Twoblock clustering trees with coskewness-based dimension reduction: recovering piecewise multivariate linear regimes

Sven Serneels

REVIEW 5 major objections 4 minor 24 references

A regression tree that splits on the coskewness of predictor–response scores recovers the piecewise-linear regimes that covariance-based splits miss; decoupling the split from the leaf model lets it match black-box regressors.

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2026-08-01 09:28 UTC pith:GXJHWKZB

load-bearing objection A genuinely new coskewness twoblock estimator and a clean isolation study, but the regime-recovery headline rests on a confounded Table 1 and five seeds with no error bars. the 5 major comments →

arxiv 2607.20760 v1 pith:GXJHWKZB submitted 2026-07-22 stat.ME stat.ML

Twoblock clustering trees with coskewness-based dimension reduction: recovering piecewise multivariate linear regimes

Sven Serneels This is my paper

classification stat.ME stat.ML MSC 62H3062H2562J0562-08

keywords twoblock clustering treecoskewnessthird-order momentshigher-order power methodregression treesmultivariate responseregime recoverypiecewise linear models

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 introduces the twoblock clustering tree (tbtree), a deterministic decision tree for multivariate responses that places a local multivariate linear model in each leaf. Its central claim is that regime boundaries in piecewise-linear data are third-moment phenomena: a node spanning two regimes is a non-Gaussian mixture, and a split objective that maximizes the coskewness of the predictor and response scores separates those regimes more cleanly than the usual covariance objective. Because third moments are built into the split but not into prediction, the paper recommends a decoupled tree — a coskewness or combined split, with ordinary covariance models refit in the leaves — and shows it recovers simulated four-regime structure with higher adjusted Rand index while using fewer leaves. On two real benchmarks the decoupled tree matches or beats a 200-tree random forest and a temporal convolutional network on scale-weighted error while remaining a single tree of 10–16 readable local models.

Core claim

On its own terms, the paper establishes that the first predictor score of a twoblock model fitted to maximize coskewness — rather than covariance — is a better ordering for cutting a regression tree when the response surface is piecewise linear across several regimes. In a four-regime simulation, the coskewness objective with a k-means cut in the (t1,u1) score plane reaches an adjusted Rand index of 0.52–0.64 against 0.40–0.47 for the covariance objective, and the advantage widens as noise increases, while producing about six leaves for four regimes instead of ten to twelve. The paper then shows that the same tree, with leaves refit under the covariance objective, matches or beats every blac

What carries the argument

The load-bearing mechanism is the coskewness twoblock decomposition, a joint low-dimensional reduction of predictor and response blocks: instead of maximizing the covariance E[tu] between predictor score t and response score u, the latent directions solve max E[t²u] (predictor side) and max E[tu²] (response side), or a convex combination controlled by γ. The paper solves this with a matrix-free, block-symmetric higher-order power method that never materializes the third-moment tensor, so each direction costs O(n(p+q)) and is initialized deterministically from the covariance warm start plus coordinate starts. The tree consumes only the first X-score t1=Xw1 as a scalar ordering, searches a cut

Load-bearing premise

The load-bearing premise is that a node containing several regimes has third-order joint moments that separate the regimes more strongly than covariances do, and that the local power method finds that maximizing direction; the paper states this as an assumption and never verifies the moment dominance analytically for its simulation design.

What would settle it

Run the Section 6 four-regime design with regimes mirrored so the mixture has zero skewness but nonzero covariance structure; if a coskewness-split tree still beats a covariance-split tree on adjusted Rand index, the gain is not coming from third moments and the central mechanism is falsified.

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

If this is right

For piecewise multivariate linear data, splitting on coskewness finds the true regimes more reliably than covariance splits, and the advantage grows as noise increases (ARI 0.61 vs 0.40 at the highest noise level in the slope-only scenario).
A single decoupled tbtree can reach ensemble-level accuracy: best scale-weighted error on gas-turbine emissions among all compared methods, and energy-efficiency R² within noise of a 200-tree random forest.
The resulting model is a shallow tree of 10–16 leaf models whose coefficients can be read to see which predictor drives the response in each operating regime — something black-box ensembles cannot provide.
Decoupling matters: using coskewness in the leaves hurts prediction; using it only to choose the split preserves regime recovery while keeping covariance-based leaf regressions accurate.
The combined objective with γ≈0.5 is the recommended compromise on smooth nonlinear data; pure coskewness is best on sharply piecewise data, and covariance remains the default on small samples.

Where Pith is reading between the lines

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

An independent attribution test not run in the paper: compare a covariance-split tree using the k-means (t1,u1) cut against a coskewness-split tree using the exhaustive variance cut. The paper reports best configurations, so part of the ARI gain could come from the cut heuristic rather than the objective.
The balanced four-regime design (n_k=100 per regime) is roughly symmetric in its mixture structure, where odd moments can partially cancel. An asymmetric or unequal-size version would test whether coskewness remains the regime signal or whether the reported gain is specific to the design.
The solver's reliance on deterministic multi-start means the extracted direction depends on warm starts; a sensitivity analysis that perturbs the starts would show how stable the split directions are, especially on wide data.
If the mechanism is right, a natural bridge not explored in the paper is using coskewness-based twoblock scores for unsupervised regime discovery when the response block is replaced by auxiliary variables.

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

A genuinely new coskewness twoblock estimator and a clean isolation study, but the regime-recovery headline rests on a confounded Table 1 and five seeds with no error bars.

read the letter

I read this one carefully because the novelty is real. The coskewness-based twoblock decomposition — maximizing the third-order cross-moment of the score pair instead of the covariance — is a genuine addition to the dimension-reduction literature, and the matrix-free HOPM that never forms the moment tensor is a real algorithmic improvement. Nesting that in a deterministic regression tree with decoupled split and leaf objectives is a thoughtful design: third-moment routing for regime separation, covariance leaves for predictive accuracy.

The best part of the paper is the Section 4 isolation study. The antithetic sampling zeroes out every odd sample moment of z exactly, so cov(z,y)=0 by construction while E[z²y]=2. Showing that the covariance objective cannot rank the planted direction above the decoy, and that coskewness loads on it at |w|≈0.99, is transparent and honest. The paper also discloses its own failures — the predictive cost of skewness-aligned splits, the small-n concrete-slump degradation, the local convergence of the solver, the coupled-variant inferiority. That all earns credit.

The soft spots sit in the empirical core, and the stress-test note has the right one. Table 1 changes two factors at once: the covariance objective is paired with the exhaustive variance cut, while the coskewness objective is paired with the k-means (t1,u1) split. Figure 3 shows the k-means search lifts ARI for the covariance objective too, and the text concedes k-means 'pairs particularly well' with coskewness. So part of the headline margin plausibly belongs to the cut search, not the third-moment objective. Since regime recovery is the central claim, the paper needs the full objective×cut-search interaction table before the attribution is clean. The fix is easy — report coskewness under the exhaustive cut — but as presented, the comparison is confounded.

The rest is the usual statistical laundry: five seeds and no error bars anywhere, single train/test splits on both real benchmarks, undocumented baseline tuning, and code that is promised but not yet public. The mixture-skewness mechanism is asserted rather than derived, and a balanced four-regime mixture can have low overall skewness; the paper never checks that the third-order moments actually dominate at the tested noise levels. All fixable, all worth fixing.

I agree with the CONDITIONAL verdict. No load-bearing error or circularity; the claims are just under-supported as presented. The paper is worth a serious referee — send it out, and require the interaction table, error bars or repeated-split variance, baseline disclosure, and public code.

Referee Report

5 major / 4 minor

Summary. The paper proposes twoblock clustering trees (tbtree), a deterministic regression tree for multivariate responses in which each node fits a twoblock dimension-reduction model that defines the split direction, and each leaf carries a local multivariate linear twoblock model. A new coskewness-based twoblock decomposition is introduced, solved by a matrix-free higher-order power method. The paper argues through a controlled simulation that the coskewness objective recovers planted non-normal structure that covariance is blind to, and through a four-regime simulation that it recovers piecewise-linear regimes better than the covariance objective underlying PLS-Trees. The default configuration decouples the split objective (coskewness or combined) from the leaf objective (covariance), and two real-world benchmarks (energy efficiency, gas-turbine emissions) are used to claim that this decoupled tree matches or beats black-box ensembles while remaining a readable single tree.

Significance. If the empirical claims hold, the contribution is significant: a single, interpretable regression tree with local linear models approaching ensemble-level predictive accuracy, and the first twoblock dimension reduction based on coskewness, would be a useful addition to the interpretable-ML toolbox. The paper is honest about limitations (the small-n failure, the coupled variant's inferiority, and the predictive cost of coskewness splits are all disclosed) and ships reproducible notebooks. The matrix-free solver avoids materializing O(p^2q) tensors, which is a genuine algorithmic strength. However, the central attribution of regime-recovery gains to the coskewness objective is currently confounded with the cut-search heuristic, and the supporting simulation evidence lacks error bars. These issues are fixable with additional analysis and experiments, so the manuscript merits a major revision rather than rejection.

major comments (5)

[Section 6.2, Table 1] The headline comparison changes two factors simultaneously: the objective (covariance vs coskewness) and the cut-search (exhaustive variance cut on t1 vs k-means partition in the (t1,u1) plane). Figure 3 shows that the k-means variant also lifts ARI for the covariance objective, so the margin in Table 1 cannot be cleanly attributed to the third-moment objective. To support the claim that 'the coskewness objective recovers the regimes markedly better than the covariance objective,' report the full factorial design (all objectives × both cut-search heuristics), or at least coskewness+exhaustive and covariance+k-means, and quantify the interaction. Without this, the attribution is not established.
[Sections 6.1–6.2] The simulation conclusions rest on five seeds with no standard errors, confidence intervals, or per-seed values. Some of the reported ARI differences (e.g., 0.52 vs 0.47 at sigma_eps=0.1 in Scenario A) are small relative to what seed-to-seed variability could plausibly be. Report variability over seeds (or increase the number of seeds) for all cells of the design, and make the per-seed results available. This is load-bearing because the main claim is a comparative one based on these margins.
[Section 3.2, Eq. (5)] The combined objective is a convex combination of E[tu] and E[t^2u]. These two terms have different units and can differ by orders of magnitude even after autoscaling, so gamma is not a scale-invariant knob across datasets. The sharp basin transition in Figure 1(b) may be specific to that design. Specify how the two moments are normalized in the combined objective (e.g., dividing each term by a norm or a reference scale), or justify explicitly why autoscaling alone makes gamma interpretable. This affects the interpretation of all 'combined' results in Sections 6–7.
[Section 5.2, k-means partition] The k-means partition in the (t1,u1) score plane is central to the claimed ARI advantage in Table 1, yet the manuscript only says it is 'catalogued in the Supplementary Material' and does not describe the algorithm, initialization, or how its boundary is snapped to a threshold on t1. No supplementary material is provided with the manuscript. This is missing support for a load-bearing component. Provide the algorithmic details, or remove the reliance on an unavailable reference.
[Section 7.2, Table 3] The black-box baselines (TCN, Cubist, random forest, MLP, CART) are reported without hyperparameter settings, tuning protocol, or repeated evaluation, and the temporal split is a single realization. The claim that tbtree 'matches or beats every black-box baseline on scale-weighted error' needs either a clear statement of default configurations and a sensitivity analysis, or a more complete benchmark protocol. At minimum, specify the software versions, the random seeds, and the hyperparameters for each baseline.

minor comments (4)

[Throughout] Several typos and misspellings need correction: 'peicewise' (Abstract), 'thrid' (Section 3), 'twolock clusterin' and 'decison' (Section 5.1), 'bne explained' (Section 3.1), 'prtoposed' (Section 1), 'of of' (Section 2).
[Section 7.1, Figure 4] The text says the combined-split tree has eleven leaves, while the figure caption says '10 leaves' and appears to show ten leaves. Verify the count and make the text and caption consistent.
[Section 5.2] The sentence 'because the k-means partition ... is selected by a separate flag; it turns out to pair particularly well with the coskewness objective' is vague. If the k-means variant is recommended, state how it is chosen in practice and whether it is used in the real-data benchmarks.
[Section 9] The software section says the package 'will be released to the public in the upcoming release.' For reproducibility, please provide a versioned release (e.g., DOI or tag) or indicate that the examples/notebooks can be run with the current development version.

Circularity Check

0 steps flagged

No circularity: central claims rest on controlled simulations and external benchmarks; the Table 1 two-factor comparison is a confound, not a circular reduction.

full rationale

I find no step in which a claimed prediction or derivation reduces by construction to a fitted input or to a self-citation. Section 4's planted-skew design is a transparent controlled demonstration: the data are explicitly constructed with E[z^2 y]=2 and cov(z,y)=0 by antithetic sampling, and the simulation then shows the optimizer recovers the planted direction. That is a definitional sanity check of the objective, not a fitted target dressed as a prediction. Section 5's motivation for the third-moment split is explicitly stated as an assumption ('lead us to assume that including a third order comoment into the split impurity criterion is conducive to a more efficient separation'), not derived from the simulation as a theorem. Section 6's Table 1 does compare the coskewness objective combined with a k-means cut against the covariance objective with an exhaustive cut, so the comparison changes two factors at once; but that is a threat to causal attribution of the ARI gain, not a circular reduction, and Figure 3 discloses the full grid. The real-world benchmarks are external and self-contained: fixed test splits, disclosed defaults (gamma=0.5, A=B=0.3, tau_cv=0.01), and comparisons against independent baselines. The optimization algorithm is cited to De Lathauwer et al., not to the author. Self-citations (Serneels 2019, 2025, 2026a/b) support optional extensions or historical context and are not load-bearing for the central claims. The paper's own caveats about local convergence of HOPM and small-sample degradation of third moments are limitations, not circularity. Thus no circular step meets the evidentiary bar required by the review rules.

Axiom & Free-Parameter Ledger

5 free parameters · 4 axioms · 0 invented entities

The paper contributes a new estimator built on several hand-set knobs (γ, A, B, τcv) and two structural assumptions. The knobs are disclosed defaults rather than fitted values, which limits but does not eliminate their influence on the reported results. The core untested premise is the mixture-skewness mechanism, explicitly flagged as an assumption in Section 5; the HOPM multi-start adequacy and the autoscaling convention are additional domain assumptions. No invented physical entities (new particles/forces) are introduced; the 'coskewness decomposition' is a statistical estimator, not an entity.

free parameters (5)

γ (coskewness weight in combined objective) = 0.5 in the four-regime simulation; 0/1 at extremes; dataset-specific tuning recommended
Interpolates between covariance (γ=0, Eq. 2) and pure coskewness (γ=1, Eq. 4) in objective (5). Chosen by hand for the experiments, not estimated; it directly controls regime sensitivity vs predictive sharpness of the split.
Impurity weights A, B = defaults A=B=0.3
Weights of the within-child response term and the t1-homogeneity term in the Section 5.2 impurity; set as defaults, no sensitivity analysis reported.
CV-gate threshold τcv = 0.01 default
Split-acceptance threshold in Section 5.3 (Eq. 8); hand-set default, no sensitivity analysis.
Tree hyperparameters d_max, n_min = depth 4 in simulation; leaf counts 10–16 in benchmarks
Standard tree hyperparameters; the leaf count (hence 'number of regimes') is an emergent output, but the depth cap and minimum child size are user choices.
Sparsity penalties ηx, ηy = not used in the experiments
Optional soft-thresholding knobs for the sparse variant (Serneels 2025); part of the framework but no values are fit or reported here.

axioms (4)

domain assumption A node straddling two regimes is a mixture whose between-regime separation surfaces in the third-order joint moments (coskewness) before the second-order ones
Stated explicitly in Sections 3.1 and 5 ('lead us to assume...'); never verified analytically for the four-regime design, whose balanced (n_k=100) construction can produce zero skewness in symmetric equal-proportion mixtures.
domain assumption The matrix-free HOPM with deterministic multi-start reaches the relevant global maximizer of the non-convex objective (5)
Section 3.3 concedes 'the ascent being only locally convergent'; the coordinate restarts are heuristic, with no guarantee for wide p.
domain assumption Autoscaling makes the non-scale-invariant coskewness objective comparable across nodes
Section 3.2: 'coskewness... is not scale invariant; one has to take care that both blocks are always autoscaled.' Scaling a mixture node with pooled variance may blur regime differences; no analysis of this effect.
standard math Standard eigen-decomposition and power-iteration convergence facts for symmetric indefinite matrices
Used in Algorithm 1's power iteration on A(u)=X^T diag(u) X / n (De Lathauwer et al. 2000b); accepted background, though the sign-flip handling for the odd-in-v objective is heuristic.

pith-pipeline@v1.3.0-alltime-deepseek · 14281 in / 26211 out tokens · 203296 ms · 2026-08-01T09:28:38.063662+00:00 · methodology

0 comments

read the original abstract

The twoblock clustering tree (\tbtree) is introduced as a highly interpretable regression tree for multivariate responses. Twoblock trees are deterministic decision trees that have local multivariate linear models as their leaves and use dense or sparse twoblock dimension reduction as local leaf models and in the impurity. The resulting models are both computationally efficient and can be highly interpretable. Beyond proposing the decision tree estimator itself, this paper also introduces an estimator for the twoblock dimension reduced space based on maximizing coskewness, which facilitates identification of non-normal clusters in the data. The tree inherently produces a set of local linear models and is therefore apt to recover peicewise linear regimes, which is illustrated in a simulation. However, two real world data examples illustrate that twoblock trees are also capable of modeling more complexly nonlinear dependencies and can perform on par with black box modeling techniques, such as random forests. At each point, both the twoblock models that generate the splits, as well as the ones in the leaves, can be inspected and interpreted.

Figures

Figures reproduced from arXiv: 2607.20760 by Sven Serneels.

**Figure 2.** Figure 2: Schematic of a depth-one tbtree. The root (ellipse) fits a twoblock model under the coskewness split objective; its first X-score t1 = x ⊤ i w1 orders the rows, and a threshold τ routes each row to one of two leaves (rectangles). Every leaf carries its own covariance twoblock model yˆi = x ⊤ i Bℓ . The bar charts show, for a hypothetical six-variable predictor block, the coskewness split weights w1 and th… view at source ↗

**Figure 3.** Figure 3: Coskewness and regime recovery on the four-regime simulation (Scenario A): [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: UCI energy efficiency: a depth-four tbtree. The dendrogram exposes the hierarchical regime structure of the simulated buildings, with the deepest low-cost split separating the two overall-height levels. Bracket heights are the impurity costs at which each split was committed. almost as accurate as state-of-the-art black-box regressors, while only using a tiny fraction of the latter’s parameters. 7.2 Gas-t… view at source ↗

discussion (0)

Reference graph

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Z., and Koltun, V

Bai, S., Kolter, J. Z., and Koltun, V. (2018). An empirical evaluation of generic convolutional and recurrent networks for sequence modeling. arXiv:1803.01271

Pith/arXiv arXiv 2018

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and Dunn, J

Bertsimas, D. and Dunn, J. (2017). Optimal classification trees. Machine Learning , 106(7):1039--1082

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Bertsimas, D., Dunn, J., and Yang, Y. (2021). Optimal regression trees. Operations Research Letters , 49(5):715--720

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Breiman, L. (2001). Random forests. Machine Learning , 45(1):5--32

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H., Olshen, R

Breiman, L., Friedman, J. H., Olshen, R. A., and Stone, C. J. (1984). Classification and Regression Trees . Wadsworth and Brooks/Cole, Monterey, CA

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D., Forzani, L., and Liu, L

Cook, R. D., Forzani, L., and Liu, L. (2023). Partial least squares for simultaneous reduction of response and predictor vectors in regression. Journal of Multivariate Analysis , 196:105163

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de Jong, S. (1993). SIMPLS : An alternative approach to partial least squares regression. Chemometrics and Intelligent Laboratory Systems , 18(3):251--263

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De Lathauwer, L., De Moor, B., and Vandewalle, J. (2000a). An introduction to independent component analysis. Journal of Chemometrics , 14(3):123--149

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De Lathauwer, L., De Moor, B., and Vandewalle, J. (2000b). On the best rank-1 and rank- (r_1,r_2, ,r_n) approximation of higher-order tensors. SIAM Journal on Matrix Analysis and Applications , 21(4):1324--1342

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P., MacMillan, Y

Driscoll, S. P., MacMillan, Y. S., and Wentzell, P. D. (2020). Sparse projection pursuit analysis: an alternative for exploring multivariate chemical data. Analytical Chemistry , 92(2):1755--1762

2020

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Eriksson, L., Trygg, J., and Wold, S. (2009). PLS -trees, a top-down clustering approach. Journal of Chemometrics , 23(11):569--580

2009

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Goodfellow, I., Bengio, Y., and Courville, A. (2016). Deep learning. MIT Press

2016

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and Arabie, P

Hubert, L. and Arabie, P. (1985). Comparing partitions. Journal of Classification , 2(1):193--218

1985

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Kaya, H., T \"u fekci, P., and Uzun, E. (2019). Predicting CO and NOx emissions from gas turbines: Novel data and a benchmark PEMS . Turkish Journal of Electrical Engineering and Computer Sciences , 27(6):4783--4796

2019

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J., Serneels, S., and Verdonck, T

Menvouta, E. J., Serneels, S., and Verdonck, T. (2022). Sparse dimension reduction based on energy and ball statistics. Advances in Data Analysis and Classification , 16(4):951--975

2022

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Quinlan, J. R. (1992). Learning with continuous classes. In Proceedings of the 5th Australian Joint Conference on Artificial Intelligence , pages 343--348, Singapore. World Scientific

1992

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Quinlan, J. R. (1993). Combining instance-based and model-based learning. In Proceedings of the Tenth International Conference on Machine Learning , pages 236--243, San Mateo, CA. Morgan Kaufmann

1993

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Serneels, S. (2019). Projection pursuit based generalized betas accounting for higher order co-moment effects in financial market analysis. In JSM Proceedings, Business and Economic Statistics Section , pages 3009--3035, Alexandria, VA. American Statistical Association. Offprint available on arXiv: https://arxiv.org/abs/1908.00141

Pith/arXiv arXiv 2019

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Serneels, S. (2025). Sparse twoblock dimension reduction: a versatile alternative to sparse PLS2 and CCA . Journal of Chemometrics , 39(8):e70051

2025

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Serneels, S. (2026a). Cellwise robust twoblock dimension reduction. Submitted for publication; preprint available at https://arxiv.org/abs/2604.15106

Pith/arXiv arXiv

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Serneels, S. (2026b). Robust twoblock simultaneous dimension reduction. Submitted for publication; preprint available at https://arxiv.org/abs/2603.24820

Pith/arXiv arXiv

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and Zhang, J

Shao, X. and Zhang, J. (2014). Martingale difference correlation and its use in high-dimensional variable screening. Journal of the American Statistical Association , 109(507):1302--1318

2014

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and Xifara, A

Tsanas, A. and Xifara, A. (2012). Accurate quantitative estimation of energy performance of residential buildings using statistical machine learning tools. Energy and Buildings , 49:560--567

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