Alternatives to the Laplacian for Scalable Spectral Clustering with Group Fairness Constraints

Iv\'an Ojeda-Ruiz; Leonardo Cambisaca; Malcolm Dickens; Young Ju Lee

arxiv: 2510.20220 · v3 · submitted 2025-10-22 · 💻 cs.LG · cs.NA· math.NA

Alternatives to the Laplacian for Scalable Spectral Clustering with Group Fairness Constraints

Iv\'an Ojeda-Ruiz , Young Ju Lee , Malcolm Dickens , Leonardo Cambisaca This is my paper

Pith reviewed 2026-05-18 04:24 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NA

keywords spectral clusteringgroup fairnessLagrangian reformulationSherman-Morrison-Woodbury identitybalance constraintsscalable algorithmsnetwork clusteringalgorithmic bias

0 comments

The pith

Fair spectral clustering runs twice as fast with new Laplacian choices

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

The paper develops Fair-SMW to make spectral clustering with group fairness constraints more scalable. It reformulates the problem using the Lagrangian method and applies the Sherman-Morrison-Woodbury identity to handle the constraints efficiently. Three alternative Laplacian matrices are introduced, each with distinct spectral gaps, to create variations of the algorithm. Tests on synthetic Stochastic Block Models and real datasets such as LastFM, FacebookNet, Deezer, and German show that the approach delivers clustering balance comparable to prior work but with substantially better runtime, reported as twice as fast and capable of twice the balance. This matters for applying fair clustering to larger networks where previous methods become impractical due to computation cost.

Core claim

Fair-SMW reformulates the constrained optimization problem using a new formulation derived from the Lagrangian method and the Sherman-Morrison-Woodbury identity, resulting in three variants based on alternatives to the Laplacian matrix that achieve clustering solutions with comparable balance to existing algorithms while offering improved runtime performance.

What carries the argument

The Sherman-Morrison-Woodbury identity applied to the Lagrangian reformulation, enabling efficient updates to solve the fairness-constrained eigenvalue problem using smaller matrices.

If this is right

Clustering solutions achieve balance comparable to state-of-the-art methods.
Computation time improves by a factor of two over existing fair spectral clustering algorithms.
The approach is flexible enough to achieve up to twice as much balance in some configurations.
Performance is validated on both synthetic Stochastic Block Model data and real-world networks including LastFM, FacebookNet, Deezer, and German.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This method could be extended to other fairness notions beyond group balance if similar Lagrangian forms can be derived.
The choice of which alternative Laplacian to use might provide control over the trade-off between clustering quality and fairness in untested scenarios.
Scalability improvements may make fair clustering feasible for very large graphs where current methods time out.
Connections to other low-rank update techniques in numerical linear algebra could yield further efficiency gains in related machine learning tasks.

Load-bearing premise

The alternative Laplacian matrices preserve the spectral properties required for accurate clustering and for satisfying the fairness balance constraints when the Sherman-Morrison-Woodbury identity is applied to the Lagrangian reformulation.

What would settle it

Experiments showing that on the tested datasets the runtime of Fair-SMW is not at least twice as fast as the state-of-the-art or that the balance achieved is not comparable or better.

Figures

Figures reproduced from arXiv: 2510.20220 by Iv\'an Ojeda-Ruiz, Leonardo Cambisaca, Malcolm Dickens, Young Ju Lee.

**Figure 4.1.** Figure 4.1: Example of GF = 0 By applying the Sherman-Morrison-Woodbury formula, we obtain (G −1 + µF F T ) −1 = G − µGF(I + µF T GF) −1F T G = G − GF(µ −1 I + F T GF) −1F T G. For µ ≫ 1, we note that the asymptotic behavior is the following: (G −1 + µF F T ) −1 → G − GF(F T GF) −1F (4.10) T G = G(I − F(F T GF) −1F (4.11) T G) We now arrive at the following optimization problem (4.12) max HT H=I HT G(I − F(F T GF) −… view at source ↗

**Figure 7.1.** Figure 7.1: (Left) Balance, (Center) Algorithm Run Time, (Right) Eigensolver Run [PITH_FULL_IMAGE:figures/full_fig_p017_7_1.png] view at source ↗

**Figure 7.2.** Figure 7.2: Comparison of eigenvalue gaps, runtimes, and scalability for SMW and S [PITH_FULL_IMAGE:figures/full_fig_p021_7_2.png] view at source ↗

read the original abstract

Recent research has focused on mitigating algorithmic bias in clustering by incorporating fairness constraints into algorithmic design. Notions such as disparate impact, community cohesion, and cost per population have been implemented to enforce equitable outcomes. Among these, group fairness (balance) ensures that each protected group is proportionally represented within every cluster. However, incorporating balance as a metric of fairness into spectral clustering algorithms has led to computational times that can be improved. This study aims to enhance the efficiency of spectral clustering algorithms by reformulating the constrained optimization problem using a new formulation derived from the Lagrangian method and the Sherman-Morrison-Woodbury (SMW) identity, resulting in the Fair-SMW algorithm. Fair-SMW employs three alternatives to the Laplacian matrix with different spectral gaps to generate multiple variations of Fair-SMW, achieving clustering solutions with comparable balance to existing algorithms while offering improved runtime performance. We present the results of Fair-SMW, evaluated using the Stochastic Block Model (SBM) to measure both runtime efficiency and balance across real-world network datasets, including LastFM, FacebookNet, Deezer, and German. We achieve an improvement in computation time that is twice as fast as the state-of-the-art, and also flexible enough to achieve twice as much balance.

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.

Desk Editor's Note private letter to a colleague

Fair-SMW speeds up group-fair spectral clustering via SMW on alternative Laplacians, but the impact on eigenvector quality for actual clustering needs checking.

read the letter

This paper's main contribution is a faster algorithm for spectral clustering under group fairness constraints. By recasting the problem with a Lagrangian that includes balance terms and then applying the Sherman-Morrison-Woodbury identity, they avoid some of the heavy matrix operations in prior approaches. They also test three different Laplacian matrices with varying spectral gaps instead of sticking to the usual normalized or unnormalized version. The result is called Fair-SMW, and the abstract claims it runs about twice as fast as current methods while allowing more flexibility in achieving balance. The experiments look reasonable for what they set out to do. Using the Stochastic Block Model lets them control the graph structure and measure both speed and how well the fairness constraints are met. They also run on real social network data from LastFM, FacebookNet, Deezer, and German, which covers some variety in size and density. If those runtime numbers hold up with proper baselines, it could be a useful drop-in improvement for anyone already using spectral methods on graphs where fairness matters. One area that could use more attention is whether the alternative Laplacians still produce eigenvectors that capture the natural clusters after the SMW correction is applied. Spectral clustering works because the bottom eigenvectors reflect the graph's connected components or communities, but swapping the Laplacian changes the eigenvalues and possibly the eigenspace. The fairness constraints are linear and get handled through the reformulation, but if that distorts the Rayleigh quotient or the null space in uncontrolled ways, the final partitions might not be as good as the balance numbers suggest. It would help to see more on how they measure overall clustering performance, not just balance and time. Readers who work on fair machine learning for networks or who need scalable clustering with constraints would get the most out of this. It is not a complete overhaul of the field, but the practical speed angle is worth checking out. I would send this to peer review. The method is straightforward to understand and the empirical results address a real bottleneck, so referees can help tighten the analysis on the spectral side.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes the Fair-SMW algorithm for spectral clustering under group fairness (balance) constraints. It reformulates the constrained eigenproblem via the Lagrangian and applies the Sherman-Morrison-Woodbury identity to obtain a scalable solver, then introduces three alternative Laplacian matrices with differing spectral gaps to produce algorithm variants. Experiments on the stochastic block model and real networks (LastFM, FacebookNet, Deezer, German) are used to claim that the resulting clusterings achieve balance comparable to prior fair spectral methods while delivering roughly twice the speed and greater flexibility in the balance-runtime trade-off.

Significance. If the central claims hold, the work would offer a practical route to fair clustering on larger graphs by reducing the cost of enforcing linear balance constraints. The idea of swapping in alternative Laplacians to modulate the spectral gap is a potentially useful degree of freedom. However, the significance hinges on whether the reported speed-ups and balance improvements are reproducible and whether the spectral properties required for both clustering quality and constraint satisfaction survive the SMW update; without that, the efficiency gains may not translate to reliable fair partitions.

major comments (2)

[§3.2 and §4.1] §3.2 (alternative Laplacian definitions) and §4.1 (Lagrangian + SMW reformulation): the manuscript asserts that the three proposed Laplacians preserve eigenvectors suitable for partitioning while still allowing the fairness constraints to be enforced after the SMW correction, yet provides neither a perturbation analysis nor a bound on the change in the Rayleigh quotient or null-space. Because the clustering correctness and the reported balance improvements rest on this preservation, the absence of such analysis is load-bearing.
[Experimental section] Experimental section (SBM and real-network results): the abstract and results claim a factor-of-two improvement in runtime and flexibility to achieve twice the balance, but the reported figures lack error bars, explicit baseline implementations, dataset statistics (e.g., number of nodes/edges, group sizes), and ablation isolating the effect of each alternative Laplacian. These omissions prevent verification that the quantitative claims are independent of the chosen parameter regimes or datasets.

minor comments (2)

[§3] Notation for the three alternative Laplacians is introduced without a compact table comparing their definitions, eigenvalues, or spectral gaps; adding such a table would improve readability.
[Related work] The manuscript cites prior fair spectral clustering work but does not discuss how the SMW-based solver differs from existing constrained-eigenvalue techniques that also avoid full matrix inversion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments on our manuscript. We address each of the major concerns below, providing clarifications and indicating the specific revisions we will make to improve the rigor and reproducibility of the work.

read point-by-point responses

Referee: [§3.2 and §4.1] §3.2 (alternative Laplacian definitions) and §4.1 (Lagrangian + SMW reformulation): the manuscript asserts that the three proposed Laplacians preserve eigenvectors suitable for partitioning while still allowing the fairness constraints to be enforced after the SMW correction, yet provides neither a perturbation analysis nor a bound on the change in the Rayleigh quotient or null-space. Because the clustering correctness and the reported balance improvements rest on this preservation, the absence of such analysis is load-bearing.

Authors: We thank the referee for identifying this gap. The three alternative Laplacians are constructed by adding or scaling specific rank-one or low-rank updates to the standard normalized Laplacian, which by design preserves the kernel (the all-ones vector for connected components) while increasing the algebraic connectivity. The SMW identity is then applied to the Lagrangian-augmented system, so the fairness constraints remain linear and are satisfied exactly at the solution of the corrected eigenproblem. We agree that an explicit perturbation argument would make this preservation clearer. In the revised manuscript we will add a short subsection in §3.2 that derives a first-order bound on the change to the Rayleigh quotient induced by the Laplacian modification and shows that the null-space perturbation remains O(ε) where ε is the size of the added update; we will also note how this bound interacts with the SMW correction to keep the fairness constraints feasible. revision: yes
Referee: [Experimental section] Experimental section (SBM and real-network results): the abstract and results claim a factor-of-two improvement in runtime and flexibility to achieve twice the balance, but the reported figures lack error bars, explicit baseline implementations, dataset statistics (e.g., number of nodes/edges, group sizes), and ablation isolating the effect of each alternative Laplacian. These omissions prevent verification that the quantitative claims are independent of the chosen parameter regimes or datasets.

Authors: We acknowledge that the current experimental presentation is insufficient for full verification. In the revision we will (i) report all runtime and balance metrics with error bars computed over 10 independent random seeds, (ii) add a table listing the number of nodes, edges, and protected-group sizes for every dataset (SBM instances and the four real networks), (iii) describe the baseline fair spectral clustering implementations in greater detail, including the exact solver used for the original constrained eigenproblem, and (iv) include an ablation table that isolates the contribution of each of the three alternative Laplacians to both runtime and achieved balance. These additions will demonstrate that the reported speed-up and balance flexibility are robust across the tested regimes. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses standard Lagrangian + SMW on proposed alternative Laplacians, with empirical validation

full rationale

The paper reformulates the fairness-constrained spectral clustering problem via Lagrangian and the Sherman-Morrison-Woodbury identity to obtain Fair-SMW, then substitutes three alternative Laplacian matrices chosen for their spectral gaps. Clustering quality and balance are evaluated empirically on SBM synthetic graphs and real networks (LastFM, FacebookNet, Deezer, German), reporting runtime and balance metrics. No equation reduces a claimed prediction to a fitted parameter by construction, no load-bearing uniqueness theorem is imported via self-citation, and the central spectral-preservation assumption is presented as a hypothesis tested on held-out data rather than asserted by definition. The reported speed and balance gains are therefore independent empirical outcomes, not tautological restatements of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard linear-algebra identities and domain assumptions about graph Laplacians and fairness constraints; no new entities are introduced.

axioms (2)

standard math The Sherman-Morrison-Woodbury identity can be applied to the matrix updates arising from the Lagrangian formulation of the fairness-constrained problem.
Invoked to obtain the efficient Fair-SMW algorithm.
domain assumption Alternative Laplacian matrices with different spectral gaps still support valid spectral clustering while allowing the fairness balance constraints to be satisfied.
Central modeling choice that enables the three Fair-SMW variants.

pith-pipeline@v0.9.0 · 5765 in / 1261 out tokens · 38880 ms · 2026-05-18T04:24:45.511234+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Fair-SMW employs three alternatives to the Laplacian matrix with different spectral gaps... Gsym = D^{-1/2}W D^{-1/2} + 2I, Grw = D^{-1}W + 2I, Gaff = W + nI
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We achieve an improvement in computation time that is twice as fast as the state-of-the-art

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 4 internal anchors

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J. Shi and J. Malik , Normalized cuts and image segmentation , IEEE Transactions on pattern analysis and machine intelligence, 22 (2000), pp. 888–905

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W ang, D

J. W ang, D. Lu, I. Davidson, and Z. Bai , Scalable spectral clustering with group fairness constraints, 2023, https://arxiv.org/abs/2210.16435. This manuscript is for review purposes only. ALTERNATIVE TO THE LAPLACIAN FOR FAIR CLUSTERING 21 Eigenvalue Index (sorted)1 1.5 2 2.5 30 0.5 1 1.5 2 2.5 3 3.5 4Eigenvalue Magnitude #10-3 S-Fair-SC Relative Eigeng...

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Unsupervised Extraction of Representative Concepts from Scientific Literature

S. Barocas, M. Hardt, and A. Narayanan , Fairness in machine learning: A survey , arXiv preprint arXiv:1710.02271, (2017)

work page internal anchor Pith review Pith/arXiv arXiv 2017

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D. P. Bertsekas , Nonlinear programming, vol. 2, Athena Scientific, 1999

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Boyd and L

S. Boyd and L. V andenberghe , Convex optimization, Cambridge university press, 2004

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A. Chhabra, K. Masalkovait ˙e, and P. Mohapatra , An overview of fairness in cluster- ing, IEEE Access, 9 (2021), pp. 130698–130720, https://doi.org/10.1109/ACCESS.2021. 3114099

work page doi:10.1109/access.2021 2021

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Fair Clustering Through Fairlets

F. Chierichetti, R. Kumar, S. Lattanzi, and S. V assilvitskii , Fair clustering through fair- lets, 2018, https://arxiv.org/abs/1802.05733

work page internal anchor Pith review Pith/arXiv arXiv 2018

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F. R. Chung , Spectral graph theory, vol. 92, American Mathematical Soc., 1997

work page 1997

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Certifying and removing disparate impact

M. Feldman, S. Friedler, J. Moeller, C. Scheidegger, and S. Venkatasubramanian , Certifying and removing disparate impact , 2015, https://arxiv.org/abs/1412.3756

work page internal anchor Pith review Pith/arXiv arXiv 2015

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R. A. Horn and C. R. Johnson , Matrix analysis, Cambridge University Press, 2013

work page 2013

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Hotelling, Analysis of a complex of statistical variables into principal components., Journal of Educational Psychology, 24 (1933), pp

H. Hotelling, Analysis of a complex of statistical variables into principal components., Journal of Educational Psychology, 24 (1933), pp. 498–520

work page 1933

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Guarantees for Spectral Clustering with Fairness Constraints

M. Kleindessner, S. Samadi, P. A wasthi, and J. Morgenstern , Guarantees for spectral clustering with fairness constraints , 2019, https://arxiv.org/abs/1901.08668

work page internal anchor Pith review Pith/arXiv arXiv 2019

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R. B. Lehoucq, D. C. Sorensen, and C. Yang , Implicitly restarted arnoldi methods for the computation of eigenvalues, SIAM Journal on Matrix Analysis and Applications, 19 (1998), pp. 1018–1032

work page 1998

[12] [12]

J. Li, Y. W ang, and A. Merchant , Spectral Normalized-Cut Graph Partitioning with Fairness Constraints, IOS Press, 2023, https://doi.org/10.3233/faia230416, http://dx.doi.org/10. 3233/FAIA230416

work page doi:10.3233/faia230416 2023

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Mitchell, A

E. Mitchell, A. D’Amour, S. Kim, and M. Singh , Algorithmic fairness in the context of criminal justice: a review of the state-of-the-art , arXiv preprint arXiv:2102.06739, (2021)

work page arXiv 2021

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Narayanan, Tutorial: 21 fairness definitions and their implications , Tutorial at the 2020 Conference on Fairness, Accountability, and Transparency (FAT* ’20), (2020)

A. Narayanan, Tutorial: 21 fairness definitions and their implications , Tutorial at the 2020 Conference on Fairness, Accountability, and Transparency (FAT* ’20), (2020)

work page 2020

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A. Y. Ng, M. I. Jordan, and Y. Weiss , On spectral clustering: Analysis and an algorithm , Advances in neural information processing systems, 14 (2002), pp. 849–856

work page 2002

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Nocedal and S

J. Nocedal and S. Wright , Numerical optimization , Springer Science & Business Media, 2006

work page 2006

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Ojeda-Ruiz and Y.-J

I. Ojeda-Ruiz and Y.-J. Lee , A fast constrained image segmentation algorithm , Results in Applied Mathematics, 8 (2020), p. 100103, https://www.sciencedirect.com/science/article/ pii/S2590037420300145. Special Issue on Recent Advances in Computational Mathematics and Applications

work page 2020

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Pothen, H

A. Pothen, H. D. Simon, and K.-P. Liou , Spectral methods for partitioning graphs and block matrices, Linear algebra and its applications, 1 (1990), pp. 1–22

work page 1990

[19] [19]

Saad, Numerical methods for large eigenvalue problems, Manchester University Press, 1992

Y. Saad, Numerical methods for large eigenvalue problems, Manchester University Press, 1992

work page 1992

[20] [20]

Shi and J

J. Shi and J. Malik , Normalized cuts and image segmentation , IEEE Transactions on pattern analysis and machine intelligence, 22 (2000), pp. 888–905

work page 2000

[21] [21]

D. C. Sorensen , Implicitly restarted arnoldi methods , SIAM Journal on Matrix Analysis and Applications, 13 (1992), pp. 357–385

work page 1992

[22] [22]

von Luxburg , A tutorial on spectral clustering , Statistics and computing, 17 (2007), pp

U. von Luxburg , A tutorial on spectral clustering , Statistics and computing, 17 (2007), pp. 395–416

work page 2007

[23] [23]

W ang, D

J. W ang, D. Lu, I. Davidson, and Z. Bai , Scalable spectral clustering with group fairness constraints, 2023, https://arxiv.org/abs/2210.16435. This manuscript is for review purposes only. ALTERNATIVE TO THE LAPLACIAN FOR FAIR CLUSTERING 21 Eigenvalue Index (sorted)1 1.5 2 2.5 30 0.5 1 1.5 2 2.5 3 3.5 4Eigenvalue Magnitude #10-3 S-Fair-SC Relative Eigeng...

work page arXiv 2023