Recovering a latent potential from observed flow on a directed graph (a discrete Poisson problem with Dirichlet boundaries) is ill-posed, and the standard fix backfires: ridge regularization shrinks toward a gauge-meaningless origin, collapsing and reversing the recovered ordering ($+0.81\to-0.42$ rank correlation against a planted ground truth). The gauge-invariant graph Dirichlet energy removes the hazard and delivers parameter-insensitivity: the estimate is stable across four orders of magnitude in $λ$, whereas ridge inverts the ordering for every $λ>0$. We prove the reduced solve is SPD and preserves dynamic range exactly where ridge collapses it, and localize absorbing boundaries from flow alone via a Poisson residual. The $H^1$ seminorm is classical; what is new is the gauge diagnosis, the parameter-insensitivity it buys, and an ablation showing the result is robust to the extraction method. On three public clickstream corpora the gauge-invariant estimate retains $28$--$41\%$ of the interior dynamic range while ridge collapses to as little as $0.2\%$. The same gauge invariance carries into graph neural networks -- neutralizing the constant mode per layer prevents the oversmoothing that collapses a deep directed GCN -- linking this classical inverse problem to a central question in graph learning.
Benoît Loucheur, P. -A. Absil, Michel Journéecs.LG math.NA math.OC
We address the low-rank matrix completion problem by incorporating graph regularization into the existing Riemannian Trust-Region Matrix Completion (RTRMC) framework. The latter uses the geometry of the low-rank constraint to remodel the problem as an unconstrained optimization problem on a single Grassmann manifold. Our approach, named Graph-Regularized RTRMC (GR-RTRMC), exploits the inherent relationships between rows and columns of the matrix. By using these relationships, we aim to improve the accuracy and robustness of matrix completion, particularly in scenarios where the underlying data exhibits strong correlations between rows or columns.
EEG-based emotion recognition is critical for mental health monitoring and affective brain-computer interfaces, yet existing deep learning approaches often treat emotion classes as isolated labels, ignoring their psychological interdependencies. We propose a graph-regularized learning framework that conceptualizes emotions as nodes in a graph where edges encode proximity based on dimensional emotion theories. We adapt three complementary regularization strategies--Graph Label Smoothing (intuitive soft labeling), Commuting distance on graph via Graph Laplacian (spectral graph theory), and Sliced Wasserstein Distance (optimal transport on graph)--ordered by increasing computational complexity. These strategies penalize model predictions that deviate from the established emotion topology. Our framework is evaluated across three representative backbone architectures: AudioTransformer (pure transformer), Conformer (CNN-transformer hybrid), and DCGNN (causal graph neural network), demonstrating architecture-agnostic benefits. Experiments on SEED-IV (4 classes) and SEED-V (5 classes) datasets show consistent improvements: best case up to +5.42% accuracy and 39% reduction in psychologically implausible misclassifications. Ultimately, our framework help raise the upper bound of performance achievable with standard approaches. Code will be released.
Self-supervised learning matches supervised accuracy from a fraction of the labels, but the labeled-sample efficiency behind this has lacked a theoretical explanation. We provide one. Data augmentation induces a similarity graph on the unlabeled data, so downstream learning on that graph is graph-Laplacian-regularized learning. We prove a fast transductive rate, $O(1/n_L)$ in the number of labels, in place of the supervised $O(1/\sqrt{n_L})$, by carrying the leave-one-out stability apparatus of Johnson and Zhang (JMLR 2007) over to the augmentation graph, and without the unrealistic assumptions of limit-based analyses (exact kernel, generalizing features). The bound makes augmentation quality explicit: the expected error is at most $C/n_L + R_{\mathrm{DA}}(y)$, where the data-augmentation alignment error $R_{\mathrm{DA}}(y)$ is the graph-cut mass of augmentations that cross a label boundary, so good augmentations let few labels suffice. The analysis uses a streamlined loss that drops the projector, negative-sample, and orthogonality overhead of standard objectives yet still recovers the top-$K$ ideal features in the infinite-data limit, the augmentation-kernel eigenspace studied by Zhai et al. The result explains the observed accuracy-versus-label-count curve rather than only bounding a generalization gap.
This paper studies how spectral geometry emerges in quantum learning models and how it can be diagnosed with physically grounded probes. In graph-regularized quantum networks, training reorganizes the output similarity graph, increases the effective spectral dimension Delta S = +0.23, and reshapes the Laplacian spectrum. Edge-resolved two-boson interference directly probes this restructuring: the bosonic enhancement Delta P_uv correlates with the Fiedler edge split |Delta v_2| (r = -0.50), linking learned spectral partitions to interference signatures. A phase diagram shows a nonmonotonic dependence of performance on coupling strength gamma and noise delta, with graph regularization improving fidelity only in a restricted regime; hardware experiments confirm the predicted interference behavior within shot-noise uncertainty. We also analyze a hybrid quantum autoencoder and introduce Bloch-space drift as a geometric diagnostic of its latent representation. With an unsupervised benign-data threshold, the model achieves high ranking performance (ROC-AUC about 0.99) and negligible false-negative rates. Absolute Bloch drift strongly discriminates anomalies (ROC-AUC at least about 0.9), while consecutive drift is near random (ROC-AUC about 0.5), showing that detection arises from persistent state-space displacement rather than local fluctuations. Through the geometry of reduced single-qubit states and associated quantum Fisher information, these results show that learning-induced spectral organization appears as measurable quantum-state structure, establishing a unified spectral-geometric framework for diagnosing quantum learning systems with bosonic and Bloch probes.
Early and accurate detection of Alzheimer's disease (AD) is important for timely intervention and disease management. Generalized Eigenvalue Proximal Support Vector Machine (GEPSVM) and its Universum-based variants have shown promising results for AD classification. However, existing methods treat Universum samples as independent points and do not consider the geometric relationships among them. This paper proposes two graph-guided Universum learning models, namely UG-GEPSVM and IUG-GEPSVM, for AD versus cognitively normal (CN) classification using structural MRI data. In the proposed framework, mild cognitive impairment (MCI) subjects are used as Universum data to provide intermediate information between AD and CN classes. A graph is constructed over the Universum samples using Gaussian similarity, Minimum Spanning Tree connectivity, and multi-hop propagation. From this graph, a Laplacian matrix is derived that captures the geometric structure of the MCI samples. This Laplacian-based regularization is incorporated into the learning process in place of the conventional independent Universum penalty term. UG-GEPSVM integrates this regularization into the generalized eigenvalue formulation, while IUG-GEPSVM extends the numerically stable improved GEPSVM framework using a standard eigenvalue formulation. Experiments on ADNI MRI dataset variants using ICA- and PCA-based features at five different noise levels show that both proposed models consistently outperform existing GEPSVM and Universum-based methods. UG-GEPSVM achieves the highest average AUC of 88.07% and maintains stable performance under increasing noise levels. Statistical tests further confirm the significance of the observed improvements.
Non-negative reduced biquaternion matrix factorization (NRBMF) uses the product of reduced biquaternion (RB) matrices to incorporate the non-negativity constraints of color image pixels into the factorization process. However, NRBMF mainly focuses on reconstruction accuracy and does not explicitly exploit the local geometric structure of image data, which may limit the discriminative ability of the obtained low-dimensional coefficient representations. To address this issue, we propose a graph regularized non-negative reduced biquaternion matrix factorization (GNRBMF) model for color image recognition. The proposed model incorporates a graph Laplacian regularizer into the reduced biquaternion coefficient matrix, encouraging nearby samples in the original space to have similar coefficient representations. Meanwhile, GNRBMF retains the non-negativity property of NRBMF in the reduced biquaternion algebra. To solve the optimization problem, a component-wise alternating projected gradient algorithm is derived, and its convergence properties are analyzed. Experimental results on three color image datasets show that the proposed GNRBMF model achieves competitive or superior recognition performance compared with several methods in most tested settings.