Graph-Based Semi-Supervised Learning

We study post-hoc refinement of frozen node classifiers: given only the graph $G$ and class distributions $Q$ predicted by a frozen model, can we improve accuracy without access to node features, model parameters, or gradients? APPNP answers this by propagating logits with a restart towards the initial predictions, minimizing the anchored Dirichlet energy. Instead, we consider the Potts energy, and decompose it into a Dirichlet term, which penalizes disagreement between neighbouring nodes, and a Gini term, which penalizes indecision within each node. This decomposition motivates Propagate, Then Sharpen (PtS), which alternates between propagation of class probabilities and node-wise, mass-preserving sharpening, with only one additional hyperparameter selected using labelled validation nodes. Across nine homophilic graphs, with a frozen MLP backbone, PtS improves mean test accuracy over independently tuned APPNP by $1.71$ percentage points on clean inputs and $3.90$ under severe Gaussian feature corruption. Gains over APPNP become smaller, but remain positive with frozen GCN and GraphSAGE backbones. Sharpening also removes most of the accuracy loss of deep propagation: on clean inputs without restart, accuracy falls by $2.2$ points between $2$ and $100$ propagation steps under PtS, compared with $33.8$ for APPNP.

Graph Neural Networks (GNNs) achieve state-of-the-art performance on graph-structured data, but training and inference on large graphs are often bottlenecked by memory constraints and sparse linear-algebra workloads. Quantum computing offers an alternative set of primitives that may improve scalability for graph learning. Building on the quantum graph neural network (QGNN) framework of Liao \textit{et al.}, this work implements two representative architectures --- the Simplified Graph Convolution (SGC) and Linear Graph Convolution (LGC) models --- and evaluates them on open benchmark graph datasets and semi-supervised learning tasks using quantum simulation. We compare predictive performance and optimization behavior against classical baselines, showing that the quantum models achieve competitive performance with fewer parameters. Finally, we present a cost gradient analysis that identifies the tasks for which the models showcased are trainable. This is followed by a classical simulability study to find regimes in which the proposed circuits remain robust during training.

Kernelized graph methods - spectral clustering, diffusion maps, and sparse kernel -regression graphs - that use Gaussian kernels depend on the choice of Gaussian bandwidth sigma, which governs the spectral character of the local kernel operator. When sigma is too small, the kernel overestimates local complexity and treats each sample as an independent direction; when sigma is too large, the kernel collapses multiple directions together, the condition number diverges, and all geometric discrimination is lost. We propose a choice of scale to make the spectral complexity of the kernel consistent with the intrinsic complexity of the underlying manifold. We propose a per-node bandwidth criterion that operationalizes this principle by jointly matching the kernel's effective rank to the local intrinsic dimension estimated via minimum spanning tree, anchoring the search in the manifold-consistent log-log scaling regime. We evaluate SSL embeddings from six encoders on CIFAR-100, showing that adaptive bandwidth consistently improves leave-one-out (LOO) classification and label propagation (LP) accuracy over fixed-bandwidth methods and competing adaptive methods.

Two active learning algorithms for hyperspectral image (HSI) classification are proposed that combine density-aware Fermat distances with Poisson-reweighted harmonic label propagation. Our methods actively query points using an uncertainty-based acquisition function, extending Poisson ReWeighted Laplace Learning (PWLL). Our first algorithm, Fermat Active Laplace Learning (FALL), builds an affinity matrix using Fermat distances between all data points. Then, PWLL is run with a diagonal perturbation using the minimum-norm acquisition function. In contrast, Approximate FALL (A-FALL) computes Fermat distances between each data point and landmark pixels selected via farthest-point sampling and constructs the affinity matrix using landmark multidimensional scaling. After several query rounds, A-FALL selects the Fermat exponent $p$ using a leave-one-out cross-validation variant. FALL and A-FALL leverage Fermat distances and subsequent harmonic label propagation to provide a density-aware estimation of the data manifold, improving labeling accuracy. Experiments on Salinas A and Pavia show the effectiveness of FALL and the scalability of A-FALL to large HSI scenes.

Machine learning is transforming molecular sciences by accelerating property prediction, simulation, and the discovery of new molecules and materials. Acquiring labeled data in these domains is often costly and time-consuming, whereas large collections of unlabeled molecular data are readily available. Standard semi-supervised learning methods often rely on label-preserving augmentations, which are challenging to design in the molecular domain, where minor changes can drastically alter properties. In this work, we show that semi-supervised methods that rely on an ensemble consensus can boost predictive accuracy across a diverse range of molecular datasets, task types, and graph neural network architectures. We find that training with an ensemble consensus objective increases robustness in models and exhibits an effect similar to knowledge distillation; an individual member of an ensemble trained this way outperforms a full ensemble trained in a traditional supervised fashion in almost all cases. In addition, this type of semi-supervised training reduces calibration error.

The monitoring of business conduct risk is hindered by sparse, uneven, and visibility-biased data. Prior studies show that business conduct risk information and media coverage propagate through supply chain, peer, and corporate structure networks, yet incident records remain incomplete for many firms. As a result, the absence of reported events could reflect limited coverage rather than the absence of underlying business conduct risk. This paper examines whether inter-firm relationships can improve the prediction of future recorded conduct related incidents, particularly among firms with limited prior visibility. We formulate the task as Positive--Unlabeled node classification on a corporate ownership graph, where firms with recorded incidents are treated as labeled positives and firms without recorded incidents remain unlabeled. We then propose a visibility- and relation-aware GCNII framework that combines relation specific message passing with non-negative Positive--Unlabeled learning to account for positive contamination in the unlabeled set. In a forward-looking evaluation, the proposed approach achieved the strongest observed ranking performance relative to non-graph- and simple graph-based benchmarks. The results further show that graph-based inference retains its predictive value among firms without prior recorded incidents. These findings demonstrate the value of inter-firm relational structure as a complementary source of information for extending risk prioritization

Graph fraud detection plays a pivotal role in safeguarding the security and integrity of modern digital ecosystems. Graph Neural Networks (GNNs) are commonly adopted for graph fraud detection. However, the practical performance of existing GNN-based detectors is severely hindered by incomplete node attributes and extreme class imbalance within graphs. To mitigate these limitations, this paper proposes a novel framework for Graph Fraud Detection with Grouped attribute completion and Confidence-aware Contrastive learning, named GFD-GC. Specifically, it first imitates heterogeneous neighborhood structures to implement group-wise aggregation, which obtains informative complete node features by capturing fine-grained graph contextual patterns. Further, it introduces a confidence-aware supervised contrastive learning strategy to augment scarce labeled fraud nodes with high confidence pseudo-fraud nodes, which enhances the compactness of fraud representations and their separability from non-fraud nodes. Extensive experiments demonstrate the superiority of the proposed GFD-GC over state-of-the-art baselines on the graph fraud detection task, thereby providing an effective solution for real-world fraud scenarios.

While the growing availability of image data has driven significant advances, labeling datasets remains costly and time-consuming. Therefore, semi-supervised approaches such as Graph Convolutional Networks (GCNs), which learn from both labeled and unlabeled data, have emerged as a promising solution. One of the primary challenges in applying GCNs to image classification is graph construction, since, unlike in citation networks or similar domains, images typically do not come with a predefined structural representation. For visual data, most studies construct graphs based on the similarity between feature vectors from pretrained deep learning backbones, typically by employing kNN or reciprocal kNN algorithms. Although Large Language Models (LLMs) have shown remarkable capability in capturing high-level semantics, their integration with GCNs for image classification remains underexplored. Aiming to fill this gap, our approach uses a Vision Language Model (VLM) to generate textual image descriptions, which are then processed by an LLM to estimate semantic similarity scores between connected images. These scores guide the pruning of edges in kNN and reciprocal kNN graphs, filtering out semantically irrelevant neighbors. Experimental results reveal that leveraging LLMs for graph refinement can improve classification accuracy, particularly for kNN graphs and some backbones. The source code is publicly available at http://gcnllm.lucasvalem.com.

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 proportional to 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 bound gives a mechanistic account of the accuracy-versus-label-count curve through augmentation quality, verified in a controlled model where the constants are known.

Graph-based semi-supervised learning (SSL) propagates a few labels over a similarity graph by minimizing a Dirichlet-type energy. The standard quadratic ($p=2$) energy reduces to a single graph-Laplacian solve, but it degenerates exactly where SSL is most useful when labels are scarce: gathering more unlabeled data drives the $p=2$ estimate to a near-constant function whenever $d\ge2$ (Nadler-Srebro-Zhou). Well-posedness requires the nonlinear $p$-Laplacian energy with $p>d$. Existing solvers reduce this to a sequence of weighted Laplacian solves, but their reference implementations use a direct sparse factorization or ichol-preconditioned CG instead. Plugging a near-linear Laplacian solver is not straightforward: at large $p$ the conductance weights degenerate near flat-gradient edges, making the system nearly singular and causing stagnation without a damped outer iteration. We close this gap. Recasting $p$-Laplacian SSL as a source-form nonlinear Laplacian flow $Bρ_p(B^\top x)=b$ and solving by damped chord-Newton continuation in $p$, every linearized system stays well-conditioned and can be delegated to a near-linear Laplacian engine. On size-scaled graph families the wall-clock is empirically $m^{0.96}$-$m^{1.02}$ per family (approximate Cholesky default), and a pooled fit across 228 SuiteSparse graphs gives $m^{1.19}$ vs.\ $m^{1.45}$ for direct factorization; the solver handles a $6.8\times10^7$-edge social network in minutes. Memory is the binding constraint: Cholesky fill reaches $10$-$280\times$ the graph nonzeros vs.\ our $O(m)$ hierarchy. Against the released FCL solver we are $1.5$-$14\times$ faster at matched accuracy. On MNIST $10$-NN, $p=3$ scores $64\%$ at one label per class vs.\ $36\%$ for $p=2$. Code: https://github.com/orenlivne/np.

Semi-supervised learning (SSL) on data streams is challenging due to the continuous evolution of high-volume data and the scarcity of labels. Existing methods are limited in leveraging the intrinsic relationships among samples because they typically rely on fixed similarity measures or static graph structures, which cannot capture how relationships evolve over time. We propose SLeDGe, an SSL method for data streams that jointly learns a predictive model and an adaptive graph structure under strict memory and label constraints. SLeDGe maintains compact labeled and unlabeled memories using distinct update strategies, balancing rapid adaptation to novel features with the retention of historical consistency. In addition, by encouraging sparsity in the relational graph, SLeDGe filters out spurious connections and enables effective propagation of label supervision. Across 12 datasets, SLeDGe outperforms state-of-the-art competitors, achieving average relative accuracy gains of 31.7% with 0.1% labels and 14.8% with 1% labels.

In biomedical and neurodegenerative disorders, accurate and early disease identification remains challenging due to the scarcity of labeled data and the complexity of imaging patterns. To address these challenges, we introduce ARMA-C3, a unified unsupervised and semi-supervised graph learning framework for node classification based on contrastive learning and graph-cut regularization to learn structurally meaningful and discriminative representations. By modeling samples or images as graph nodes and exploiting inter-sample relationships, the proposed framework captures subject-level dependencies that conventional machine learning methods typically overlook. We conduct extensive binary classification experiments across five clinically relevant datasets: the Alzheimer's Disease Neuroimaging Initiative (ADNI), the Neuroimaging in Frontotemporal Dementia (NIFD) dataset, and three medical imaging benchmarks (BreastMNIST, PneumoniaMNIST, and a liver ultrasound dataset). Experimental results demonstrate that ARMA-C3 achieves competitive and frequently superior performance compared to classical clustering techniques, state-of-the-art machine learning models, and existing graph-based deep learning approaches across multiple evaluation settings, particularly under limited supervision and severe class imbalance. The proposed framework further demonstrates robust representation learning and strong cross-modal generalization across diverse biomedical imaging modalities.

The exponential growth of data has intensified the gap between the availability of unlabeled data and the high cost of manual annotation. Graph Neural Networks (GNNs) have emerged as a promising solution, as they exploit relational structures and learn from both labeled and unlabeled data, performing semi-supervised learning. A crucial component of many of these models is degree-based normalization, which influences message propagation but typically assumes uniform importance among neighboring nodes. In image classification, graphs are usually constructed from feature similarity, where treating all neighbors equally may overlook important variations in relevance. Motivated by this gap, we propose GRaNDe (Gaussian Rank-based Neighborhood Degree). This novel degree measure integrates neighborhood ranking with Gaussian distance weighting to better capture node importance. Experiments on five public image classification datasets show consistent accuracy improvements and competitive or superior results compared to state-of-the-art methods.

In the transductive setting, where the full graph is observed but node labels are only partially available, progress in semi-supervised node classification has largely focused on architectural innovation. In this paper, we revisit an orthogonal axis: the training objective. We start from a simple observation: transductive models produce predictions for every node during training, including nodes without labels. These unlabeled-node predictions may contain useful training signal, but standard supervised objectives discard them because no ground-truth labels are available. Inspired by the decomposition of cross-entropy into a label-dependent alignment term and a label-independent entropy term, we propose prediction confidence as a natural way to extract this signal in the absence of labels. This motivates Transductive Sharpening (TS): a loss-level modification that minimizes prediction entropy on unlabeled nodes while counterbalancing this effect on labeled nodes. We evaluate Transductive Sharpening across a wide range of node-classification benchmarks and observe consistent performance improvements without requiring any changes to the backbone architecture. Code is available at https://github.com/transductive-sharpening/tunedGNN.

Semi-supervised learning faces significant challenges in realistic scenarios where labeled data is scarce and unlabeled data follows unknown, arbitrary distributions. We formalize this critical yet under-explored paradigm as Universal Semi-supervised Learning (UniSSL). Existing methods typically leverage unlabeled data via pseudo-labeling. However, they often rely on the idealized assumption of a uniform unlabeled data distribution or require sufficient labeled data to estimate it. In the UniSSL setting, such dependencies lead to numerous erroneous pseudo-labels, thereby triggering representation confusion. Fortunately, we observe that inter-sample relations captured by representations are more reliable than pseudo-labels. Leveraging this insight, we shift our focus to representation-level structural inference to bypass distribution estimation. Accordingly, we propose Simplex Anchored Graph-state Equipartition (SAGE), which captures high-order inter-sample dependencies to establish structural consensus for guiding representation learning. Meanwhile, to mitigate representation confusion, we employ vectors that satisfy a simplex equiangular tight frame to serve as a coordinate frame for guiding inter-class representation separation. Finally, we introduce a weighting strategy based on distribution-agnostic metrics to prioritize reliable pseudo-labels and an auxiliary branch to isolate potentially erroneous pseudo-labels. Evaluations on five standard benchmarks show that SAGE consistently outperforms state-of-the-art methods, with an average accuracy gain of \textbf{8.52%}.

We develop graph-based methods for semi-supervised learning based on label propagation on a data similarity graph. When data is abundant or arrive in a stream, the problems of computation and data storage arise for any graph-based method. We propose a fast approximate online algorithm that solves for the harmonic solution on an approximate graph. We show, both empirically and theoretically, that good behavior can be achieved by collapsing nearby points into a set of local representative points that minimize distortion. Moreover, we regularize the harmonic solution to achieve better stability properties. We also present graph-based methods for detecting conditional anomalies and apply them to the identification of unusual clinical actions in hospitals. Our hypothesis is that patient-management actions that are unusual with respect to the past patients may be due to errors and that it is worthwhile to raise an alert if such a condition is encountered. Conditional anomaly detection extends standard unconditional anomaly framework but also faces new problems known as fringe and isolated points. We devise novel nonparametric graph-based methods to tackle these problems. Our methods rely on graph connectivity analysis and soft harmonic solution. Finally, we conduct an extensive human evaluation study of our conditional anomaly methods by 15 experts in critical care.

This paper proposes an algorithm for real-time learning without explicit feedback. The algorithm combines the ideas of semi-supervised learning on graphs and online learning. In particular, it iteratively builds a graphical representation of its world and updates it with observed examples. Labeled examples constitute the initial bias of the algorithm and are provided offline, and a stream of unlabeled examples is collected online to update this bias. We motivate the algorithm, discuss how to implement it efficiently, prove a regret bound on the quality of its solutions, and apply it to the problem of real-time face recognition. Our recognizer runs in real time, and achieves superior precision and recall on 3 challenging video datasets.

This paper proposes a novel algorithm for semisupervised learning. This algorithm learns graph cuts that maximize the margin with respect to the labels induced by the harmonic function solution. We motivate the approach, compare it to existing work, and prove a bound on its generalization error. The quality of our solutions is evaluated on a synthetic problem and three UCI ML repository datasets. In most cases, we outperform manifold regularization of support vector machines, which is a state-of-the-art approach to semi-supervised max-margin learning.

We introduce Sparse-HFS, a scalable algorithm that can compute solutions to SSL problems using only O(n polylog(n)) space and O(m polylog(n)) time.

Traditional loss functions, including cross-entropy, contrastive, triplet, and su pervised contrastive losses, used for fine-tuning pre-trained language models such as BERT, operate only within local neighborhoods and fail to account for the global semantic structure. We present G-Loss, a graph-guided loss function that incorporates semi-supervised label propagation to use structural relationships within the embedding manifold. G-Loss builds a document-similarity graph that captures global semantic relationships, thereby guiding the model to learn more discriminative and robust embeddings. We evaluate G-Loss on five benchmark datasets covering key downstream classification tasks: MR (sentiment analysis), R8 and R52 (topic categorization), Ohsumed (medical document classification), and 20NG (news categorization). In the majority of experimental setups, G-Loss converges faster and produces semantically coherent embedding spaces, resulting in higher classification accuracy than models fine-tuned with traditional loss functions.

Semi-supervised node classification is a foundational task in graph machine learning, yet state-of-the-art Graph Neural Networks (GNNs) are hindered by significant computational overhead and reliance on strong homophily assumptions. Traditional GNNs require expensive iterative training and multi-layer message passing, while existing training-free methods, such as Label Propagation, lack adaptability to heterophilo-us graph structures. This paper presents \textbf{F$^2$LP-AP} (Fast and Flexible Label Propagation with Adaptive Propagation Kernel), a training-free, computationally efficient framework that adapts to local graph topology. Our method constructs robust class prototypes via the geometric median and dynamically adjusts propagation parameters based on the Local Clustering Coefficient (LCC), enabling effective modeling of both homophilous and heterophilous graphs without gradient-based training. Extensive experiments across diverse benchmark datasets demonstrate that \textbf{F$^2$LP-AP} achieves competitive or superior accuracy compared to trained GNNs, while significantly outperforming existing baselines in computational efficiency.

Hyperspectral tree species classification is challenging due to limited and imbalanced class labels, spectral mixing (overlapping light signatures from multiple species), and ecological heterogeneity (variability among ecological systems). Addressing these challenges requires methods that integrate biological and structural characteristics of vegetation, such as canopy architecture and interspecific interactions, rather than relying solely on spectral signatures. This paper presents a biologically informed, semi-supervised deep learning method that integrates multi-sensor Earth observation data, specifically hyperspectral imaging (HSI) and airborne laser scanning (ALS), with expert, ecological knowledge. The approach relies on biologically inspired pseudo-labelling over a precomputed canopy graph, yielding accurate classification at low training cost. In addition, ecological priors on species cohabitation are automatically derived from reliable sources using large language models (LLMs) and encoded as a cohabitation matrix with likelihoods of species occurring together. These priors are incorporated into the pseudo-labelling strategy, effectively introducing expert knowledge into the model. Experiments on a real-world forest dataset demonstrate 5.6% improvement over the best reference method. Expert evaluation of cohabitation priors reveals high accuracy with differences no larger than 15%.