Cluster Exploration using Informative Manifold Projections

John Y. Goulermas; Stavros Gerolymatos; Vladimir Gusev; Xenophon Evangelopoulos

REVIEW 1 major objections 2 minor 40 references

A manifold optimization of contrastive PCA combined with kurtosis projection pursuit generates embeddings that discount known structure while exposing remaining clusters.

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 →

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T0 review · grok-4.3

2026-05-24 07:25 UTC

load-bearing objection The paper combines contrastive PCA and kurtosis projection pursuit on a manifold to discount prior structure in DR, but the single mixing weight is an unanalyzed hyperparameter that controls the outcome. the 1 major comments →

arxiv 2309.14857 v3 submitted 2023-09-26 cs.LG cs.HC

Cluster Exploration using Informative Manifold Projections

Stavros Gerolymatos , Xenophon Evangelopoulos , Vladimir Gusev , John Y. Goulermas This is my paper

classification cs.LG cs.HC

keywords dimensionality reductioncontrastive PCAprojection pursuitmanifold optimizationcluster explorationprior knowledgeembeddingsvisual exploration

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.

The paper presents a dimensionality reduction approach designed to incorporate and then remove prior knowledge about a dataset so that any leftover patterns become visible in low-dimensional views. It does this by optimizing a linear mix of two goals: contrastive PCA removes the effects of the supplied prior information, while kurtosis projection pursuit pushes the projection toward directions that separate the data points. The optimization is carried out on a manifold to respect the geometry of the projection directions. Experiments on multiple datasets with three different kinds of prior information are used to show that the resulting embeddings support visual cluster exploration. An automated iterative framework is also supplied so that users can successively refine their views of the data.

Core claim

The paper claims that informative low-dimensional embeddings arise from solving a manifold optimization problem whose objective is a linear combination of contrastive PCA, which subtracts structure tied to supplied prior knowledge, and kurtosis projection pursuit, which promotes separation among the remaining data points; the resulting embeddings therefore factor out the known structure and reveal any underlying cluster organization that remains.

What carries the argument

Linear combination of contrastive PCA and kurtosis projection pursuit, posed and solved as a manifold optimization problem.

Load-bearing premise

The linear combination of contrastive PCA and kurtosis projection pursuit, once optimized on the manifold, will produce projections that separate the data after the prior structure has been removed.

What would settle it

On standard benchmark datasets supplied with explicit priors, the two-dimensional embeddings obtained by the method show no clearer separation of the remaining clusters than those produced by ordinary PCA or by contrastive PCA alone.

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

If this is right

The embeddings discount structure associated with different kinds of prior knowledge supplied by the user.
Any remaining underlying structure becomes visible for cluster exploration in the low-dimensional space.
An automated iterative framework allows successive refinement of the visual exploration process.
Empirical results across datasets with three distinct prior types support the utility of the approach.

Where Pith is reading between the lines

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

The same manifold formulation could be tested with nonlinear extensions of the two objectives to handle more complex data manifolds.
Successive applications of the method might allow systematic removal of multiple layers of known structure in a single pipeline.
The approach may be combined with interactive user feedback to refine the choice of prior at each iteration.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

The paper combines contrastive PCA and kurtosis projection pursuit on a manifold to discount prior structure in DR, but the single mixing weight is an unanalyzed hyperparameter that controls the outcome.

read the letter

The core contribution is a linear blend of two established techniques—contrastive PCA to remove known structure and kurtosis-based projection pursuit to find other separations—optimized on the manifold. They apply it to three kinds of prior knowledge and add an iterative visual exploration loop. That synthesis for this specific use case is new, and the empirical checks across datasets give a practical sense of how it behaves in standard settings. The automated framework is a useful addition for people who actually do visual cluster hunting with side information. The experiments appear to support the basic claim that the method can surface remaining structure once the prior is discounted. The soft spot is exactly the one in the stress-test note: the weighting scalar between the two objectives is a free parameter with no derivation, selection rule, or sensitivity study. Results therefore hinge on post-hoc choice of that coefficient, which weakens the guarantee that the output reliably isolates the intended remaining clusters rather than an arbitrary compromise. No load-bearing circularity or invented entities show up. The work is aimed at practitioners in interactive data analysis who already use DR tools and have partial labels or groupings they want to set aside. A reader in that niche can get value from the concrete implementation and the three-prior test cases. It is coherent enough on its own terms to deserve a serious referee who can check the optimization details and the parameter sensitivity in the full experiments.

Referee Report

1 major / 2 minor

Summary. The paper proposes a dimensionality reduction method for cluster exploration that incorporates prior knowledge by linearly combining contrastive PCA (to discount known structure) with kurtosis projection pursuit (to promote separation), formulated and solved as a manifold optimization problem. It claims this yields embeddings revealing remaining underlying structure and provides an automated iterative visual exploration framework, validated empirically on multiple datasets with three types of prior knowledge.

Significance. If the combined objective reliably isolates structure orthogonal to the supplied prior without strong dependence on manual tuning, the method would offer a practical extension of existing DR techniques for informed exploratory analysis. The manifold optimization framing and empirical coverage across prior types are positive elements, but the approach's utility hinges on robustness to its free parameter.

major comments (1)

[Method section (objective formulation)] Method section (formulation of the objective): the linear combination weight between contrastive PCA and kurtosis projection pursuit is introduced as an explicit scalar hyperparameter with no derivation, automatic selection rule, sensitivity analysis, or worst-case guarantee. Because the central claim requires that the optimized embedding isolates remaining structure after discounting the prior, and the skeptic correctly notes that this weight directly controls the trade-off, the absence of guidance on its value makes the headline result dependent on post-hoc tuning whose effect is unquantified.

minor comments (2)

[Abstract] The abstract states that the task is formulated as a manifold optimization problem but does not specify the manifold, retraction, or solver used; this detail should be added for reproducibility.
[Experiments] Figure captions and experimental tables should explicitly state the value(s) of the weighting parameter used for each reported result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments. We address the single major comment below.

read point-by-point responses

Referee: [Method section (objective formulation)] Method section (formulation of the objective): the linear combination weight between contrastive PCA and kurtosis projection pursuit is introduced as an explicit scalar hyperparameter with no derivation, automatic selection rule, sensitivity analysis, or worst-case guarantee. Because the central claim requires that the optimized embedding isolates remaining structure after discounting the prior, and the skeptic correctly notes that this weight directly controls the trade-off, the absence of guidance on its value makes the headline result dependent on post-hoc tuning whose effect is unquantified.

Authors: We agree that the scalar weight λ is introduced as a hyperparameter without an automatic selection procedure, theoretical derivation, or worst-case analysis in the submitted manuscript. Values were chosen empirically per dataset and prior type to achieve a practical balance between the two terms. To directly address the concern that the central claim depends on unquantified tuning, we will add a dedicated sensitivity analysis subsection (including quantitative metrics of cluster separation across a range of λ values) in the revised Method and Experiments sections. This will supply the missing guidance and robustness quantification. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper formulates the task as direct manifold optimization of an explicit linear combination of two existing objectives (contrastive PCA to discount prior structure and kurtosis projection pursuit for separation), with empirical validation on multiple datasets and prior types. No equations, self-citations, or fitted parameters are shown that reduce any claimed prediction or result to the inputs by construction; the central premise remains an optimization problem whose outputs are assessed independently via experiments rather than tautologically.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review is abstract-only so the ledger is necessarily incomplete; the approach rests on standard manifold optimization and the effectiveness of the two cited DR objectives without introducing new entities.

free parameters (1)

weighting parameter for linear combination of objectives
A linear combination of two objectives is described; the relative weight is a free parameter whose value is not specified in the abstract.

axioms (1)

standard math Manifold optimization can be applied to solve the combined contrastive PCA plus kurtosis objective
The task is explicitly formulated as a manifold optimization problem in the abstract.

pith-pipeline@v0.9.0 · 5694 in / 1423 out tokens · 25900 ms · 2026-05-24T07:25:20.606795+00:00 · methodology

0 comments

read the original abstract

Dimensionality reduction (DR) is one of the key tools for the visual exploration of high-dimensional data and uncovering its cluster structure in two- or three-dimensional spaces. The vast majority of DR methods in the literature do not take into account any prior knowledge a practitioner may have regarding the dataset under consideration. We propose a novel method to generate informative embeddings which not only factor out the structure associated with different kinds of prior knowledge but also aim to reveal any remaining underlying structure. To achieve this, we employ a linear combination of two objectives: firstly, contrastive PCA that discounts the structure associated with the prior information, and secondly, kurtosis projection pursuit which ensures meaningful data separation in the obtained embeddings. We formulate this task as a manifold optimization problem and validate it empirically across a variety of datasets considering three distinct types of prior knowledge. Lastly, we provide an automated framework to perform iterative visual exploration of high-dimensional data.

discussion (0)

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages · 1 internal anchor

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K. Puolamäki, E. Oikarinen, B. Kang, J. Lijffijt, and T. De Bie. Interac- tive visual data exploration with subjective feedback: An information- theoretic approach. In 2018 IEEE 34th International Conference on Data Engineering (ICDE), pages 1208–1211, 2018

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K. Puolamäki, E. Oikarinen, and A. Henelius. Guided visual exploration of relations in data sets.Journal of Machine Learning Research, 22(96): 1–32, 2021

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S. Roweis and L. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science (New York, N.Y.), 290:2323–6, 2001

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D. A. Senanayake, W. Wang, S. H. Naik, and S. K. Halgamuge. Self- organizing nebulous growths for robust and incremental data visualiza- tion. IEEE Transactions on Neural Networks and Learning Systems, 32: 4588–4602, 2019

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Siyuan and P

H. Siyuan and P. Wentzell. Re-centered kurtosis as a projection pursuit index for multivariate data analysis.Journal of Chemometrics, 28, 2014

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[34]

Smith, B

A. Smith, B. Laubach, I. Castillo, and V . M. Zavala. Data analysis using riemannian geometry and applications to chemical engineering. Computers & Chemical Engineering, 168:108023, 2022

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J. Tang, J. Liu, M. Zhang, and Q. Mei. Visualizing large-scale and high- dimensional data. In Proceedings of the 25th International Conference on World Wide Web. International World Wide Web Conferences Steer- ing Committee, apr 2016

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J. B. Tenenbaum, V . de Silva, and J. C. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290(5500): 2319, 2000

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F. J. Theis, T. P. Cason, and P. A. Absil. Soft dimension reduction for ica by joint diagonalization on the stiefel manifold. In T. Adali, C. Jutten, J. M. T. Romano, and A. K. Barros, editors, Independent Component Analysis and Signal Separation , pages 354–361. Springer Berlin Hei- delberg, 2009

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J. Townsend, N. Koep, and S. Weichwald. Pymanopt: A python toolbox for optimization on manifolds using automatic differentiation. Journal of Machine Learning Research, 17(137):1–5, 2016

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Sandal" and

L. van der Maaten and G. Hinton. Visualizing data using t-sne. Journal of Machine Learning Research, 9(86):2579–2605, 2008. A Dirichlet Process Gaussian Mixture Model A Gaussian mixture model with K components can be described by p(z|{θl}K l=1) = KX l=1 wlN (z|ml, C−1 l ), (6) where θl = {wl, ml, C−1 l } is the set of parameters for component l, wl are th...

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[1] [1]

A. Abid, M. J. Zhang, V . K. Bagaria, and J. Zou. Exploring patterns en- riched in a dataset with contrastive principal component analysis. Na- ture Communications, 9(1):2134, May 2018

work page 2018

[2] [2]

Absil, R

P.-A. Absil, R. Mahony, and R. Sepulchre. Optimization Algorithms on Matrix Manifolds. Princeton University Press, 2007

work page 2007

[3] [3]

TriMap: Large-scale Dimensionality Reduction Using Triplets

E. Amid and M. K. Warmuth. TriMap: Large-scale Dimensionality Re- duction Using Triplets. arXiv preprint arXiv:1910.00204, 2019

work page Pith review arXiv 1910

[4] [4]

J. R. Anderson. The adaptive nature of human categorization. Psycho- logical Review, 98(3):409–429, 1991

work page 1991

[5] [5]

Becker and R

B. Becker and R. Kohavi. Adult. UCI Machine Learning Repository,

work page

[6] [6]

DOI: https://doi.org/10.24432/C5XW20

work page doi:10.24432/c5xw20

[7] [7]

C. M. Bishop. Pattern Recognition and Machine Learning (Information Science and Statistics). Springer-Verlag, 2006

work page 2006

[8] [8]

Cavallo and Ç

M. Cavallo and Ç. Demiralp. Clustrophile 2: Guided visual clustering analysis. IEEE Transactions on Visualization and Computer Graphics, 25(1):267–276, 2019

work page 2019

[9] [9]

H. E. Cetingul and R. Vidal. Intrinsic mean shift for clustering on stiefel and grassmann manifolds. In 2009 IEEE Conference on Computer Vi- sion and Pattern Recognition, pages 1896–1902. IEEE, 2009

work page 2009

[10] [10]

Chiang, C.-I

S.-S. Chiang, C.-I. Chang, and I. Ginsberg. Unsupervised target detec- tion in hyperspectral images using projection pursuit. IEEE Transac- tions on Geoscience and Remote Sensing, 39(7):1380–1391, 2001

work page 2001

[11] [11]

Driscoll, Y

S. Driscoll, Y . MacMillan, and P. Wentzell. Sparse projection pursuit analysis: An alternative for exploring multivariate chemical data. Ana- lytical Chemistry, 12 2019

work page 2019

[12] [12]

Dua and C

D. Dua and C. Graff. UCI machine learning repository, 2019. URL http://archive.ics.uci.edu/ml

work page 2019

[13] [13]

Friedman and J

J. Friedman and J. Tukey. A projection pursuit algorithm for exploratory data analysis,” ieee transactions on computers, c-23, 881-889. Comput- ers, IEEE Transactions on, C 23:881 – 890, 10 1974

work page 1974

[14] [14]

Heiter, B

E. Heiter, B. Kang, R. Seurinck, and J. Lijffijt. Revised conditional t- sne: Looking beyond the nearest neighbors. In Advances in Intelligent Data Analysis XXI, page TBD, Cham, 2023. Springer International Pub- lishing

work page 2023

[15] [15]

Hotelling

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

work page 1933

[16] [16]

Hou and P

S. Hou and P. Wentzell. Fast and simple methods for the optimization of kurtosis used as a projection pursuit index. Analytica chimica acta, 704:1–15, 2011

work page 2011

[17] [17]

Huang and S

L.-K. Huang and S. Pan. Communication-efficient distributed PCA by Riemannian optimization. In H. D. III and A. Singh, editors, Proceed- ings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pages 4465–4474. PMLR, 13–18 Jul 2020

work page 2020

[18] [18]

Hyvärinen and E

A. Hyvärinen and E. Oja. Independent component analysis: algorithms and applications. Neural networks : the official journal of the Interna- tional Neural Network Society, 13 4-5:411–30, 2000

work page 2000

[19] [19]

P. Jaccard. Distribution de la flore alpine dans le bassin des dranses et dans quelques régions voisines. Bulletin de la Societe Vaudoise des Sciences Naturelles, 37:241–72, 01 1901

work page 1901

[20] [20]

B. Kang, D. G. García, J. Lijffijt, R. Santos-Rodríguez, and T. de Bie. Conditional t-sne: More informative t-sne embeddings. In 2021 IEEE 8th International Conference on Data Science and Advanced Analytics (DSAA), pages 1–2, 2021

work page 2021

[21] [21]

Krizhevsky and G

A. Krizhevsky and G. Hinton. Learning multiple layers of features from tiny images. Master’s thesis, Department of Computer Science, Univer- sity of Toronto, 2009

work page 2009

[22] [22]

P. C. Mahalanobis. On the generalized distance in statistics. Proceed- ings of the National Institute of Sciences (Calcutta), 2:49–55, 1936

work page 1936

[23] [23]

UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction

L. McInnes and J. Healy. Umap: Uniform manifold approximation and projection for dimension reduction. ArXiv, abs/1802.03426, 2018

work page internal anchor Pith review Pith/arXiv arXiv 2018

[24] [24]

R. M. Neal. Markov chain sampling methods for dirichlet process mix- ture models. Journal of Computational and Graphical Statistics, 9:249 – 265, 2000

work page 2000

[25] [25]

Pedregosa, G

F. Pedregosa, G. Varoquaux, A. Gramfort, V . Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V . Dubourg, J. Vander- plas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duch- esnay. Scikit-learn: Machine learning in Python. Journal of Machine Learning Research, 12:2825–2830, 2011

work page 2011

[26] [26]

Peltonen, W

J. Peltonen, W. Xu, T. Nummenmaa, and J. Nummenmaa. Fair neighbor embedding. In Proceedings of the 40th International Conference on Machine Learning, ICML’23. JMLR.org, 2023

work page 2023

[27] [27]

Peña and F

D. Peña and F. J. Prieto. Multivariate outlier detection and robust co- variance matrix estimation. Technometrics, 43(3):286–300, 2001

work page 2001

[28] [28]

Puolamäki, E

K. Puolamäki, E. Oikarinen, B. Kang, J. Lijffijt, and T. De Bie. Interac- tive visual data exploration with subjective feedback: An information- theoretic approach. In 2018 IEEE 34th International Conference on Data Engineering (ICDE), pages 1208–1211, 2018

work page 2018

[29] [29]

Puolamäki, E

K. Puolamäki, E. Oikarinen, and A. Henelius. Guided visual exploration of relations in data sets.Journal of Machine Learning Research, 22(96): 1–32, 2021

work page 2021

[30] [30]

C. E. Rasmussen. The infinite gaussian mixture model. In Proceedings of the 12th International Conference on Neural Information Processing Systems, NIPS’99, page 554–560. MIT Press, 1999

work page 1999

[31] [31]

Roweis and L

S. Roweis and L. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science (New York, N.Y.), 290:2323–6, 2001

work page 2001

[32] [32]

D. A. Senanayake, W. Wang, S. H. Naik, and S. K. Halgamuge. Self- organizing nebulous growths for robust and incremental data visualiza- tion. IEEE Transactions on Neural Networks and Learning Systems, 32: 4588–4602, 2019

work page 2019

[33] [33]

Siyuan and P

H. Siyuan and P. Wentzell. Re-centered kurtosis as a projection pursuit index for multivariate data analysis.Journal of Chemometrics, 28, 2014

work page 2014

[34] [34]

Smith, B

A. Smith, B. Laubach, I. Castillo, and V . M. Zavala. Data analysis using riemannian geometry and applications to chemical engineering. Computers & Chemical Engineering, 168:108023, 2022

work page 2022

[35] [35]

E. Stiefel. Richtungsfelder und fernparallelismus in n-dimensionalen mannigfaltigkeiten. Commentarii mathematici Helvetici , 8:305–353, 1935

work page 1935

[36] [36]

J. Tang, J. Liu, M. Zhang, and Q. Mei. Visualizing large-scale and high- dimensional data. In Proceedings of the 25th International Conference on World Wide Web. International World Wide Web Conferences Steer- ing Committee, apr 2016

work page 2016

[37] [37]

J. B. Tenenbaum, V . de Silva, and J. C. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290(5500): 2319, 2000

work page 2000

[38] [38]

F. J. Theis, T. P. Cason, and P. A. Absil. Soft dimension reduction for ica by joint diagonalization on the stiefel manifold. In T. Adali, C. Jutten, J. M. T. Romano, and A. K. Barros, editors, Independent Component Analysis and Signal Separation , pages 354–361. Springer Berlin Hei- delberg, 2009

work page 2009

[39] [39]

Townsend, N

J. Townsend, N. Koep, and S. Weichwald. Pymanopt: A python toolbox for optimization on manifolds using automatic differentiation. Journal of Machine Learning Research, 17(137):1–5, 2016

work page 2016

[40] [40]

Sandal" and

L. van der Maaten and G. Hinton. Visualizing data using t-sne. Journal of Machine Learning Research, 9(86):2579–2605, 2008. A Dirichlet Process Gaussian Mixture Model A Gaussian mixture model with K components can be described by p(z|{θl}K l=1) = KX l=1 wlN (z|ml, C−1 l ), (6) where θl = {wl, ml, C−1 l } is the set of parameters for component l, wl are th...

work page 2008