arxiv: 2605.01691 · v1 · submitted 2026-05-03 · 💻 cs.LG

Recognition: 3 theorem links

· Lean Theorem

Complex Diffusion Maps with ω-Parameterized Kernels Revealing Inherent Harmonic Representations

Tongzhen Dang , Weiyang Ding , Michael K. Ng

Authors on Pith no claims yet

Pith reviewed 2026-05-08 19:30 UTC · model grok-4.3

classification 💻 cs.LG

keywords complex diffusion mapsomega parameterized kernelscomplex harmonicsdiffusion operatorangular structurespectral embeddingfMRI analysisEEG classification

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The pith

Complex Diffusion Maps use ω-parameterized complex kernels to reveal dominant harmonic representations that preserve angular structure in high-dimensional data.

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

The paper proposes Complex Diffusion Maps to extract the main complex harmonics from data using a tunable family of complex-valued kernels. These kernels trade off local and nonlocal connections, drawing from both heat and Schrödinger equations to define a diffusion operator. The resulting maps are designed to keep angular information intact, which helps distinguish similar points better than real-valued approaches. This framework is shown to be robust in noisy environments and effective for analyzing brain activity signals.

Core claim

The central discovery is that a unified family of ω-parameterized complex-valued kernels defines diffusion operators whose spectra yield complex harmonic maps. These maps preserve angular structure in the complex plane rather than relying on magnitudes alone, and the complex kernel form amplifies distinctions among confusable samples. The approach establishes well-defined diffusion distances and provides an optimization interpretation for the embeddings.

What carries the argument

The ω-parameterized complex-valued kernel, which generates a diffusion operator and its complex eigenmaps that maintain angular relationships.

Load-bearing premise

The complex kernels produce diffusion operators with spectra that give meaningful complex harmonic maps preserving angular structure beyond real-valued kernels.

What would settle it

Observing no improvement in sample separation or eigengap clarity when applying the method to synthetic high-noise data with known confusable groups compared to standard diffusion maps would challenge the claim.

Figures

Figures reproduced from arXiv: 2605.01691 by Michael K. Ng, Tongzhen Dang, Weiyang Ding.

**Figure 1.** Figure 1: Motivation of the w-parameterized complex kernels and the interpretation of optimal embeddings. The Gaussian kernel in Diffusion Maps (DM) is related to the heat equation and mainly captures the local similarity of data. The complex-valued kernel induced by the Schr¨odinger equation has been shown to encode nonlocal connections through distance-dependent phase, allowing nonzero coupling and thus generating… view at source ↗

**Figure 2.** Figure 2: Illustration of the three-point toy example. (A) Euclidean distances (left), real-valued Gaussian kernel (middle), and angles between global connections (right). Points 1 (blue) and 2 (yellow) are close and indistinguishable under the real-valued kernel, while complex weights enhance their differences via angle information. (B) The first two embeddings from DM (left), the real parts of CDM (middle), and th… view at source ↗

**Figure 3.** Figure 3: Comparison of CDM with linear and nonlinear real-valued methods on an artificial dataset constructed via amplitude–phase coupling (see Appendix B for details). (A) Heatmap of the data matrix (left), corresponding Euclidean distances (middle), and angle distribution (right). Classes 1 and 3 exhibit similar distance structures and are difficult to distinguish, while Class 2 shows distinct intra-class distanc… view at source ↗

**Figure 4.** Figure 4: Low-dimensional representations of DM and CDM for a simulated time series under different noise levels, low noise ε = 0.1 (SNR ≈ 17 dB), medium noise ε = 0.5 (SNR ≈ 3 dB), and high noise ε = 1 (SNR ≈ −3 dB). Under low noise, the first two dimensions of DM capture only low-frequency components, whereas CDM reveals a clear four-cluster structure. As noise increases, the embeddings of both methods become more… view at source ↗

**Figure 5.** Figure 5: Eigenvalues and the first three dominant maps of DM and CDM for the simulated time series under high noise ε = 1 (SNR ≈ −3 dB). The eigenvalue gaps of CDM are more pronounced than those of DM, with the first three components contributing a larger proportion of the total variance. The corresponding three-dimensional embeddings reveal a clear four-cluster structure. In contrast, the real-valued kernel produc… view at source ↗

**Figure 6.** Figure 6: Comparison of functional connectivity (FC) reconstruction and edge-centric metastability (ECM) derived from resting-state fMRI signals of 100 participants from the Human Connectome Project [46] under PCA, DM, and CDM frameworks. (A) Correlation and reconstruction error between reconstructed and source FC matrices in the learned low-dimensional manifolds, evaluated over kernel bandwidth σ 2 and diffusion st… view at source ↗

**Figure 7.** Figure 7: Clustering results on the ISRUC-S3 dataset [49]. The dataset contains overnight polysomnography (PSG) recordings from 10 subjects, annotated into five sleep stages: Wake, N1, N2, N3, and rapid eye movement (REM). The evaluation metrics ACCclu, ARI, and NMI are described in D.3.1. (A) Mean clustering scores of CDM and DM under varying embedding dimensions and stacking orders p. CDM consistently achieves hig… view at source ↗

**Figure 8.** Figure 8: 10-fold cross-validation results of CDM under the intra-subject setting on the ISRUC-S3 dataset [49]. (A) Test accuracy ACCcls for one fold at stacking order p = 4, where dim denotes the embedding dimension. The best result is marked with a star. (B) Best test accuracy of each fold under different stacking orders p, with the mean accuracy indicated by hollow markers, showing the optimal order at p = 4. (C)… view at source ↗

**Figure 9.** Figure 9: Display of runtime on intra-subject setting. Runtime refers to the average runtime of 10-fold experiments with their respective optimal parameters. (A) Comparison of the runtime of CDM with baseline neural network-based methods. (B) Breakdown of CDM’s runtime across its four phases, Order-p data stacking, CDM computation, embedding alignment, and SVM classification. The results show that CDM outperforms se… view at source ↗

**Figure 10.** Figure 10: 10-fold cross-validation results of CDM under the cross-subject setting on the ISRUC-S3 dataset [49]. (A) Test accuracy ACCcls of one fold under the stacking order p = 15. The variable dim denotes the dimension of the low-dimensional embeddings used for SVM [72] classification. The best performance is marked with a star. (B) The best test accuracy of each fold under different order p, where colored scatte… view at source ↗

**Figure 11.** Figure 11: Display of runtime on cross-subject setting. Runtime refers to the average runtime of 10-fold experiments with their respective optimal parameters. (A) Comparison of the runtime of CDM with baselines. (B) Breakdown of CDM’s runtime across its four phases, Order-p data stacking, CDM computation, embedding alignment, and SVM classification. Combined with the lower part of view at source ↗

**Figure 12.** Figure 12: Display of runtime on intra-subject and cross-subject view at source ↗

read the original abstract

In this paper, we propose Complex Diffusion Maps (CDM), a novel diffusion mapping framework that aims to reveal the dominant complex harmonics of high-dimensional data. Inspired by the local Gaussian kernel relevant to the heat equation and the nonlocal Schr\"odinger kernel relevant to the Schr\"odinger equation, we propose a unified family of $\omega$-parameterized complex-valued kernels for the trade-off between local and nonlocal connections. We establish the theoretical foundation based on the operator spectrum theory, where the corresponding diffusion operator, diffusion distance, and complex harmonic maps are well-defined. An optimization-based interpretation of the maps is also developed, aiming to preserve angular structure in the complex diffusion space rather than relying solely on real-valued magnitude. We extensively evaluate CDM on both synthetic and real-world datasets. The complex-valued kernel amplifies differences among easily confusable samples, improving discriminative power over both linear and nonlinear methods based on real-valued kernels. CDM remains robust in high-noise settings, yielding a clearer eigengap that enhances spectral separation. For resting-state fMRI data, CDM captures more strongly correlated and nonlocal spatiotemporal dynamics. Without task-specific tuning, CDM achieves competitive performance on a public EEG sleep dataset, while maintaining high computational efficiency compared with both traditional machine learning and deep neural network approaches, highlighting its generality and practical value.

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.

Desk Editor's Note private letter to a colleague

Complex diffusion maps add ω-tuned kernels and angular preservation but the non-self-adjoint operator issue looks unresolved from the abstract.

read the letter

The main new piece is the ω-parameterized family of complex kernels that interpolates between local Gaussian and nonlocal Schrödinger behavior, plus the claim that the resulting complex harmonic maps preserve angular structure in the embedding. They apply this to fMRI and EEG data and report better separation in noisy settings and competitive sleep staging without heavy tuning. That is a concrete extension of the diffusion map toolkit, and the neuroscience examples show it can surface nonlocal correlations that real-valued versions miss. The optimization view they sketch for the maps is also a useful reframing. The theory section is the soft spot. The abstract invokes operator spectrum theory to declare the diffusion operator, distance, and maps well-defined, yet complex kernels generally produce non-self-adjoint integral operators. Without an explicit inner-product change, symmetrization step, or proof that the operator stays normal and compact in the complex setting, the standard spectral guarantees do not carry over, and claims about clearer eigengaps or angular fidelity rest on an unverified extension. The empirical claims are also hard to weigh because the abstract supplies no numbers, error bars, or direct baseline comparisons. This is aimed at people working on spectral methods for manifold learning or neuroimaging time series. A reader already familiar with diffusion maps could pick up the kernel idea and test it, but would have to redo the spectral analysis themselves. I would send it to peer review so the authors can supply the missing derivations and full quantitative results; the direction is worth referee time even if the current write-up needs substantial tightening on the operator side.

Referee Report

2 major / 2 minor

Summary. The paper proposes Complex Diffusion Maps (CDM) using a unified family of ω-parameterized complex-valued kernels that interpolate between local Gaussian kernels (heat equation) and nonlocal Schrödinger kernels. It claims to establish a theoretical foundation via operator spectrum theory under which the associated diffusion operator, diffusion distance, and complex harmonic maps are well-defined, with an optimization-based interpretation that preserves angular structure in the complex plane. Empirical evaluations on synthetic data, real-world datasets, resting-state fMRI, and a public EEG sleep dataset report improved discriminative power over real-valued linear and nonlinear methods, greater robustness in high-noise regimes with clearer eigengaps, and competitive performance with high computational efficiency.

Significance. If the extension of diffusion-map theory to complex kernels is rigorously justified and the reported empirical advantages hold under replication, the work could provide a practical tool for datasets exhibiting phase or angular structure, such as neuroimaging time series, while offering a parameter-controlled trade-off between local and nonlocal connectivity. The optimization view of angular preservation and the unified kernel family are potentially generalizable strengths.

major comments (2)

[Abstract] Abstract: The claim that 'the corresponding diffusion operator, diffusion distance, and complex harmonic maps are well-defined' via operator spectrum theory is load-bearing for all downstream assertions (discriminative power, eigengap clarity, angular preservation). For a general complex kernel K(x,y) the induced integral operator need not be self-adjoint or normal on L², so the classical spectral theorem supplying a countable orthonormal basis of eigenfunctions with real eigenvalues does not apply directly; no explicit inner-product adjustment, Hermitian symmetrization, or compactness/normalcy proof is referenced to restore these properties.
[Theoretical development (presumed §3–4)] Theoretical development (presumed §3–4): The optimization-based interpretation that the maps 'preserve angular structure in the complex diffusion space' presupposes that the complex spectrum yields geometrically meaningful harmonics beyond magnitude; without a concrete demonstration that the ω-parameterized kernel produces a compact normal operator (or equivalent) whose eigenfunctions retain angular interpretability, the superiority over real-valued kernels in confusable-sample and high-noise settings remains an unverified extension.

minor comments (2)

[Abstract and Experiments] The abstract and evaluation sections would benefit from explicit statements of the precise values of ω used in each experiment and any sensitivity analysis, as the single free parameter is central to the claimed trade-off.
[Experiments] Quantitative tables or figures reporting eigengap sizes, classification accuracies, or correlation strengths with error bars or statistical tests are needed to substantiate claims of 'clearer eigengap' and 'more strongly correlated' dynamics.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and commit to revisions that will strengthen the theoretical justifications as requested.

read point-by-point responses

Referee: [Abstract] Abstract: The claim that 'the corresponding diffusion operator, diffusion distance, and complex harmonic maps are well-defined' via operator spectrum theory is load-bearing for all downstream assertions (discriminative power, eigengap clarity, angular preservation). For a general complex kernel K(x,y) the induced integral operator need not be self-adjoint or normal on L², so the classical spectral theorem supplying a countable orthonormal basis of eigenfunctions with real eigenvalues does not apply directly; no explicit inner-product adjustment, Hermitian symmetrization, or compactness/normalcy proof is referenced to restore these properties.

Authors: We appreciate this observation. While our ω-parameterized kernels are constructed to satisfy the Hermitian symmetry condition K(y, x) = conjugate(K(x, y)), which ensures the integral operator is self-adjoint on the complex L² space, we acknowledge that the manuscript does not explicitly state the compactness and normality arguments. In the revised version, we will add a dedicated proposition in Section 3 (or an appendix) proving that, under standard assumptions on the data manifold being compact and the kernel being continuous and bounded, the operator is compact and self-adjoint, thereby justifying the application of the spectral theorem. This will include the explicit verification for the boundary cases (ω=0 corresponding to real Gaussian and ω=1 to Schrödinger-like) and the interpolation. We believe this addresses the concern without altering the core claims. revision: yes
Referee: [Theoretical development (presumed §3–4)] Theoretical development (presumed §3–4): The optimization-based interpretation that the maps 'preserve angular structure in the complex diffusion space' presupposes that the complex spectrum yields geometrically meaningful harmonics beyond magnitude; without a concrete demonstration that the ω-parameterized kernel produces a compact normal operator (or equivalent) whose eigenfunctions retain angular interpretability, the superiority over real-valued kernels in confusable-sample and high-noise settings remains an unverified extension.

Authors: We agree that a more explicit link between the operator properties and the angular interpretability would strengthen the paper. The optimization view is derived from viewing the embedding as minimizing a complex-valued loss that penalizes deviations in argument (phase) as well as magnitude. To demonstrate this, we will include in the revision a short subsection or example showing that for the proposed kernels, the eigenfunctions are complex-valued with phases that correspond to harmonic oscillations, and that the diffusion distance incorporates both real and imaginary parts. This will be supported by the spectral properties established in the new proposition. Regarding the empirical superiority, the experiments on synthetic data with confusable samples and high-noise regimes already illustrate the benefits, but we will add a note clarifying that these rely on the well-defined spectrum. We do not claim the superiority is solely due to the theory but is observed empirically when using the complex maps. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained.

full rationale

The paper defines a family of ω-parameterized complex kernels, invokes standard operator spectrum theory to define the diffusion operator/distance/maps, and provides an optimization view for angular preservation. All claims (eigengap, discriminative power, noise robustness) rest on these definitions plus empirical evaluation rather than reducing to fitted parameters renamed as predictions or to self-citation chains. No equation equates a derived quantity to its own input by construction, and no load-bearing uniqueness theorem is imported from the authors' prior work. The framework extends real-valued diffusion maps without internal circular reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim depends on the existence of a well-defined spectrum for the complex diffusion operator and on the interpretability of angular preservation in the embedding space.

free parameters (1)

ω
Tunable parameter balancing local Gaussian and nonlocal Schrödinger-type kernel contributions; its specific value selection is not detailed in the abstract.

axioms (1)

domain assumption Operator spectrum theory applies to the diffusion operator induced by the ω-parameterized complex kernel.
Invoked to define diffusion distance and complex harmonic maps.

invented entities (1)

Complex harmonic maps no independent evidence
purpose: Embeddings that preserve angular structure in complex diffusion space.
New concept introduced to interpret the output of the complex operator beyond real magnitude.

pith-pipeline@v0.9.0 · 5536 in / 1257 out tokens · 58983 ms · 2026-05-08T19:30:47.433756+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost.lean (J = ½(x+x⁻¹)−1) and Foundation/AlphaCoordinateFixation.lean cost_alpha_one_eq_jcost / J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose a family of ω-parameterized complex-valued kernels that unify and generalize the Gaussian kernel reflecting local similarity and the Schrödinger kernel reflecting nonlocal connections to obtain a trade-off via the parameter ω.
IndisputableMonolith/Foundation/AlphaDerivationExplicit.lean alphaProvenanceCert (parameter-free derivation) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Both the kernel bandwidth σ and the complex parameter ω are selected via grid search.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

K(x_i, x_j) = exp(-ω ||x_i - x_j||² / σ²), where ω = e^{iθ}, θ ∈ [-π/2, 0].

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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