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arxiv: 2606.17531 · v1 · pith:QM2KGBDBnew · submitted 2026-06-16 · 💻 cs.LG · cs.CG· math.AT

Non-negative Matrix Factorisation with Topological Regularisation

Matias de Jong van Lier , Shizuo Kaji , Keunsu Kim This is my paper

Pith reviewed 2026-06-27 01:24 UTC · model grok-4.3

classification 💻 cs.LG cs.CGmath.AT
keywords non-negative matrix factorisationpersistent homologytopological regularisationinterpretable basesspatial coherenceperiodic structuresgraph signals
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The pith

Persistent homology supplies stable topological scores that regularise non-negative matrix factorisation without thresholds or discreteness.

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

The paper shows how to add topological regularisation to non-negative matrix factorisation by using persistent homology on the support of the basis functions. This produces bases whose structure matches the expected properties of the data domain. The scores enter the objective directly and remain differentiable, so standard continuous optimisation applies to image patches, time series and graphs alike. A sympathetic reader would care because the same mechanism handles spatial coherence, periodicity and clique structure in one language.

Core claim

We employ persistent homology as a stable, threshold-free topological quantifier and design topological scores that integrate into the NMF objective as regularisers. The resulting framework encompasses spatially coherent image components, periodic time-series structures, and clique-like graph signals within a unified modelling language.

What carries the argument

Persistent homology computed on the support of each basis function, converted into differentiable regularisation terms added to the NMF loss.

If this is right

  • Spatially coherent components emerge for image data without extra post-processing.
  • Periodic structures are favoured in time-series factorisation.
  • Clique-like signals are recovered in graph-valued data.
  • All three cases are handled by the same regularisation term inside one optimisation loop.

Where Pith is reading between the lines

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

  • The same regulariser could be dropped into other non-negative decompositions such as topic models.
  • Synthetic benchmarks with controlled topology would isolate how much each persistent-homology feature contributes.
  • The approach might reduce the need for hand-crafted smoothness penalties in domains where topology is the dominant prior.

Load-bearing premise

The topology of a basis function's support is intrinsically linked to the quality of that basis in a way that persistent homology scores can measure and enforce during continuous optimisation.

What would settle it

Run the regularised NMF on a dataset whose ground-truth bases have known topology; if the learned bases show no improvement in topological fidelity or reconstruction quality over plain NMF, the claim fails.

Figures

Figures reproduced from arXiv: 2606.17531 by Keunsu Kim, Matias de Jong Van Lier, Shizuo Kaji.

Figure 1
Figure 1. Figure 1: Conceptual overview of Top-NMF. Observations are treated as non-negative func￾tions on a structured domain Ω, allowing images, time series, and graph signals to be expressed in a common language. Standard NMF variables W and V are retained, but each basis row vj is interpreted as a function on Ω. A domain￾appropriate filtration is built from vj , summarised by persistent homology, and converted into a scal… view at source ↗
Figure 2
Figure 2. Figure 2: Two signals with the same 0-dimensional lifetime but different structural inter￾pretations. The finite class on the left has (b, d) = (0.50, 0.20), while the one on the right has (b, d) = (0.95, 0.65); in both cases the persistence is b − d = 0.30, so the unweighted contribution to TP(0) is 0.302 = 0.09. For p = 1, however, the weighted contributions are (1 − 0.20)0.302 = 0.072 and (1 − 0.65)0.302 = 0.0315… view at source ↗
Figure 3
Figure 3. Figure 3: One-dimensional unimodal-peak experiment. Top: the four ground-truth Gaussian atoms. Bottom: the six observed non-negative mixtures. Each observation contains several peaks, so reconstruction alone does not force the learned bases to correspond to individual unimodal components. 7.1.1 Synthetic one-dimensional signal: unimodal peaks This experiment applies the support-filtration viewpoint to a one-dimensio… view at source ↗
Figure 4
Figure 4. Figure 4: Basis comparison for the one-dimensional signal experiment. The Top-NMF bases are all unimodal, whereas standard NMF spreads several peaks across each basis. The number of detected modes shown in the panel titles is computed by peak detection with relative height and prominence thresholds. image grid. The ground-truth atoms comprise six binary single-bar images: three horizontal bars (rows 2, 6, 9) and thr… view at source ↗
Figure 5
Figure 5. Figure 5: Synthetic bar-atom experiment. Left: the six ground-truth atoms, each a single connected bar on the 12 × 12 grid. Right: a random selection of training samples, formed by summing three to five atoms with randomly perturbed weights. Standard NMF bases Top-NMF bases [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Bases learned on the synthetic bar data with rank r = 6. Standard NMF mixes overlapping atoms and leaks “ghost” intersection pixels into individual bases, whereas Top-NMF drives every basis toward a single connected bar that aligns with one ground-truth atom. average Betti numbers of their superlevel supports (ground-truth β0 = 1, β1 = 0): Top-NMF and sparse NMF attain β0 = 1.0, matching the ground truth a… view at source ↗
Figure 7
Figure 7. Figure 7: Reconstruction RMSE as a function of the number of components r on the synthetic bar data. ㄱ,ㄷ,ㄹ (consonant) ㅏ,ㅑ (vowel) [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Hangul image experiment. Left: schematic decomposition into consonant and vowel parts. Right: the six 64 × 64 images used to form the data matrix. gap as the bar experiment: Top-NMF reaches β0 = 1.0, exactly the connectedness target expected for individual strokes, whereas both standard NMF and sparse NMF leave the bases fragmented at β0 = 1.8 and 1.6 respectively. The reconstruction RMSEs are Top-NMF 0.03… view at source ↗
Figure 9
Figure 9. Figure 9: Basis comparison for the Hangul experiment. Standard NMF produces bases that mix consonant and vowel regions, whereas Top-NMF favours connected components aligned with the intended parts. 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 1.00 1.00 3.00 2.00 2.00 2.00 2.00 2.00 1 3 2 4 5 6 7 8 9 Graph data 1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 2.00 3.00 1.00 1.00 1.00 1.00 1.00 1 3 2 4 5 … view at source ↗
Figure 10
Figure 10. Figure 10: Sample observations for the synthetic graph experiment. [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Basis graphs learned with r = 3. Standard NMF mixes edges from different latent cliques. Top-NMF separates the clique-like components [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Effect of α in the graph experiment with r = 5 basis vectors. From top to bottom: small α = 0.1 favours fragmented small structures, moderate α = 1.0 recovers the intended cliques, and large α = 5.0 favours larger structures. 7.2.2 Human contact graphs: SocioPatterns high school As a real-world demonstration, we apply the same edge-filtration construction to the SocioPatterns high-school face-to-face cont… view at source ↗
Figure 13
Figure 13. Figure 13: Learned contact-graph bases on SocioPatterns. Node colours encode known school classes; only the displayed high-weight edges above the 0.85 quantile of positive edge weights in each component are shown. Standard NMF retains a small but visible number of cross-class high-weight edges, whereas Top-NMF concentrates edges of every basis inside a class block. 0 2 4 6 Time 0.0 0.5 1.0 1.5 2.0 Amplitude GT atom:… view at source ↗
Figure 14
Figure 14. Figure 14: Periodic time-series experiment. Left three panels: the ground-truth atoms. Right four panels: the four non-negative mixtures used as a dataset. u2(t) = tri(2t) + 1 generated by a sawtooth with width parameter 0.5, and an aperiodic trend u3(t) = t. We then form a dataset consisting of four non-negative mixtures, namely u1 + u3, u1 + 0.4u2 + 0.3u3, 0.9u1 + 1.2u2 + 0.3u3, and u2 + u3, and normalise each row… view at source ↗
Figure 15
Figure 15. Figure 15: Learned bases for the periodic time-series experiment, sorted by the periodicity score PerSM,τ shown in each panel title. Top-NMF respects the target vector (a1, a2, a3) = (0, 1, 1): one basis collapses to the trend (PerS ≈ 0.04) while the remaining two recover the cosine and the triangle wave as separate periodic components (PerS ≈ 0.91). Standard NMF leaves all three bases at intermediate periodicity, m… view at source ↗
Figure 16
Figure 16. Figure 16: Fourier spectra of the learned bases. ventricular contractions (PVCs). A PVC is an ectopic beat whose morphology and timing depart from the dominant rhythm. We use this setting as a post hoc anomaly-ranking task: after factorisation, a window should receive a high score if it is poorly explained by the learned rhythm-like periodic bases. The NMF objective does not use the PVC labels. Beat annotations are … view at source ↗
Figure 17
Figure 17. Figure 17: MIT-BIH record 200 ECG excerpt after non-negative preprocessing. Vertical red lines indicate annotated PVC beats; the annotations are used only for window construction and post hoc evaluation. 0.0 0.2 0.4 0.6 0.8 1.0 Standard NMF amplitude basis 3 PerS=0.351 basis 2 PerS=0.315 basis 0 PerS=0.214 basis 1 PerS=0.165 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 seconds 0.0 0.2 0.4 0.6 0.8 1.0 Top-NMF amplitude basis … view at source ↗
Figure 18
Figure 18. Figure 18: Learned ECG bases sorted by the periodicity score PerSM,τ . Top-NMF produces two high-periodicity rhythm templates (PerS = 0.684 and 0.646) and two low￾periodicity residual bases (PerS = 0.056 and 0.073), whereas standard NMF leaves all bases at intermediate periodicity. Finally, we use the post hoc PVC labels to test whether this separation is informative at the window level. The pipeline is deliberately… view at source ↗
Figure 19
Figure 19. Figure 19: Post hoc PVC evaluation on MIT-BIH record 200. The plotted score is the periodic-only reconstruction RMSE over a one-minute interval. PVC-centred windows, shown in red, concentrate at high scores even though PVC labels are not used in the factorisation objective. obtains AUROC 0.955, AUPRC 0.953, and top-k precision 0.897, compared with AUROC 0.105, AUPRC 0.205, and top-k precision 0.080 for standard NMF … view at source ↗
read the original abstract

We investigate the learning of interpretable bases in non-negative matrix factorisation (NMF) by regularising the topology of the learned basis functions. Our approach is motivated by the observation that many data modalities can be viewed as non-negative functions on a structured domain, where the quality of a basis is intrinsically linked to its topology. However, naive methods for incorporating the topology of the support are often hindered by discreteness and threshold dependence, rendering them unsuitable for continuous optimisation. We address these challenges by employing persistent homology as a stable, threshold-free topological quantifier and by designing topological scores that integrate into the NMF objective as regularisers. The resulting framework encompasses spatially coherent image components, periodic time-series structures, and clique-like graph signals within a unified modelling language.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper proposes integrating persistent homology into the NMF objective as regularizers to enforce topological properties (spatial coherence for images, periodicity for time series, clique structure for graphs) on the learned non-negative basis functions. The central claim is that persistent homology supplies a stable, threshold-free topological quantifier that can be turned into differentiable scores suitable for gradient-based optimization of the factorization.

Significance. If the claimed differentiable topological scores can be constructed without reintroducing effective thresholds or extra hyperparameters, the work would supply a unified modelling language for topology-aware NMF across modalities. The absence of any parameter-free derivation or machine-checked verification in the supplied abstract, however, leaves the practical utility of the framework unestablished.

major comments (2)
  1. [Abstract] Abstract (and §3–4, assuming standard placement of the loss): the claim that the topological scores are 'threshold-free' and 'integrate into the NMF objective' is load-bearing for the central contribution, yet the abstract supplies neither the explicit regularizer term nor its gradient with respect to the basis matrix. Without this construction it is impossible to verify that the persistent-homology approximation remains compatible with continuous optimisation.
  2. [Abstract] The weakest assumption identified in the stress-test note is not discharged by the abstract: standard persistent homology is combinatorial; any practical regulariser must employ an approximation whose stability under perturbation of the basis functions is demonstrated. No such verification (e.g., Lipschitz bound on the score or empirical gradient check) appears in the provided text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation of the framework.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and §3–4, assuming standard placement of the loss): the claim that the topological scores are 'threshold-free' and 'integrate into the NMF objective' is load-bearing for the central contribution, yet the abstract supplies neither the explicit regularizer term nor its gradient with respect to the basis matrix. Without this construction it is impossible to verify that the persistent-homology approximation remains compatible with continuous optimisation.

    Authors: We agree that the abstract, as a concise summary, does not include the explicit regularizer expression or gradient formula. Sections 3 and 4 of the full manuscript derive the persistent-homology-based topological scores, show their incorporation as differentiable regularizers in the NMF objective, and detail the gradient computation with respect to the basis matrix. To address the concern directly, we will revise the abstract to include a brief statement of the regularizer term and note its differentiability for gradient-based optimization. revision: yes

  2. Referee: [Abstract] The weakest assumption identified in the stress-test note is not discharged by the abstract: standard persistent homology is combinatorial; any practical regulariser must employ an approximation whose stability under perturbation of the basis functions is demonstrated. No such verification (e.g., Lipschitz bound on the score or empirical gradient check) appears in the provided text.

    Authors: We acknowledge that neither the abstract nor the current text provides an explicit stability analysis such as a Lipschitz bound or empirical gradient check. The framework relies on differentiable approximations to persistent homology to enable continuous optimization. In the revised version we will add a subsection discussing the stability properties of the scores, including an empirical gradient verification and analysis of the approximation error under perturbations of the basis functions. revision: yes

Circularity Check

0 steps flagged

No circularity; topological regularisers presented as external inputs to NMF

full rationale

The abstract and provided context describe persistent homology as an independent, stable quantifier imported to regularize the NMF objective, addressing discreteness and threshold issues via new score designs. No equations, self-citations, or fitted parameters are shown reducing the claimed predictions or framework to the inputs by construction. The motivation links topology to basis quality but does not define the regulariser in terms of the NMF result itself. This is the common case of an externally supported addition, warranting score 0 rather than any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the premise that persistent homology can be turned into a differentiable regulariser without introducing new optimisation difficulties. No free parameters, axioms, or invented entities are explicitly named in the abstract.

pith-pipeline@v0.9.1-grok · 5663 in / 1174 out tokens · 18734 ms · 2026-06-27T01:24:30.964375+00:00 · methodology

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