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REVIEW 4 major objections 5 minor 45 references

GBFRS: Robust Fuzzy Rough Sets via Granular-ball Computing

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Replacing sample points with granular-balls makes fuzzy rough set feature selection more robust under label and attribute noise.

desk verdict A promising granular-ball fuzzy rough set idea with a broken monotonicity guarantee and impossible error bars—revision needed before it can be trusted. read the letter →

arxiv 2501.18413 v1 pith:2YTJMX6O submitted 2025-01-30 cs.AI cs.LG

Shuyin Xia , Xiaoyu Lian , Binbin Sang , Guoyin Wang , Xinbo Gao This is my paper · ORCID
classification cs.AIcs.LG
keywords granular-ballcomputingfuzzyroughsetsfeatureselectionattributereductionnoiserobustnessgranulardependency
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to cure the noise sensitivity of fuzzy rough set feature selection. Instead of computing fuzzy similarities between individual samples, it first covers the dataset with granular-balls—groups of nearby points summarized by a center and radius and labeled by the majority class—and then runs the fuzzy rough set machinery on those balls. The resulting granular-ball fuzzy rough set (GBFRS) defines lower and upper approximations, a weighted fuzzy dependency function, and a forward attribute-selection algorithm built on it. Across ten UCI datasets, GBFRS holds up better than point-based fuzzy rough set baselines as label noise rises to 30%, and it is best or second-best on most datasets under attribute perturbation. If the claim holds, practitioners get a feature-selection method that trades a modest approximation for a large gain in robustness.

What carries the argument

The load-bearing object is the granular-ball, a cluster of samples represented only by its center $c$ and mean radius $r$, with quality controlled by a purity threshold on the majority class. Fuzzy similarity between balls uses only the Euclidean distance between centers, $GBR_a(GB_i, GB_j) = 1 - \Delta^a_2(c_i,c_j)/\sqrt{C}$, and the weighted granular-ball fuzzy dependency $W\partial_B(D) = \sum_i |GB_i| \cdot GBPOS_B(D)(GB_i) / |U|$ is what the forward search maximizes. Theorem 2's identity $W\partial^{i+1}_B = \sqrt{C_i/C_{i+1}} \, W\partial^i_B$ is the computational hinge: it turns a change in the distance parameter $C$ into a scalar rescaling, so attribute significance can be updated incrementally rather than recomputed from scratch.

What would settle it

A synthetic dataset with two classes separated only by feature variance, not by feature mean, would settle it: if GBFRS never selects the variance-only feature while a point-based fuzzy rough set does, the center-only ball representation is discarding discriminative information.

Watch

Extended reading notes

Core claim

The paper's central discovery is that fuzzy rough set reasoning can be carried out on granular-balls rather than points without losing the formal structure that makes dependence-based attribute reduction work. It defines fuzzy similarity between balls using only the Euclidean distance between their centers (Eq. 3), reproduces the upper and lower approximations in that setting (Definition 5), and introduces a weighted granular-ball fuzzy dependency $W\partial_B(D)$ (Definition 6) that weights each ball's positive-domain membership by the number of original samples it contains. This weighted dependency reduces exactly to the classical fuzzy dependency when every ball is a singleton (Theorem 1), is monotone with attribute-set inclusion (Property 3), and satisfies $W\partial^{i+1}_B = \sqrt{C_i/C_{i+1}} \, W\partial^i_B$ (Theorem 2), letting the forward search reuse prior computations. The empirical claim is that majority-label balls absorb minority noise, so the lower approximation—the part of the model that decides which features matter—stays accurate under heavy label noise and attribute perturbation.

Load-bearing premise

The load-bearing premise is that each granular-ball's center and majority label preserve the information needed for feature selection, so the ball's internal spread, shape, or label distribution can be ignored.

Editorial extensions

If this is right

  • Under label noise up to 30%, GBFRS degrades more gracefully than the point-based fuzzy rough set baselines in the reported experiments, often keeping the highest kNN accuracy.
  • With 10% attribute perturbation, GBFRS is either the best or the second-best feature selector on most of the ten UCI datasets tested.
  • Because the weighted dependency collapses to the classical fuzzy dependency when each ball holds one sample, GBFRS is a strict generalization of point-based fuzzy rough set attribute reduction.
  • The scaling identity in Theorem 2 lets the forward search update dependencies after a change in the distance parameter $C$ by a scalar factor, avoiding full recomputation.
  • Replacing $n$ sample points with $k$ granular-balls shrinks the similarity matrix from $n \times n$ to $k \times k$, so the method's per-iteration cost depends on the number of balls rather than the number of samples.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same ball-substitution principle should transfer to other instance-based and prototype-based learners, because the robustness mechanism—majority labels inside a ball absorbing minority noise—is not specific to fuzzy sets.
  • A stress test the paper does not run: if a feature's signal lives in within-ball variance rather than the center, GBFRS should fail to select it, which would expose the exact boundary of the center-only approximation.
  • The paper's closing limitation—that granular-ball computing is not fully adaptive because of the purity threshold—implies the threshold $T$ must be tuned per dataset, and an automatic rule for $T$ is a natural next step.
  • The experiments validate selected features only with kNN, so whether the robustness carries over to other classifiers remains an open check.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes GBFRS, a fuzzy rough set model in which sample points are replaced by granular-balls. It defines fuzzy similarity between balls from their centers, introduces upper/lower approximations and a weighted dependency function, and presents a forward-search feature-selection algorithm. The authors claim that the coarse-grained representation improves robustness to label and attribute noise, and they report experiments on UCI datasets comparing GBFRS with five fuzzy-rough/neighborhood feature-selection methods. The main theoretical result is a reduction to the classical dependency when each ball contains one point, and the main empirical claim is that GBFRS achieves the highest accuracy under increasing noise levels.

Significance. The idea of using granular-balls as the atomic units in a fuzzy rough set model is coherent and could be valuable: if the framework is sound, it may offer a principled way to improve the efficiency and noise robustness of fuzzy-rough feature selection while retaining a well-defined dependency measure. Theorem 1 is correct, and the authors provide a public code/data link, which is a strength for reproducibility. However, the significance is currently undercut by load-bearing problems: the central monotonicity argument relies on an inconsistent treatment of the distance parameter C, the convergence theorem has an invalid proof, the pseudocode for the feature-selection algorithm is not executable as stated, and the reported standard deviations in the experiments are impossible for accuracy values from a 5-fold cross-validation. These issues affect both the theoretical guarantee and the empirical support for the paper's central claim.

major comments (4)
  1. [Definition 4, Eq. (3), Property 1, and Theorem 3 (Section III-B/C)] The distance parameter C is used inconsistently. Definition 4 and Eq. (3) call C a fixed distance parameter, but the paragraph immediately after Definition 4 says 'C is a fixed distance parameter: the number of attribute sets B,' and Theorem 3 sets C = i = |B|. Property 1's proof compares r_B and r_A with the same C, which is valid only if C does not change with the attribute subset. Theorem 3 changes C when going from B to B ∪ {a}, so the monotonicity of Property 3 no longer follows. A concrete failure is: take two balls whose centers differ only in attribute a1 by distance 1 and are equal in a2; for B = {a1}, C = 1 gives similarity 0, while for A = {a1,a2}, C = 2 gives similarity 1 - 1/sqrt(2), violating Property 1. Since the forward-search algorithm in Algorithm 1 and the significance formula in Eq. (19) rest on Property 3, the theoretical guarantee for the feature-selection procedure is unsupported.
  2. [Theorem 4 (Section III-C)] The proof of Theorem 4 is invalid. The step asserting W∂Ci_B(D) ≤ W∂Ci_{B∪{ajk}}(D) ≤ W∂Ci_{B∪{ajk}∪{ajk+1}}(D) and the diminishing-returns inequality W∂Ci_{B∪{ajk}}(D) - W∂Ci_B(D) ≥ W∂Ci_{B∪{ajk}∪{ajk+1}}(D) - W∂Ci_{B∪{ajk}}(D) is introduced as 'classical fuzzy rough set theory' without proof, and it is not a consequence of any property established earlier in the paper. Additionally, the conclusion 'bounded monotonic, therefore converges to 0' is false: a decreasing sequence bounded below converges to some limit, but not necessarily to 0. The proof also compares significances of different attributes relative to different current sets (a_jk relative to B versus a_jk+1 relative to B∪{a_jk}), which does not establish convergence of SIG(a,B,D) for a fixed attribute a. Therefore Theorem 4, which is used to justify the stopping criterion of the feature-selection algorithm, is not proven.
  3. [Algorithm 1, lines 15-27] The pseudocode for feature selection contains logical errors. Line 15 gives the loop condition as 'while B' = ∅ or max_W∂ ≤ W∂ do'; with the initial values max_W∂ = 0 and W∂ = 0 this condition is true, and if no attribute improves the dependency, max_W∂ remains 0, W∂ remains 0, and the loop never terminates. Inside the loop, line 19 tests 'if W∂ > W∂''B' and then overwrites W∂ at line 20, so W∂ no longer holds the dependency of the current best subset; the comparison at line 24 between max_W∂ and W∂ is therefore comparing against a value that has already been reassigned. Line 17 already adds every candidate attribute to B'' during the inner loop, and line 25 adds Ni again, so the output set B'' can contain all attributes regardless of whether they improve the dependency. These issues make the algorithm non-reproducible and prevent verification of the claimed forward-search behavior.
  4. [Tables II and III (Section IV)] The reported standard deviations in Tables II and III are impossible for accuracy values. Accuracy is a proportion in [0,1], and for a 5-fold cross-validation the sample standard deviation across the five fold accuracies cannot exceed about 0.55; yet the tables report values such as 0.7871 ± 0.7888 (lymphography, FAR_FIE, 0% noise), 0.5116 ± 1.9454 (lymphography, FAR_FIE, 20% noise), and multiple entries in Table III exceeding 1.0. These entries cannot arise from the described experimental procedure. Since the central empirical claim of improved robustness rests on these tables, the experimental support is not credible as reported.
minor comments (5)
  1. [Abstract] There is a typo in the abstract: 'Muiti-garanularty' should be 'Multi-granularity'.
  2. [Property 2 proof (Section III-B)] The proof of Property 2 says 'according to Property 2, we have ΔB1 ≤ ΔB2,' which is circular; the reference should be to Property 1 or to the definition of the distance.
  3. [Definition 4, Eq. (3)] The text after Eq. (3) says C places GBRa in the interval '[0.1]'; this should be '[0,1]'.
  4. [Section IV, Table I] The text says 'we randomly select 9 UCI data sets,' but Table I lists 10 datasets; the count should be corrected.
  5. [Section I] The sentence 'Since fuzzy rough sets are very sensitive to noise in uncertainty data, the evaluation of uncertainty is not accurate' appears twice in the introduction; one occurrence should be removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: GBFRS's robustness claim is empirically benchmarked and its dependency definitions are derived in-paper; the C-parameter inconsistency is a rigor issue, not a circular reduction.

full rationale

The paper's central claim—that replacing sample points with granular-balls improves robustness to label and attribute noise—is supported by experiments against external baselines (FNRS, HANDI, FAR_FIE, FS_NDEM, FRDMAR) on UCI datasets, not by fitting a parameter to the target accuracy. The weighted dependency W∂B and the granular-ball fuzzy approximations are defined in the manuscript itself (Definitions 5 and 6), and the consistency with classical fuzzy dependency is explicitly checked in Theorem 1 rather than assumed. The authors' prior granular-ball work is cited for background and for the general granular-ball generation model (Eq. 2), but the fuzzy similarity relation (Eq. 3), approximations (Eqs. 8–10), and monotonicity properties are developed in this paper rather than imported as a black-box uniqueness or existence theorem. I find no load-bearing self-citation and no prediction that is forced by construction. The main caveats are correctness risks, not circularity: Definition 4 calls C a fixed distance parameter while Theorem 3 sets C = |B|, and the claim that SIG_i(a,B,D) > 0 for every a ∉ B is asserted from Definition 8's reduction condition rather than established from the model; the conclusion also notes that granular-ball computing is not fully adaptive due to the purity threshold. These issues weaken the rigor of the forward-search guarantee, but they do not make the derivation equivalent to its inputs.

Assumptions & free parameters 3 free parameters · 3 assumptions · 1 invented entities

The method rests on the authors' prior granular-ball computing model and on the choice to represent balls by centers only. The purity threshold and initial ball count are hand-set or tuned, and the distance parameter C is equated with attribute-subset size, which is an ad hoc choice central to the theory. No new physical entities are introduced; the GBFRS construct is testable via downstream accuracy.

free parameters (3)
  • purity threshold T = optimized in [0.6, 1], step 0.05
    Controls granular-ball generation; tuned per dataset, affecting the number and size of balls and thus the downstream dependency values.
  • initial number of granular-balls = sqrt(n) (square root of sample count)
    Chosen by the algorithm (Algorithm 1 line 1) as the k for initial k-means; a hand-set heuristic, not derived.
  • distance parameter C = |B| (cardinality of attribute subset)
    Set equal to the number of attributes in the current subset in Theorem 3; a modeling choice that makes similarity depend on feature-set size and is required for the monotonicity theorems.
assumptions (3)
  • domain assumption Granular-ball computing model (Eq. 2) and its quality constraint (purity threshold) are valid.
    Adopted from prior self-cited work; the paper assumes the standard granular-ball generation and its quality measure transfer to FRS.
  • domain assumption Fuzzy similarity between granular-balls can be computed from Euclidean distance between centers only.
    Definition 4 and Eq. (3); radii and internal distribution are ignored in the similarity relation.
  • ad hoc to paper The distance parameter C can be set to |B| without changing the nature of the fuzzy rough set model.
    Introduced in Theorem 3; no independent justification is given for this coupling, which is central to the convergence proofs.
invented entities (1)
  • Granular-ball fuzzy rough set (GBFRS) approximations and weighted dependency independent evidence
    purpose: Replaces sample points with balls to define lower/upper approximations and a weighted dependency function for feature selection.
    The construct is testable through downstream classification accuracy on noisy data, which provides an external handle beyond the definitions.

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Cite this review

Pith. "Pith review of GBFRS: Robust Fuzzy Rough Sets via Granular-ball Computing." pith.science (2026). https://pith.science/paper/2YTJMX6O

@misc{pith2026250118413,
  author       = {Pith},
  title        = {Pith review of: GBFRS: Robust Fuzzy Rough Sets via Granular-ball Computing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2YTJMX6O}},
  note         = {Machine review of arXiv:2501.18413}
}
read the original abstract

Fuzzy rough set theory is effective for processing datasets with complex attributes, supported by a solid mathematical foundation and closely linked to kernel methods in machine learning. Attribute reduction algorithms and classifiers based on fuzzy rough set theory exhibit promising performance in the analysis of high-dimensional multivariate complex data. However, most existing models operate at the finest granularity, rendering them inefficient and sensitive to noise, especially for high-dimensional big data. Thus, enhancing the robustness of fuzzy rough set models is crucial for effective feature selection. Muiti-garanularty granular-ball computing, a recent development, uses granular-balls of different sizes to adaptively represent and cover the sample space, performing learning based on these granular-balls. This paper proposes integrating multi-granularity granular-ball computing into fuzzy rough set theory, using granular-balls to replace sample points. The coarse-grained characteristics of granular-balls make the model more robust. Additionally, we propose a new method for generating granular-balls, scalable to the entire supervised method based on granular-ball computing. A forward search algorithm is used to select feature sequences by defining the correlation between features and categories through dependence functions. Experiments demonstrate the proposed model's effectiveness and superiority over baseline methods.

Figures

Figures reproduced from arXiv: 2501.18413 by the authors.

Figure 1
Figure 1. The comparison of traditional fuzzy rough sets and granular-ball fuzzy [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. First, the data is preprocessed, including normalizing [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 2
Figure 2. Attribute reduction process of fuzzy rough sets. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figures from the paper (5 more)
Figure 3
Figure 3. Figure 3: Take the data set “fourclass” as an example, the result of granular [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]
Figure 4
Figure 4. Figure 4: The flowchart of granular-balls generation. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: The illustration is that traditional fuzzy rough sets are not robust. [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: Schematic diagram of the robustness of granular-ball fuzzy rough [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: Taking the dataset “wine, audit_risk” as an example, the accuracy [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

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