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REVIEW 3 major objections 5 minor 81 references

FADEx gives every original feature a single importance score per instance, explaining why a dimensionality-reduction method places each point where it does — using only the already-computed embedding, with no need to rerun the DR algorithm.

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2026-08-01 07:33 UTC pith:7ZPOJTFP

load-bearing objection A genuinely useful DR explanation framework with a real scale-dependence problem in its core attribution score; worth reviewing carefully. the 3 major comments →

arxiv 2607.27463 v1 pith:7ZPOJTFP submitted 2026-07-29 cs.LG cs.HC

FADEx: Feature Attribution and Distortion-based Explanation of Dimensionality Reduction

Lucas Greff Meneses , Evandro S. Ortigossa , Claudio Silva , Luis Gustavo Nonato This is my paper
classification cs.LG cs.HC
keywords feature attributiondimensionality reduction explainabilityJacobian estimationsingular value decompositionlocal linear approximationt-SNEUMAPdistortion metric
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.

FADEx sets out to make any dimensionality-reduction embedding explainable at the level of individual points and features. The idea is to approximate the DR mapping around each point by a local linear map — a Jacobian — fit with weighted least squares to the point's nearest neighbors and their known projections, so the method never has to rerun t-SNE, UMAP, Isomap, or LLE on new samples. A singular value decomposition of that local operator yields one attribution score per feature, a measure of local stretching or shrinking, and directional influence vectors, all from the same computation. If the method works as claimed, an analyst can ask not just where an embedding puts a point, but which original features put it there and how much the projection distorts the neighborhood. The paper supports the claim with synthetic ground-truth data, ablation tests against other attribution methods, and case studies, arguing FADEx matches or beats cluster-based and gradient-based alternatives while being faster and model-agnostic.

Core claim

The central claim is that a single scalar per feature, φ_j = Σ_i (λ_i/λ_1)|v_{ij} x_j|, computed from the SVD of a locally estimated Jacobian, captures how much each original feature drives an instance's placement in a DR embedding. That Jacobian is fitted by Gaussian-weighted ridge least squares over each point's neighbors and their known embeddings, removing the need for out-of-sample mapping and making the method agnostic to the DR algorithm. The same operator's columns yield the Feature Influence Vectors and its largest singular value yields the SND distortion score — attribution, distortion, and directional analysis from one computation. Ablation tests argue the scores are load-bearing:

What carries the argument

Load-bearing is the estimated Jacobian Ĵ_M(x) — the local linear map of the DR transformation at each point — fitted with Gaussian-weighted ridge least squares (Eq. 11) over displacement pairs of neighbors, in the original and projected spaces. Its SVD, Ĵ = UΣV^T, splits the local map into projection directions v_i, singular-value scalings λ_i, and output directions u_i. The attribution φ_j = Σ_i (λ_i/λ_1)|v_{ij} x_j| reads feature j's importance off the right singular vectors, normalized by the dominant singular value and weighted by the raw feature value x_j; the operator's columns are the Feature Influence Vectors; and the spectral norm λ_1 is the SND distortion score. One operator carrie

Load-bearing premise

Equation (6) assumes a feature's importance scales with its raw value: the sensitivity terms v_{ij} are multiplied by x_j, so on features with different units or scales the attribution can be dominated by large-magnitude features even when the DR mapping barely uses them — a modeling choice the paper makes without testing.

What would settle it

Construct a dataset with two features of very different scales whose true DR mapping depends only on the small-scale feature, compute FADEx attributions, and check whether the large-scale feature still receives the higher score. Simpler: rerun FADEx after rescaling every feature to unit variance and see whether the per-instance ranking changes; if it does, Eq. (6)'s dependence on raw |x_j| is driving the result rather than the mapping's actual sensitivity.

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

If this is right

  • FADEx explains any embedding produced by any DR method, including t-SNE and UMAP, which cannot map out-of-sample points; only the original data and its embedding are required.
  • Each feature receives exactly one attribution per instance, avoiding the two-scores-per-feature ambiguity of dimension-wise surrogate models and the per-cluster scores of cluster-based Shapley methods.
  • Ablating FADEx's top-ranked features degrades cluster structure in UMAP and t-SNE layouts far more than random ablation, and comparably to methods that use predefined clusters, indicating the scores mark features the embedding actually depends on.
  • The SND distortion score — the local operator's largest singular value — correlates with an established distance-preservation ratio across datasets, giving a per-point expansion/shrinkage map at no extra computational cost.
  • Per-instance explanations are about an order of magnitude faster and more memory-efficient than the compared methods on larger datasets, making the analysis practical at scale.

Where Pith is reading between the lines

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

  • The attribution multiplies sensitivity by the raw feature value |x_j|, so on data with features on different scales the ranking may reflect feature magnitudes rather than the mapping's true sensitivity; rescaling features to a common scale before computing φ_j is a natural variant the paper neither tests nor discusses.
  • The Dry Bean case study finds strong agreement in feature importance across t-SNE, UMAP, Isomap, and LLE (Spearman 0.65–0.84); if that agreement is general, FADEx scores on a fixed dataset could predict which features any DR method will rely on — a claim the paper leaves open.
  • As the paper notes in its limitations, when local PCA does not reduce dimensionality without distorting neighborhood structure, the estimated Jacobian and everything derived from it inherit that error; the method's reliability is therefore conditioned on neighborhoods being locally reducible.
  • The Feature Influence Vectors are, in effect, a per-feature displacement field; they could power an interactive perturb-and-see tool — or a test of attribution stability across random restarts of stochastic DR methods — neither of which the paper builds.

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

3 major / 5 minor

Summary. FADEx is a local feature-attribution method for explaining dimensionality reduction (DR) mappings. It estimates the Jacobian of the DR map at each instance by a weighted ridge least-squares fit over high-dimensional neighbors and their embeddings, avoiding out-of-sample mappings. The estimated Jacobian is decomposed by SVD, and the attribution score for feature j is defined as φ_j = Σ_{i=1}^d (λ_i/λ_1)|v_{ij} x_j| (Eq. 6). The same Jacobian also yields feature influence vectors (columns of the Jacobian) and a distortion metric SND (spectral norm). The authors claim that this single per-feature score is model-agnostic, requires no OOS data, is computationally efficient, and reliably identifies features that drive instance placement in the embedding. Validation includes a synthetic ground-truth experiment, comparisons with LXDR, ClusterShapley, and Corbugy, an ablation study, distortion validation against δ=d/D, and three case studies on tabular and image data.

Significance. If the central attribution claim holds, FADEx would be a practically valuable contribution: it is model-agnostic, avoids the out-of-sample problem that limits several DR explanation methods, assigns a single attribution per feature, and integrates attribution with distortion and directional analysis in one framework. The paper also ships an open-source implementation, which aids reproducibility. However, the central formula in Eq. (6) is an ad-hoc modeling choice that couples feature importance to raw feature magnitude. The validation evidence does not yet separate sensitivity from magnitude, and the real-data case studies use features with very different scales (e.g., Breast Cancer). Since the abstract and conclusion claim 'robust and reliable explanations,' this scale dependence is a load-bearing issue. The paper notes in Sec. 6 that explanations degrade when local PCA is unreliable, but the sensitivity of attributions to the pipeline's hyperparameters (α, PCA threshold, k) is not quantified. Thus the contribution is promising but the central claim needs additional support or a reformulated attribution rule.

major comments (3)
  1. [§3.1, Eq. (6)] The attribution formula is not derived from the Taylor expansion; it is introduced by the unstated step 'we set h_j = x_j.' This makes the importance score scale linearly with the raw feature value. The natural local-sensitivity-based measure for a perturbation along feature j would involve the norm of the j-th column of the Jacobian (e.g., sqrt(Σ_i λ_i^2 v_ij^2)|h_j|), not the L1-style weighted sum with λ_i/λ_1 and an additional factor |x_j|. Under feature rescaling x_j → c x_j, φ_j changes by |c| even when the embedding (e.g., on standardized data) is unchanged. No invariance argument, axiomatic justification, or test with heterogeneous feature scales is provided. The synthetic validation in §4.1 uses features all drawn from [0,1] with identical shift sizes, so the confound is invisible. The ablation study in §4.3.1 is also confounded: ablating high-magnitude features removes high-vari
  2. [§4.1 / §3.3] The experimental validation does not isolate sensitivity from feature magnitude. In the synthetic experiment, all ground-truth-relevant features are shifted by the same amount and initialized on the same interval, so any scale-dependent heuristic trivially recovers the 'important' features. The Jacobian approximation quality study (§3.3) only measures directional agreement of displacement vectors, not whether the estimated Jacobian's singular vectors yield correct attribution rankings. The paper should include an experiment where the ground-truth important feature has a deliberately small raw magnitude (or where features are standardized after ground truth is defined), and show that Eq. (6) still recovers it. Without such a test, Figures 3 and 6 do not establish that FADEx 'correctly identifies' the driving features beyond the same-scale scenario.
  3. [§3.2 / §6] The FADEx pipeline introduces several free parameters — Gaussian kernel scale σ (set to median distance), ridge term α=0.1, constant-feature variance threshold 10^{-6}, local PCA explained-variance threshold 0.95, and neighborhood size k (default 'roughly 10% of the dataset'). The paper acknowledges in Sec. 6 that the method's reliability depends on the PCA step, but no sensitivity analysis of the final attributions with respect to these parameters is reported. The supplementary partially analyzes k, but α and the PCA threshold are fixed without discussion. Since these are part of the method, the 'robust' claim requires at least a stability analysis over these hyperparameters, especially for high-dimensional datasets where local PCA is always applied.
minor comments (5)
  1. [§3.1] The notation 'h = h_j = [0,...,0,h_j,0,...,0]' is confusing: h is used as both a vector and a scalar component. Write h = h_j e_j for clarity.
  2. [Fig. 2] The vertical red line in panel (A) is not described in the caption. Please state what it marks (the 'optimum' neighborhood size) and how it was chosen.
  3. [§4.3 / Table 3] The statement that FADEx is 'nearly one order of magnitude faster' than the other methods is not supported for Breast Cancer, where FADEx and Corbugy both report 0.02 s. Please qualify the claim or report full distributions.
  4. [Table 1] The 'Speed' entries 'Fast' and 'Faster' are vague; define them operationally (e.g., relative to Table 3) or remove the column.
  5. [§5.3] The image case study would benefit from specifying how VGG16 fine-tuning was performed (learning rate, epochs, which layers were updated) and whether the attribution maps were quantitatively evaluated or only qualitatively inspected.

Circularity Check

0 steps flagged

No significant circularity: FADEx's attribution is a stated definition validated on external ground truth, with no fitted parameter or self-citation doing load-bearing work.

full rationale

FADEx's derivation chain is self-contained rather than circular. The Jacobian is estimated by weighted ridge least squares (Eq. 11) from local displacements in the original and embedded spaces, and the approximation quality is checked in Sec. 3.3 by a hold-out protocol that predicts held-out neighbor displacements and measures cosine similarity; this is a genuine external prediction, not a refit of the target. The attribution score φ_j in Eq. 6 is introduced as an explicit definition ('we set h_j = x_j... From this scenario, we define the importance/attribution'), not as a quantity fitted to reproduce some predetermined explanation. Although the h_j = x_j choice is a scale-dependent heuristic that deserves scrutiny, it is a stated modeling choice rather than a circular reduction: the formula does not presuppose the validation outcomes. The synthetic validation in Sec. 4.1 uses independently constructed ground-truth features, and the ablation study in Sec. 4.3.1 and supplementary Sec. G compares FADEx against random ablation and other methods, providing an external behavioral check. The SND distortion metric is validated against the independent distance-ratio measure δ (Sec. 4.2). Self-citations in the paper (e.g., refs. [48], [50], [52], [73]) are contextual literature references and are not load-bearing for the method's derivation; no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled in via citation. The main legitimate concerns—feature-scale sensitivity of Eq. 6 and the confound in the synthetic data—are correctness/robustness issues, not circularity.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The method relies on standard linear-algebra tools and a locally smooth manifold assumption. The main non-standard ingredient is the attribution formula itself, which is postulated rather than derived, and the hand-chosen hyperparameters for neighborhood size, ridge regularization, and PCA thresholds. No new physical entities are introduced.

free parameters (5)
  • neighborhood size k (n_neighbors) = 10% of dataset in most experiments
    Controls locality of the Jacobian fit; approximation quality changes substantially with k (Fig. 13).
  • ridge regularization α = 0.1 default
    Hand-set to stabilize the weighted least-squares system in Eq. (11).
  • local PCA explained-variance threshold = 0.95
    Retained variance for the local PCA preprocessing when dimensionality exceeds 50.
  • constant-feature variance threshold = 10^-6
    Features with below-threshold local variance are discarded before PCA/Jacobian estimation.
  • Gaussian kernel scale σ = median of neighbor distances
    Data-dependent bandwidth for the locality weights in Eq. (10).
axioms (5)
  • domain assumption The DR mapping M is locally smooth enough for a first-order Taylor expansion to hold in a neighborhood of each point.
    Required for Eq. (1); the authors test this empirically but it can fail for strongly nonlinear embeddings or poorly chosen neighborhoods.
  • domain assumption k-nearest neighbors in the original high-dimensional space form a local neighborhood suitable for fitting the Jacobian.
    The weighted least-squares fit in Eq. (11) assumes these neighbors are close enough and representative of the local geometry.
  • domain assumption The weighted ridge least-squares solution yields an accurate estimate of the true Jacobian.
    The method substitutes the estimated Jacobian for the true derivative; accuracy is measured only indirectly through cosine similarity and Pearson correlation.
  • ad hoc to paper Feature importance can be represented by φ_j = Σ_i (λ_i/λ_1)|v_ij x_j|.
    This definition is introduced in Eq. (6) without derivation and makes attributions depend on raw feature magnitudes; it is a modeling choice rather than a mathematical consequence.
  • domain assumption Local PCA preserves the local structure relevant to the DR mapping when dimensionality exceeds 50.
    Needed for the chain-rule Jacobian reconstruction; the authors themselves list this as a limitation in Sec. 6.

pith-pipeline@v1.3.0-daily-deepseek · 29038 in / 9714 out tokens · 103000 ms · 2026-08-01T07:33:26.758201+00:00 · methodology

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read the original abstract

Dimensionality Reduction (DR) is a fundamental tool for high-dimensional data exploration, reducing the complexity of latent spaces of machine learning models, and assisting in the explanation of complex opaque models. However, non-linear DR techniques often function as opaque transformations themselves, making it challenging to understand how individual features influence instance positioning in the reduced space. This lack of transparency complicates the analysis and interpretation of structural patterns, hindering the ability to reason about the organization of high-dimensional data based on the projected layout. In order to address this challenge, dimensionality reduction explanation methods have shown promise in improving the understanding of the observed groups and cluster structures. Unfortunately, existing DR explanation approaches tend to suffer from limitations such as multiple attributions per feature and restricted applicability to specific dimensionality reduction methods, which hinder their use. In this work, we propose FADEx, a novel local per-instance feature attribution method that leverages local linear approximation via first-order Taylor expansion and Singular Value Decomposition to provide explanations. FADEx computes the local linear models via weighted least squares, eliminating the need for out-of-sample data mapping, making it agnostic to the DR method, while simultaneously providing local feature attributions and distortion analysis. Through qualitative and quantitative evaluations, comparisons with existing methods, and case studies, we demonstrate FADEx's effectiveness and versatility in providing explanations and analytical resources for analyzing the behavior of DR methods. The results indicate FADEx yields robust and reliable explanations, outperforming existing approaches in several aspects.

Figures

Figures reproduced from arXiv: 2607.27463 by Claudio Silva, Evandro S. Ortigossa, Lucas Greff Meneses, Luis Gustavo Nonato.

Figure 1
Figure 1. Figure 1: The dimensionality reduction method transforms instances from [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Approximation quality of the local Jacobian for the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FADEx’s SND distortion visualization and its relation with neigh [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Ablation test. The left-most column presents the original [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FADEx’s explanation of the UMAP embedding shows that [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparing explanations across DR algorithms. Explanation vectors from all DR methods are clustered in the explanation space, and [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: t-SNE embedding of the Cats and Dogs dataset with FADEx [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Decomposing the DR mapping as a combination of PCA [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: (A) Projected space produced by t-SNE on the MNIST dataset. [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: (A) Interlocked rings artificial dataset visualized in its original [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 15
Figure 15. Figure 15: Ablation experiment of Section 4.3.1 applied to the Dermatol [PITH_FULL_IMAGE:figures/full_fig_p015_15.png] view at source ↗
Figure 14
Figure 14. Figure 14: Ablation sweep: silhouette of the ablated cluster as a function [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗
Figure 13
Figure 13. Figure 13: (A), (B) and (C) present the mean cosine similarity (top row) and Pearson correlation (bottom row) between predicted and observed [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 16
Figure 16. Figure 16: t-SNE mapping of 512-dimensional features extracted from [PITH_FULL_IMAGE:figures/full_fig_p017_16.png] view at source ↗

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