radius of the type-affinity matrix (or its continuous integral-operator analogue). That controlled expansion lets a multiscale landmark scheme place a few carefully sized sets of reference nodes so that both the lower- and upper-bound distance estimators stay within (1±ε) of the true distances, yet the total embedding dimension shrinks to only Ω(n^{1-ε} log n). The same dimension-distortion trade-off extends from finite-type models to arbitrary L² kernels by a metric-sandwiching argument that approximates any continuous kernel by two nearby step-function kernels. Global averages over all connected pairs concentrate as well, and a GNN trained on small random instances can replace exact landmark distances while transferring to large real networks.

Core claim

For supercritical inhomogeneous random graphs (finite types or general L² kernels), landmark-based embeddings achieve (1±ε)-distortion of shortest-path distances with embedding dimension Ω(n^{1-ε} log n)—a polynomial improvement over the classical worst-case requirements—because neighborhood expansion is governed by a multi-type branching process whose growth rate is the spectral radius of the affinity operator.

What carries the argument

Metric sandwiching: any L² kernel is squeezed between two finite step-function kernels whose spectral radii stay within O(δ) of the original; the finite-type distortion theorems then pass to the continuum limit as the partition is refined.

Load-bearing premise

The affinity operator must stay uniformly supercritical: its leading eigenvalue is bounded away from 1 by a fixed positive gap that does not shrink with n. Without that gap the exponential neighborhood growth that powers the dimension saving collapses.

What would settle it

Generate a sequence of supercritical IHGs whose spectral radius approaches 1 from above, run the same multiscale landmark scheme with dimension o(n^{1-ε} log n), and check whether the fraction of pairs whose distortion exceeds (1±ε) stays bounded away from zero.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies landmark-based distance-preserving embeddings on inhomogeneous random graphs (IHGs) with type-dependent edge probabilities. Using multi-type branching-process approximations of neighborhood expansion, it proves that in the uniformly supercritical regime a multi-scale landmark scheme achieves (1±ε)-distortion with embedding dimension Ω(n^{1-ε} log n), a polynomial improvement over classical worst-case bounds. The same trade-off is lifted to global averages over connected pairs and, via a metric-sandwiching construction that approximates an arbitrary L² kernel by finite step-function kernels, to continuous latent-space models including heavy-tailed Chung–Lu graphs. A GNN surrogate for the local landmark-distance step is introduced and shown experimentally to transfer from small ER graphs to large synthetic and real networks while matching or exceeding exact BFS landmarks on denser instances.

Significance. If the claims hold, the work supplies the first average-case dimension–distortion guarantees for a practically used embedding method on a broad, realistic random-graph family, replacing the pessimistic polynomial exponents of Bourgain–Matoušek–Sarma with an essentially linear (yet still sub-linear in the worst-case sense) dependence that improves with the spectral gap. The sandwiching argument unifies discrete and continuous models under a single spectral mechanism and therefore covers power-law networks. Full proofs of all neighborhood-growth, intersection and distortion statements appear in §7; the accompanying GNN code is released and the transferability experiments are reproducible. These elements together give both a theoretical foundation for virtual spanners on heterogeneous networks and a concrete, transferable algorithmic realization.

major comments (2)
  1. [§3.2 Assumption 3.2 and Remark 1] Assumption 3.2 (uniform supercriticality λ₁(D)≥1+ε for a fixed ε>0 independent of n) is load-bearing for every expansion lemma (4.6–4.8), the intersection control (Prop. 4.4) and therefore Theorems 4.1–4.2 and 5.2. The paper correctly conditions all statements on this gap, yet the near-critical regime λ₁↓1 is left unexplored; a short quantitative discussion of how the dimension exponent degrades as ε→0 would clarify the modeling boundary of the claimed polynomial improvement.
  2. [§6 Experimental Setup and Experiments 1–3] All synthetic GNN training and evaluation (§6) is performed exclusively on the T=1 Erdős–Rényi special case. While the theory is developed for multi-type and continuous kernels, the empirical claim that “models trained on small-scale random graphs learn to extract universal distance-preserving features” is therefore supported only for homogeneous graphs; a multi-type synthetic experiment would strengthen the bridge between the main theorems and the GNN results.
minor comments (4)
  1. [Abstract and §1–§2] Numerous missing spaces appear throughout the extracted text (“bothlocal”, “typicallarge-scale”, “virtualgraph”, “W orst-Case”, “T ransferability”, etc.). These are almost certainly PDF-extraction artefacts but should be cleaned in the camera-ready version.
  2. [§6.1 Experiment 1] Figure 3 caption and surrounding text state that GNN depth exceeds ⌈log_λ n⌉, yet predictions still saturate; a one-sentence clarification that message-passing depth is necessary but not sufficient for long-range distances would help readers.
  3. [§2.2] The notation for the lower- and upper-bound estimators switches between d̲, d̄ and d, d̄; a single consistent pair of symbols should be fixed in §2.2 and used thereafter.
  4. [§6.3 and Table 1] Table 1 lists 16 real networks but only a subset appear in Figures 5–6; either all should be shown or the selection criterion stated.

Circularity Check

0 steps flagged

No significant circularity; distortion–dimension trade-offs are derived from first-principles multi-type branching-process neighborhood expansion and spectral radius of the affinity matrix/kernel under explicit supercriticality assumptions.

full rationale

The central claims (Theorems 4.1–4.2, Remark 1, Theorem 4.5, Theorems 5.1–5.2) rest on Lemmas 4.6–4.8 and Propositions 4.3–4.4, which bound neighborhood sizes |∂N_k(u)_t| = Θ(λ_1^k) and intersections via the multi-type branching-process approximation of IHG exploration (standard coupling to the mean matrix D or integral operator T_κ, citing Bollobás–Janson–Riordan and van der Hofstad). These are not defined in terms of the target distortion; the (1±ε) guarantees and the improved dimension Ω(n^{1-ε} log n) follow by plugging the exponential growth into the multiscale landmark sampling probabilities (exactly as in the classical Sarma et al. argument, but with the tighter expansion rate). The metric-sandwiching construction (Theorem 5.1) is an independent coupling argument that transfers the finite-type bounds; it does not presuppose the distortion result. GNN experiments and transferability citations (Ruiz et al.) are methodological and non-load-bearing for the theorems. No parameter is fitted to the claimed trade-off, no uniqueness theorem is imported from the authors, and no known empirical pattern is merely renamed. The uniform-supercriticality gap (Assumption 3.2) is an explicit modeling hypothesis, not a circular definition. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 6 axioms · 1 invented entities

The central claims rest on three modeling assumptions that define the IHG class, standard multi-type branching-process approximation theorems from the random-graph literature, and the classical existence of (1±ε) embeddings in worst-case metrics. No free parameters are fitted to data for the theoretical bounds; the GNN experiments use ordinary hyper-parameters that do not enter the mathematical claims.

free parameters (2)
  • GNN hidden widths and depths
    Nine architectures with √n nodes in first/last layers and varying hidden sizes; chosen by hand for the experimental section only, not used in any theorem.
  • landmark base M and number of repetitions R
    M>1 integer and R=Ω(n^{1-ε+ς}) are free design choices that appear in the statements of Theorems 4.1–4.2; they are not fitted to data.
axioms (6)
  • domain assumption Affinity matrix D is primitive (irreducible and aperiodic) — Assumption 3.1
    Guarantees [D^k]_{ij}=Θ(λ_{1}^k) uniformly, used throughout neighborhood-growth lemmas.
  • domain assumption Uniform supercriticality: λ_{1}(D)≥1+ε for a fixed ε>0 independent of n — Assumption 3.2
    Ensures a unique giant component and exponential neighborhood expansion at rate λ_{1}; load-bearing for all distortion theorems.
  • domain assumption Type proportions n_t/n o α_t >0 — Assumption 3.3
    Prevents vanishing type classes that would break concentration of edge counts.
  • standard math Local neighborhoods of IHGs couple to multi-type Poisson branching processes up to depth κ log_λ_{1} n (Bollobás–Janson–Riordan, van der Hofstad)
    Standard random-graph fact invoked in Lemmas 4.7–4.8 and Propositions 4.3–4.4.
  • standard math Kesten–Stigum theorem for supercritical multi-type branching processes (Grama et al. 2023)
    Supplies the almost-sure exponential growth on the survival event used in Lemma 4.7.
  • standard math Bourgain/Matoušek/Sarma worst-case dimension-distortion lower bounds
    Used only as the baseline that the IHG bounds improve upon; not needed for the positive results.
invented entities (1)
  • metric sandwiching framework (κ^±_δ step-function kernels) independent evidence
    purpose: Couples an arbitrary L^{2} kernel between two finite-type models so that shortest-path distances and spectral radii are controlled, transferring the finite-type distortion theorems to continuous latent spaces.
    The construction is new to this paper; independent evidence is the spectral-perturbation and edge-inclusion arguments given in Theorem 5.1, which rely only on standard operator theory.

pith-pipeline@v1.1.0-grok45 · 48897 in / 3609 out tokens · 31135 ms · 2026-07-14T00:36:03.114215+00:00 · methodology

0 comments
read the original abstract

Graph machine learning provides powerful tools for understanding complex networks and learning meaningful node representations. A central challenge, however, is designing embeddings with minimal distortion of both local and global functionals, such as shortest path lengths. Prior distortion guarantees for distance-preserving embeddings are worst-case in nature, producing overly pessimistic bounds that fail to capture the structure of typical large-scale networks. To address this, we analyze shortest-path approximation via landmark-based embeddings on inhomogeneous random graphs, a general model with type-dependent edge probabilities. By retaining shortest paths to a small set of reference nodes called landmarks, landmark-based methods effectively function as virtual graph spanners, where structural heterogeneity and controlled neighborhood expansion modeled via multi-type branching processes enable significantly tighter dimension-distortion trade-offs than classical worst-case bounds. We extend these guarantees to global, component-wide averages and unify the analysis across finite-type and continuous latent spaces through a novel metric sandwiching framework, establishing universal distortion bounds for general $L^2$ kernel models, including heavy-tailed and power-law networks. Finally, we introduce a GNN-augmented variant that replaces rigid, computationally expensive exact shortest-path queries with flexible, structure-aware neural surrogates. By leveraging the inherent alignment between graph neural message-passing and the dynamic programming principles of shortest-path algorithms, our approach demonstrates that models trained on small-scale random graphs learn to extract universal distance-preserving features, achieving robust generalization to large-scale, real-world networks that match or exceed the fidelity of classical, exact landmark-based embeddings.

Figures

Figures reproduced from arXiv: 2607.10074 by Luana Ruiz, My Le, Souvik Dhara.

Figure 1
Figure 1. Figure 1: Schematic depicting the computation of the lower bound d(u1, u2), where k2−k1 ≥ (1 − ε)d(u1, u2). Blue nodes are the source u1 and target u2, orange nodes are landmarks in set S, and gray nodes are arbitrary nodes. Since D is primitive, [Dk ]ij = Θ(λ k 1 ) for any type pair (i, j), so Proposition 4.3 implies that neighborhoods grow exponentially at rate λ k 1 for 1 ≪ k ≤ logλ1 n. Hence, the local step of t… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic depicting the computation of the upper bound ¯d(u1, u2), where k1 = k2 = 1+ε 2 d(u1, u2). Blue nodes are the source u1 and target u2, orange nodes are landmarks in set S, and gray nodes are arbitrary nodes. Proposition 4.4 shows that once k1 + k2 exceeds logλ1 nt , the intersection ∂Nk1 (u1)t ∩ ∂Nk2 (u2)t is non-trivial and grows as λ k1+k2 1 /nt w.h.p. In other words, once neighborhoods 12 [PIT… view at source ↗
Figure 3
Figure 3. Figure 3: plots the actual shortest path distances versus those predicted by our selected GNN architectures. Predictions for distances beyond the GNN depth saturate, indicating that GNNs cannot capture longer distances even with depth exceeding the expected path length. As expected, GNNs are not suitable for computing end-to-end shortest path distances, especially on sparser graphs with λ ∈ {3, 4}, which tend to exh… view at source ↗
Figure 4
Figure 4. Figure 4: (a)-(d) Error rates of BFS-based and GNN-based lower bounds on graphs gener￾ated by ERn(λ/n), with the GNNs trained on graphs from the same model. (e) Time required to generate all node-to-landmark distances in n-node ER graphs by NetworkX’s highly optimized BFS compared to our widest and deepest GNNs. All GCN, GraphSage, GAT, and GIN models are represented by the same color and solid lines for the same R,… view at source ↗
Figure 5
Figure 5. Figure 5: Error rates of BFS-based and GNN-based lower bounds on (a,d) test Erd˝os–R´enyi graphs generated by ERn′(λ/n′ ), (b,e) Arxiv COND-MAT collaboration network with 21,364 nodes, and (c,f) GEMSEC company network with 14,113 nodes, with the GNNs trained on graphs from ERn(λ/n). Legend is the same as in [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Additional transferability results on real networks, with the GNNs trained on graphs from ERn(λ/n). Legend is the same as in [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

285 extracted references · 2 canonical work pages

  1. [1]

    W. B. Johnson and J. Lindenstrauss and G. Schechtman , title =. Geometrical Aspects of Functional Analysis (1985/86) , series =. 1987 , doi =

    1985

  2. [2]

    J. Matou. Note on bi-Lipschitz embeddings into normed spaces , journal =

  3. [3]

    Proceedings of the Symposium on Foundations of Computer Science (FOCS) , publisher =

    Piotr Indyk , title =. Proceedings of the Symposium on Foundations of Computer Science (FOCS) , publisher =. 2001 , pages =

    2001

  4. [4]

    Riordan and N

    O. Riordan and N. Wormald , title =. Combinatorics, Probability and Computing , year =

  5. [5]

    and Gama, F

    Ruiz, L. and Gama, F. and G. Marques, A. and Ribeiro, A. Invariance-Preserving Localized Activation Functions for Graph Neural Networks. 2020

    2020

  6. [6]

    and Gama, F

    Ruiz, L. and Gama, F. and Ribeiro, A. Gated Graph Recurrent Neural Networks. 2020

    2020

  7. [7]

    and Gama, F

    Ruiz, L. and Gama, F. and Ribeiro, A. Graph Neural Networks: Architectures, Stability and Transferability. Proc. IEEE. 2021

    2021

  8. [8]

    and Chamon, L

    Ruiz, L. and Chamon, L. F. O. and Ribeiro, A. Graphon Signal Processing. 2021

    2021

  9. [9]

    and Ruiz, L

    Cervino, J. and Ruiz, L. and Ribeiro, A. Learning by Transference: Training Graph Neural Networks on Growing Graphs. 2023

    2023

  10. [10]

    and Ruiz, L

    Wang, Z. and Ruiz, L. and Ribeiro, A. Stability to Deformations in Manifold Neural Networks. arXiv [cs.LG]:2106.03725. 2021

    Pith/arXiv arXiv 2021

  11. [11]

    and Chamon, L

    Ruiz, L. and Chamon, L. F. O. and Ribeiro, A. Transferability Properties of Graph Neural Networks. IEEE Transactions on Signal Processing , year=

  12. [12]

    and Gama, F

    Ruiz, L. and Gama, F. and G. Marques, A. and Ribeiro, A. Median Activation Functions for Graph Neural Networks. 44th. 2019

    2019

  13. [13]

    and Gama, F

    Ruiz, L. and Gama, F. and Ribeiro, A. Gated Graph Convolutional Recurrent Neural Networks. 27th. 2019

    2019

  14. [14]

    and Gama, F

    Ruiz, L. and Gama, F. and Ribeiro, A. Spatial Gating Strategies for Graph Recurrent Neural Networks. 45th. 2020

    2020

  15. [15]

    and Chamon, L

    Ruiz, L. and Chamon, L. F. O. and Ribeiro, A. The G raphon F ourier T ransform. 45th. 2020

    2020

  16. [16]

    and Chamon, L

    Ruiz, L. and Chamon, L. F. O. and Ribeiro, A. Graphon Neural Networks and the Transferability of Graph Neural Networks. 34th. 2020

    2020

  17. [17]

    and Ruiz, L

    Iancu, B. and Ruiz, L. and Ribeiro, A. and Isufi, E. Graph-Adaptive Activation Functions for Graph Neural Networks. 30th Int. Workshop Mach. Learn. Signal Process. 2020

    2020

  18. [18]

    and Chamon, L

    Ruiz, L. and Chamon, L. F. O. and Ribeiro, A. Graphon Filters: Signal Processing in Very Large Graphs. 28th. 2021

    2021

  19. [19]

    and Ruiz, L

    Parada-Mayorga, A. and Ruiz, L. and Ribeiro, A. Graphon Pooling in Graph Neural Networks. 28th. 2021

    2021

  20. [20]

    and Ruiz, L

    Wang, Z. and Ruiz, L. and Ribeiro, A. Stability of Neural Networks on R iemannian Manifolds. 29th. 2021

    2021

  21. [21]

    and Wang, Z

    Ruiz, L. and Wang, Z. and Ribeiro, A. Graphon and Graph Neural Network Stability. 46th. 2021

    2021

  22. [22]

    and Gama, F

    Ruiz, L. and Gama, F. and Ribeiro, A. and Isufi, E. Nonlinear State-Space Generalizations of Graph Convolutional Neural Networks. 46th. 2021

    2021

  23. [23]

    and Ainslie, J

    Ruiz, L. and Ainslie, J. and Onta \ n \'o n, S. Iterative Decoding for Compositional Generalization in Transformers. arXiv:2110.04169 [cs.LG]. 2021

    Pith/arXiv arXiv 2021

  24. [24]

    and Chamon, L

    Ruiz, L. and Chamon, L. F. O. and Ribeiro, A. Transferable Graph Neural Networks on Large-Scale Stochastic Graphs. 55th. 2021

    2021

  25. [25]

    and Ruiz, L

    Wang, Z. and Ruiz, L. and Ribeiro, A. Stability of Neural Networks on Manifolds to Relative Perturbations. 47th. 2022

    2022

  26. [26]

    and Ruiz, L

    Wang, Z. and Ruiz, L. and Eisen, M. and Ribeiro, A. Stable and Transferable Wireless Resource Allocation Policies via Manifold Neural Networks. 47th. 2022

    2022

  27. [27]

    and Ribeiro, A

    Cervi\ no, J/ and Ruiz, L. and Ribeiro, A. Training Stable Graph Neural Networks through Constrained Learning. 47th. 2022

    2022

  28. [28]

    and Varma, R

    Chen, S. and Varma, R. and Sandryhaila, A. and Kovacevic, J. Discrete Signal Processing on Graphs: Sampling Theory. 2015

    2015

  29. [29]

    Marques, A. G. and Segarra, S. and Leus, G. and Ribeiro, A. Sampling of Graph Signals with Successive Local Aggregations. 2015

    2015

  30. [30]

    Chamon, L. F. O. and Ribeiro, A. Greedy Sampling of Graph Signals. 2017

    2017

  31. [31]

    and Eldar, Y

    Heimowitz, A. and Eldar, Y. C. A Unified View of Diffusion Maps and Signal Processing on Graphs. 2017 Int. Conf. Sampling Theory and Appl. 2017

    2017

  32. [32]

    and Moura, J

    Sandryhaila, A. and Moura, J. M. F. Discrete Signal Processing on Graphs: Graph Filters. 38th. 2013

    2013

  33. [33]

    and Moura, J

    Sandryhaila, A. and Moura, J. M. F. Discrete Signal Processing on Graphs: Frequency Analysis. 2014

    2014

  34. [34]

    Gama, F. and G. Marques, A. and Leus, G. and Ribeiro, A. Convolutional Neural Network Architectures for Signals Supported on Graphs. 2018

    2018

  35. [35]

    and Isufi, E

    Levie, R. and Isufi, E. and Leus, G. Kutyniok, G. On the Transferability of Spectral Graph Filters. arXiv:1901.10524 [cs.LG]. 2019

    Pith/arXiv arXiv 1901

  36. [36]

    Shuman, D. I. and Narang, S. K. and Frossard, P. and Ortega, A. and Vandergheynst, P. The Emerging Field of Signal Processing on Graphs: Extending high-dimensional data analysis to networks and other irregular domains. 2013

    2013

  37. [37]

    and Bresson, X

    Defferrard, M. and Bresson, X. and Vandergheynst, P. Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering. 2016

    2016

  38. [38]

    Kipf, T. N. and Welling, M. Semi-Supervised Classification with Graph Convolutional Networks. 5th. 2017

    2017

  39. [39]

    and Bengio, Y

    LeCun, Y. and Bengio, Y. and Hinton, G. Deep Learning. Nature. 2015

    2015

  40. [40]

    and Zaremba, W

    Bruna, J. and Zaremba, W. and Szlam, A. and LeCun, Y. Spectral Networks and Deep Locally Connected Networks on Graphs. arXiv:1312.6203v3 [cs.LG]. 2014

    Pith/arXiv arXiv 2014

  41. [41]

    Bronstein, M. M. and Bruna, J. and LeCun, Y. and Szlam, A. and Vandergheynst, P. Geometric Deep Learning: Going Beyond Euclidean Data. arXiv:1611.08097v2 [cs.CV]. 2017

    Pith/arXiv arXiv 2017

  42. [42]

    and Bengio, Y

    Goodfellow, I. and Bengio, Y. and Courville, A. Deep Learning. 2016

    2016

  43. [43]

    and Warde-Farley, D

    Goodfellow, I. and Warde-Farley, D. and Mirza, M. and Courville, A. and Bengio, Y. Maxout Networks. 30th. 2013

    2013

  44. [44]

    Kuo, C.-C. J. The CNN as a Guided Multilayer RECOS Transform. 2017

    2017

  45. [45]

    Kingma, D. P. and Ba, J. L. ADAM : A Method for Stochastic Optimization. 3rd. 2015

    2015

  46. [46]

    Approximation Capabilities of Multilayer Feedforward Networks

    Hornik, K. Approximation Capabilities of Multilayer Feedforward Networks. Neural Networks. 1991

    1991

  47. [47]

    Hodgson, R. M. and Bailey, D. G. and Naylor, M. J. and Ng, A. L.M. and McNeill, S.J. Properties, Implementations and Applications of Rank Filters. Image and Vision Computing. 1985

    1985

  48. [48]

    and Zhang, X

    He, K. and Zhang, X. and Ren, S. and Sun, J. Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification. 2015. 2015

    2015

  49. [49]

    and Van Vaerenbergh, S

    Scardapane, S. and Van Vaerenbergh, S. and Comminiello, D. and Uncini, A. Improving Graph Convolutional Networks with Non-Parametric Activation Functions. 26th. 2018

    2018

  50. [50]

    and Muller, A

    Guido, S. and Muller, A. Introduction to Machine Learning with Python. 2016

    2016

  51. [51]

    Segarra, S. and G. Marques, A. and Leus, G. and Ribeiro, A. Interpolation of graph signals using shift-invariant graph filters. 23rd. 2015

    2015

  52. [52]

    Gama, F. and G. Marques, A. and Mateos, G. and Ribeiro, A. Rethinking Sketching as Sampling: A Graph Signal Processing Approach. Signal Processing. 2020

    2020

  53. [53]

    Huang, W. and A. W. Bolton, T. and D. Medaglia, J. and S. Bassett, D. and Ribeiro, A. and Van De Ville, D. A Graph Signal Processing Perspective on Functional Brain Imaging. 2018

    2018

  54. [54]

    and Cammoun, L

    Hagmann, P. and Cammoun, L. and Gigandet, X. and Meuli, R. and J. Honey, C. J. Wedeen, V. and Sporns, O. Mapping the Structural Core of Human Cerebral Cortex. PLoS Biol. 2008

    2008

  55. [55]

    and Moura, J

    Sandryhaila, A. and Moura, J. M. F. Discrete Signal Processing on Graphs. 2013

    2013

  56. [56]

    Segarra, S. and G. Marques, A. and Ribeiro, A. Optimal Graph-Filter Design and Applications to Distributed Linear Network Operators. 2017

    2017

  57. [57]

    and Giannakis, G

    Shen, Y. and Giannakis, G. B. Online Identification OF Directional Graph Topologies Capturing Dynamic and Nonlinear Dependencies. 2018. 2018

    2018

  58. [58]

    and Yang, R

    Yin, L. and Yang, R. and Gabbouj, M. and Neuvo, Y. Weighted Median Filters: a Tutorial. 1996

    1996

  59. [59]

    Segarra, S. and G. Marques, A. and R. Arce, G. and Ribeiro, A. Center-Weighted Median Graph Filters. 2016. 2016

    2016

  60. [60]

    Segarra, S. and G. Marques, A. and R. Arce, G. and Ribeiro, A. Design of Weighted Median Graph Filters. 2017. 2017

    2017

  61. [61]

    The Backpropagation Algorithm

    Rojas, R. The Backpropagation Algorithm. In: Neural Networks. 1996

    1996

  62. [62]

    and Eisen, M

    Segarra, S. and Eisen, M. and Ribeiro, A. Authorship Attribution Through Function Word Adjacency Networks. 2015

    2015

  63. [63]

    and Wallace, D

    Mosteller, F. and Wallace, D. Inference and Disputed Authorship: The Federalist. 1964

    1964

  64. [64]

    Huang, W. and G. Marques, A. and Ribeiro, A. Rating Prediction via Graph Signal Processing. 2018

    2018

  65. [65]

    Monti, F. and M. Bronstein, M. and Bresson, X. Geometric Matrix Completion with Recurrent Multi-Graph Neural Networks. 2017

    2017

  66. [66]

    and Singh, Y

    Chandra, P. and Singh, Y. An Activation Function Adapting Training Algorithm for Sigmoidal Feedforward Networks. Neurocomputing. 2004

    2004

  67. [67]

    and He, R

    Ying, R. and He, R. and Chen, K. and Eksombatchai, P. and L. Hamilton, W. and Leskovec, J. Graph Convolutional Neural Networks for Web-Scale Recommender Systems. 24th ACM SIGKDD Int. Conf. on Knowledge Discovery & Data Mining. 2018

    2018

  68. [68]

    and Hinton, G

    Srivastava, N. and Hinton, G. and Krishevsky, A. and Sutskever, I. and Slakhutdinov, R. Dropout: A Simple Way to Prevent Neural Networks from Overfitting. Journal of Machine Learning Research. 2014

    2014

  69. [69]

    and Mateos, G

    Segarra, S. and Mateos, G. and Marques, A. G. and Ribeiro, A. Blind Identification of Graph Filters. 2016

    2016

  70. [70]

    Random Geometric Graphs

    Penrose, M. Random Geometric Graphs. 2007

    2007

  71. [71]

    Harper, F. M. and Konstan, J. A. The MovieLens Datasets: History and Context. ACM Trans. Interactive Intell. Syst. 2016

    2016

  72. [72]

    and Parise, F

    Avella-Medina, M. and Parise, F. and Schaub, M. and Segarra, S. Centrality Measures for Graphons: Accounting for Uncertainty in Networks. IEEE Trans. Netw. Sci. Eng. 2018

    2018

  73. [73]

    Large Networks and Graph Limits

    Lov \'a sz, L. Large Networks and Graph Limits. 2012

    2012

  74. [74]

    Notes on the sin 2 Theorem

    Seelmann, A. Notes on the sin 2 Theorem. Integral Equations and Operator Theory. 2014

    2014

  75. [75]

    Wolfe, P. J. and Olhede, S. C. Nonparametric Graphon Estimation. arXiv:1309.5936 [math.ST]. 2013

    Pith/arXiv arXiv 2013

  76. [76]

    and Naor, A

    Alon, N. and Naor, A. Approximating the Cut-Norm via G rothendieck's Inequality. Proc. 36th Annu. ACM Symp. on Theory Comput. 2004

    2004

  77. [77]

    Lax, P. D. Functional Analysis. 2002

    2002

  78. [78]

    Penrose, M. D. Connectivity of Soft Random Geometric Graphs. The Annals of Applied Probability. 2016

    2016

  79. [79]

    and Ribeiro, A

    Gama, F. and Ribeiro, A. Ergodicity in Stationary Graph Processes: A Weak Law of Large Numbers. 2019

    2019

  80. [80]

    Schaub, M. T. and Segarra, S. and Wai, H-T. Spectral Partitioning of Time-Varying Networks with Unobserved Edges. 44th. 2019

    2019

Showing first 80 references.