Learnable Similarity and Dissimilarity Guided Symmetric Non-Negative Matrix Factorization

Wenlong Lyu; Yuheng Jia

REVIEW 3 major objections 5 minor 36 references

Learnable Similarity and Dissimilarity Guided Symmetric Non-Negative Matrix Factorization

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that symmetric nonnegative matrix factorization can be made robust to unreliable large-k nearest-neighbor graphs by learning one weight per neighbor slice for a similarity graph and a second weight vector for a dual…

desk verdict The empirical recipe is solid and reproducible, but the paper's headline convergence guarantee does not hold as written because R(V) is never defined as a global objective. read the letter →

arxiv 2412.04082 v1 pith:I6IB72AT submitted 2024-12-05 cs.LG

Wenlong Lyu , Yuheng Jia This is my paper · ORCID

classification cs.LG

keywords symmetricnonnegativematrixfactorizationadaptivesimilaritylearningweightedk-nearest-neighborgraphdissimilarityorthogonalityregularizationclusteringKKTstationarity

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

The paper argues that the standard fixed k-nearest-neighbor similarity matrix in symmetric nonnegative matrix factorization (SymNMF) is both wasteful and unreliable, since later neighbor slices are mostly noise. It replaces that fixed matrix with a weighted sum of the k-th neighbor slices, learning one nonnegative weight per slice; this collapses the graph search space from O($n^{2}$) entries to n-1 weights. A second learned weight vector builds a dual dissimilarity graph from the same slices, telling the factorization which pairs are actively dissimilar rather than merely unobserved, and a new orthogonality regularizer keeps the cluster-indicator columns well separated. The paper claims an alternating algorithm with monotone convergence to a KKT point and reports the best ACC/NMI on 13 of 16 benchmark comparisons against nine clustering methods.

What carries the argument

The load-bearing object is the k-th-nearest-neighbor slice matrix A(k), with unit Frobenius norm after normalization and disjoint support from every other slice, so ⟨A(k), A(t)⟩=0 and S(w), D(p) decompose into sums of weights. That orthogonality turns the whole graph-learning task into a low-dimensional simplex problem: updating w and p is a strongly convex quadratic with simplex constraints, solved by O(n log n) projection, and the density penalties ∥w∥^2, ∥p∥^2 come directly from ∥S(w)∥^2_F=Σ $w_k^{2}$. On the factor side, the argument runs through the identity det(V^T V)=det(V_{-j}^T V_{-j}) v_j^T(I-V_{-j}V_{-j}^†)v_j, which supplies the regularizer R(v_j), the geometric reading of R(v_j) as the squared distance from column v_j to the span of the other columns, the bound 0≤R(v_j)≤∥v_j∥^2, and a column-wise update that fits the progressive hierarchical alternating least-squares (PHALS) scheme with monotone decrease and KKT convergence. Proposition 2 is the bridge between the relaxed (w,p) problem and the original nonconvex constraint w^T p=0: the relaxed global optimum is also optimal for the constrained problem whenever Σ_k w_k p_k=0, a condition the authors observe to hold in their experiments but state needs further study.

What would settle it

Run Algorithm 1 with η=0.99μ on a dataset where later neighbor slices are not predominantly wrong-class (e.g., periodic or interleaved clusters), and measure Σ_k w*_k p*_k after convergence; a clearly positive value, together with a nonzero KKT residual for (22), would refute the claimed stationarity of the original model.

Watch

Extended reading notes

Core claim

The paper's core assertion is that the k-th-nearest-neighbor slices A(k) of a kernel matrix are normalized, mutually orthogonal building blocks, so a weighted combination S(w) = Σ_k w_k A(k) spans every useful similarity graph in only n-1 dimensions. Learning those weights instead of every entry of S makes adaptive similarity practical, and the learned weights concentrate on early slices whose neighbor relations agree with true clusters. The same slices with a second weight vector p build a dual dissimilarity graph D(p), with w^T p=0 forcing every slice to serve exactly one role; a large entry of D(p) tells the factor matrix VV^T that a pair is dissimilar rather than merely unobserved. For discrimination, the paper introduces the column-wise orthogonality regularizer R(v_j)=v_j^T(I-V_{-j}V_{-j}^†)v_j, the squared length of the component of v_j orthogonal to the space spanned by the other columns, which follows from det(V^T V)=det(V_{-j}^T V_{-j})R(v_j), is bounded, and cannot vanish even when a column is driven to zero. With alternating updates — rank-one SymNMF steps for each column of V and simplex projections for w and p — the paper claims every variable reaches a stationary point satisfying the KKT conditions of the relaxed model, and empirically the method records the best ACC/NMI on 13 of 16 comparisons.

Load-bearing premise

The load-bearing premise is that the relaxed weight problem (37) returns a pair (w*,p*) with Σ_k w*_k p*_k = 0, which the paper asserts "almost always holds" without proof, because if it fails the algorithm has solved a penalized surrogate and the claimed KKT stationarity for the original model (22) does not follow.

Editorial extensions

If this is right

The search for an adaptive similarity matrix in clustering drops from O(n^2) free entries to n-1 learned weights, so adaptive graph learning becomes feasible at larger sample sizes.
The learned coefficient vector w automatically assigns high weight to reliable early neighbor slices and near-zero weight to unreliable later ones, which removes the need to hand-tune the k in k-NN for the similarity graph; p learns the opposite pattern for the dissimilarity graph.
Because each column update of V is a rank-one SymNMF step with a monotone objective and the (w,p) updates are global optima of a strongly convex problem, the whole algorithm has a convergence guarantee and an O(n^2 r + n r^2 + n log n) per-iteration cost.
The orthogonality regularizer R(v_j) is bounded and cannot collapse a column to zero, avoiding the all-zero-cluster failure mode of off-diagonal orthogonality penalties while retaining a geometric meaning as squared distance to the other columns' span.

Reading between the lines

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

If the slice decomposition carries the same benefit outside SymNMF, the weighted k-NN parameterization could replace full similarity-matrix learning in spectral clustering or graph-based semi-supervised learning, inheriting the same n-1 dimensional search space.
The paper's own open condition — Σ_k w*_k p*_k = 0 almost always holds — invites a stress test: datasets with non-monotone neighbor reliability curves (e.g., periodic cluster layouts) are exactly where the relaxation and the original model may diverge, and plotting the overlap across such datasets would show how much the unproven condition matters.
A direct check of the surrogate gap (compare the objective of the original problem (36) at the relaxed solution against a locally optimized feasible solution of (36)) would quantify how much the relaxation changes the model, something the paper does not report.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the method is worth knowing about, but the main theoretical claim needs to be retracted or rewritten. The weighted k-NN slice idea is neat: instead of learning an n×n similarity matrix, you learn n weights on fixed slices, cutting the search space to n−1 and giving a clean story about why later slices are less reliable. The dissimilarity dual is natural, and the experiments are extensive—eight datasets, nine baselines, ablations, and released code. The reported gains are consistent, so the empirical side deserves credit.

Where it falls apart is the theory. Model (22) has a term -αR(V), but the paper never defines R(V) as a function of V. Remark 1 says explicitly that it cannot be expressed as a function of V; you only get R(v_j) with V_{-j} fixed. If you take R(V) to be Σ_j R(v_j), then updating v_j changes R(v_i) for i≠j through V_{-j}†, so the column subproblem (33) is not a restriction of (22). The PHALS monotonicity from Lemma 1 applies to the surrogate, not to the original objective, and the KKT conditions in (47) are for the column-wise surrogate too. Thus Theorem 2's claim that the variables converge to a KKT point of (22) is unsupported. Theorem 1's monotone decrease has the same problem. This is not a minor missing proof; it is a structural gap in the central theoretical claim.

A second, softer issue is Proposition 2's bridge from the relaxed (37) to the constrained (36). It only works if w*^T p* = 0, which the authors say "almost always holds" without proof. Their experiments (Fig. 4) do show zero overlap, so empirically it works, but the guarantee is not established.

Minor: the performance comparisons lack significance tests, so the "best in 13/16 cases" claim is weaker than it looks.

I would not cite the convergence theorem. I would cite the method and the empirical results, since those are concrete and reproducible. A serious referee should engage with the paper, but the authors should be asked to either prove monotonicity and KKT claims for a well-defined global objective, or downgrade the theory to a weaker statement about the column-wise surrogate.

Referee Report

3 major / 5 minor

Summary. The paper proposes a SymNMF-based clustering model in which the similarity matrix is a weighted sum of k-NN slices, S(w)=Σ_k w_k A(k), with learnable nonnegative weights w summing to one, and a dual dissimilarity matrix D(p) built from the same slices with weights p. The model also includes a new column-wise orthogonality regularizer R(v_j)=v_j^T(I - V_{-j}V_{-j}^†)v_j, which the authors argue is more stable than log-det regularization. An alternating algorithm (Algorithm 1) is proposed: V is updated column-wise by a rank-one PHALS step, and (w,p) is updated by solving a strongly convex relaxation of the simplex-constrained problem. The paper claims that the variables converge to a KKT point of the constrained model (22) (Theorem 2) and reports state-of-the-art clustering ACC/NMI on 13 of 16 benchmark comparisons, with code released.

Significance. If the theoretical claims were correct, the paper would make a useful contribution: reparameterizing similarity learning in an n-1 dimensional space is an elegant way to reduce the search complexity relative to O(n^2) adaptive graph methods, and the column-wise orthogonality regularizer has a clear geometric interpretation and is potentially more practical than log-det variants. The experimental study is reasonably broad (eight datasets, nine baselines) and the ablations show consistent gains from the dissimilarity term and the orthogonality regularizer. The release of code and the reproducibility-oriented experimental description are strengths. However, the convergence and KKT guarantees -- a central advertised selling point -- are not rigorously established as written, and one of the two equivalence steps (Proposition 2) relies on an unproven empirical condition. The practical method may still be sound, but the paper's main theoretical contribution needs substantial repair.

major comments (3)

[Section III-C, Remark 1; Eqs. (32)-(34); Theorems 1-2] The regularizer R(V) is never defined as a global function of V. Remark 1 states that R(V) cannot be explicitly expressed as a function of V, and the subproblem (33) only contains R(v_j) with V_{-j} fixed. If one interprets R(V) as Σ_j R(v_j), then the column subproblem (33) is not the restriction of (22)/(32) to v_j, because the terms R(v_i) for i≠j depend on v_j through V_{-i} (and specifically through the projection V_{-i}V_{-i}^†). Lemma 1 is cited from [29] for the standard SymNMF objective, where no such coupling is present, so it does not imply monotone decrease of the full objective (22). Thus Theorem 1's claim that the objective of (22) decreases monotonically, and Theorem 2's claim that the iterates converge to a KKT point of (22), are unsupported. The authors need to either define a proper global regularizer and prove descent for it, or explicitly state that the convergence guarantee applies only to the surrogate column-wise problem (34) and analyze the discrepancy.
[Section IV-B, Proposition 2; Section V-C] The algorithm solves the relaxed problem (37), and Proposition 2 states that its global optimum (w*, p*) is also a global optimum of the constrained problem (36) only under condition (38), namely Σ_k w*_k p*_k = 0. The authors write that this condition 'almost always holds' and that 'the reason for which needs further study', which is not a proof. Section V-C provides only a post-hoc empirical observation that the learned w and p appear to have disjoint supports. If (38) fails, the algorithm solves a penalized surrogate, not the constrained model, and the KKT claim for (w,p) in Theorem 2 is not justified. The paper should either prove (38) under explicit assumptions or replace the KKT statement with a convergence result for the penalized problem (37).
[Section IV-C, Theorem 2 proof; Eq. (47)-(50)] The KKT conditions (47) are derived for the rank-one surrogate (34), not for the coupled objective (22). The proof that the PHALS update drives a stationarity gap to zero (Eq. (50)) is a restatement of the convergence result for the surrogate; it does not establish joint stationarity of (V, w, p) for the original constrained problem (22), because the V-update is not an exact minimization of (22) (the column subproblem ignores the couplings in R(V)), and the (w,p)-update solves a relaxed problem. A standard block-coordinate-descent KKT argument requires exact or sufficiently faithful subproblem solutions for the same objective, and that condition is not verified here. Consequently, the strong claim in the abstract and Theorem 2 is not supported by the presented analysis.

minor comments (5)

[Table II and Table III] Several entries in Tables II and III appear garbled, with values repeated or concatenated (e.g., the RBSMF and S3NMF rows). Please ensure the final version renders each cell exactly once with the correct values, since the current formatting makes it difficult to verify the reported '13/16 best' claim.
[Section IV-D, computational complexity] The per-iteration complexity statement is inconsistent: line 8 computes V_{-j}^† by reduced SVD for each j in O(n(r-1)^2), which summed over j gives O(n r^3), not O(n r^2) as claimed in the summary 'O(n^2 r + n r^2)'. The complexity analysis should be corrected or the SVD computed more efficiently.
[Eq. (24)] The equivalence between (22)'s V-subproblem and (24) holds only up to additive constants independent of V (specifically the term -β<S,D> + (β^2/2)||D||^2). Please state this explicitly to avoid confusion.
[Section V-A] The hyper-parameter tuning protocol is not fully specified: it is not stated whether the 'best' hyper-parameters were selected on a validation set or on the test data, and whether the reported metrics are from the same runs used for selection. Clarifying this would strengthen the reproducibility of the comparisons.
[Section I and III-A] The paper introduces D as a 'k-farthest neighbor graph' (Section I) but then defines D(p) using the same nearest-neighbor slices A(k) in Eq. (20). Since A(k) are ordered by nearest-neighbor rank, the term 'farthest' is confusing; consider clarifying that the dissimilarity weights p are learned over the same slices, with larger k acting as far-neighbor relations.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the model's weights are learned from the NMF objective, and the paper's own noted limitations are correctness gaps rather than circular reductions.

full rationale

The derivation of the weighted-slice SymNMF model (22) is self-contained: S(w) is defined as a linear combination of k-th NN slices, and the (w,p) subproblem (25) follows by direct algebra. The correct-rate curve (Fig. 1a) is used only as motivation for Assumption 1 and is not fed into the objective; the learned w is later observed to correlate with correct rate post hoc, which is not circular. The paper's own remarks flag two limitations: Remark 1 states that R(V) cannot be expressed as a single function of V, so the global objective (22) containing -αR(V) is not well-posed as written; and Section IV-B admits the condition w*^T p* = 0 required to equate the relaxed problem (37) to the constrained (36) 'almost always holds' with no proof. These are genuine correctness gaps in the claimed KKT guarantee, but they are not circularity: the PHALS theorem [29] is external prior work, and the issue is an invalid or unproven application, not a conclusion that is assumed in the input. Self-citations appear only in related work and are not load-bearing for the new model or theorems. No experimental number is a renamed fitted parameter; performance is evaluated on external benchmarks. Thus no step in the derivation reduces to its own input by construction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on two stated modeling assumptions, a linear-independence property of the k-NN slices, the PHALS convergence theorem, and an unproven condition on the relaxation of the w^T p = 0 constraint. No new physical entities are introduced.

free parameters (4)

alpha (orthogonality regularization weight) = tuned in {0.01, 0.03, 0.07, 0.1, 0.3, 0.7, 1}; default 0.1
Controls contribution of R(V); tuned per dataset on a grid.
beta (dissimilarity regularization weight) = tuned in {1,5,10,50,100,500,1000}; default 10
Controls D(p) contribution; tuned per dataset on a grid.
mu (density regularization weight) = tuned in {0.05,0.07,0.1,0.3,0.5,0.7,1}; default 0.1
Controls density of S and D; tuned per dataset on a grid.
eta (penalty for w^T p = 0) = set to 0.99*mu
Relaxation parameter; tied to mu, but the equivalence to the original constraint is not proven.

assumptions (5)

domain assumption Assumption 1: higher correct rate implies higher reliability
Used to justify weighted k-NN parameterization (Section I).
domain assumption Assumption 2: ideal S(w*) corresponds to discriminative clustering
Motivates dissimilarity term and orthogonality regularization (Section I).
domain assumption The normalized k-NN slices A(k) are mutually orthogonal with unit norm
Used to derive properties in Section III-B; can fail with ties or asymmetric neighbor definitions.
standard math PHALS convergence theorem (Theorem 2 of [29]) applies to subproblem (34)
Lemma 1 invokes this for the rank-one update; M may be asymmetric, and the equivalence via symmetrization is not fully addressed.
ad hoc to paper Condition w*^T p* = 0 holds for the relaxed problem (37)
Proposition 2 requires this for equivalence to (36); authors state it 'almost always holds' but give no proof (Section IV-B).

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Cite this review

Pith. "Pith review of Learnable Similarity and Dissimilarity Guided Symmetric Non-Negative Matrix Factorization." pith.science (2026). https://pith.science/paper/I6IB72AT

@misc{pith2026241204082,
  author       = {Pith},
  title        = {Pith review of: Learnable Similarity and Dissimilarity Guided Symmetric Non-Negative Matrix Factorization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I6IB72AT}},
  note         = {Machine review of arXiv:2412.04082}
}

abstract

Symmetric nonnegative matrix factorization (SymNMF) is a powerful tool for clustering, which typically uses the $k$-nearest neighbor ($k$-NN) method to construct similarity matrix. However, $k$-NN may mislead clustering since the neighbors may belong to different clusters, and its reliability generally decreases as $k$ grows. In this paper, we construct the similarity matrix as a weighted $k$-NN graph with learnable weight that reflects the reliability of each $k$-th NN. This approach reduces the search space of the similarity matrix learning to $n - 1$ dimension, as opposed to the $\mathcal{O}(n^2)$ dimension of existing methods, where $n$ represents the number of samples. Moreover, to obtain a discriminative similarity matrix, we introduce a dissimilarity matrix with a dual structure of the similarity matrix, and propose a new form of orthogonality regularization with discussions on its geometric interpretation and numerical stability. An efficient alternative optimization algorithm is designed to solve the proposed model, with theoretically guarantee that the variables converge to a stationary point that satisfies the KKT conditions. The advantage of the proposed model is demonstrated by the comparison with nine state-of-the-art clustering methods on eight datasets. The code is available at \url{https://github.com/lwl-learning/LSDGSymNMF}.

Figures

Figures reproduced from arXiv: 2412.04082 by the authors.

**Figure 1.** (a) Correct rate of each k-th NN slice A(k) . (b) Clustering ACC trained by standard SymNMF [7] on the ORL dataset with respect to k, where the kernel function κ(xi, xj ) is defined by the self-tuning method [12]. linearly non-separable data [9], [10], [11]. In SymNMF, S is usually constructed by k-nearest neighbor (k-NN) graph as follows: sij = ( κ(xi , xj ), if xj is a k-NN of xi 0, otherwise , (3) where xi ∈ R m … view at source ↗

**Figure 2.** The learned w∗ in (6) (left y-axis) and the correct rate (right y-axis). It can be seen that w∗ is sparse and consistent with correct rate. as a similarity matrix. The larger (V V T )ij is, the more likely xi and xj are belonging to the same class. However, as shown in [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** The geometric meaning of R(v3), which can be seen as the square of distance between v3 and the plane spanned by {v1, v2}. 3) sij (w) = 0 and dij (p) = 0, which means that the relation between xi and xj is unknown. One might think that this case would lead to (V V T )ij ≈ 0. However, V V T is a low-rank matrix, thus the impact of this case on V is negligible compared to the second case. 4) sij (w) > 0 and dij (p) > 0… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Correct rate and the learned w and p of each dataset. The x-axis represents the k-th nearest neighbors on a logarithmic scale. The upper and lower parts of the left y-axis represents the coordinates of w and p respectively, and the right y-axis is the coordinate of cor…

**Figure 5.** Figure 5: Average values of ACC of the proposed model with different values of [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Convergence curves of the proposed model ( [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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