Knowledge graph learning provides a powerful framework for representing and inferring structured knowledge, with broad practical applications. However, the scarcity of relation-specific labeled triples per entity hinders the training of expressive models, and the ad hoc design of scoring functions limits generalizability and lacks theoretical grounding. We address both issues with a theoretically grounded, end-to-end training framework that extends and subsumes existing methods. Our framework is a two-stage procedure: unsupervised pretraining over heterogeneous corpora followed by supervised learning with multiple relation types. We establish a nonasymptotic risk bound that disentangles pretraining representation error from labeled-sample complexity, formally quantifying the benefit of large-scale unlabeled data for downstream knowledge prediction. Synthetic experiments validate each theoretical component, and real-world experiments confirm the effectiveness of our approach on large-scale knowledge graph benchmarks.
Graph signal processing tasks that leverage spectral information typically assume access to the complete graph topology, which is often unavailable in practice. We propose a systematic framework for subgraph filter learning (SFL), where subgraph-supported operators approximate ambient graph filters under partial observations. We formulate SFL as a statistical learning problem in which optimal subgraph operators are inherently data-dependent. To address the difficulty of directly estimating such operators, we develop a subgraph filter algebra based on distance-aware Laplacian constructions, defining a structured and controllable class of filters for effective approximation. We further establish performance risk bounds under the least squares loss, quantifying how well the learned operator approximates the restricted ambient mapping. Experiments real-world datasets show that, for SFL tasks, the proposed algebraic models consistently outperform polynomial filters, distribution-agnostic operators, and direct numerical filter learning baselines that attempt to recover the underlying structure from data.
Parameter estimation in stochastic differential equations is a classical statistical problem of much importance in many scientific fields. Recent work of Tapia Costa et al. (2026) introduced a novel technique for estimating the drift when the diffusion parameter is known, using discrete samples from multiple trajectories. Their method treats drift estimation as a denoising problem, and leverages tools from (conditional) score-matching diffusion models. Although their experiments showed promising results across different drift classes, the question of theoretical guarantees for their estimator was left unanswered. In this note, we address this gap by exploiting techniques from diffusion model theory. More concretely, we derive an explicit risk bound for the time-averaged mean-squared error of said drift estimator. Our bound decomposes the risk into the (i) Euler-Maruyama discretization, (ii) score/denoiser approximation, (iii) noise initialization, and (iv) sampling variance, revealing the trade-offs between the different hyperparameters and sources of error in the estimator.
Junhyoung Chung, Euijong Song, Won Hwa Kim +1stat.ML cs.LG math.ST stat.ME
We introduce Convex Distance Operator Transport (CDOT), the first convex optimal transport framework that aligns distributions across heterogeneous domains by jointly preserving feature correspondence and intrinsic geometric structure. Specifically, CDOT employs an operator-based regularization that aligns aggregated distance structures by introducing distance and conditional expectation operators. Consequently, the proposed regularization improves the robustness to local geometric variations. We further prove that the resulting CDOT discrepancy is a valid pseudometric on the space of attributed compact metric-measure spaces. In addition, we characterize the relationship between CDOT and Gromov--Wasserstein (GW) through a new notion of dispersion gap, formally elucidating the geometric source of non-convexity in GW compared to the convexity of CDOT. In the finite-sample regime, we derive a non-asymptotic risk bound decomposed into optimization and statistical errors, establishing risk consistency under a globally convergent Frank--Wolfe algorithm. Experiments on synthetic point clouds, brain connectomes, and graph classification benchmarks demonstrate better performance over existing methods, with stable and reliable behavior in practice.
Softmax-routed mixture-of-experts models approach hard routing as the temperature tends to zero, but this limit is singular near routing ties. This paper studies that singularity at the population level for squared-loss MoE regression. The central object is the \emph{boundary mass}, namely the probability that the top two router scores are separated by only a small margin. Under smoothness and transversality assumptions on the router and input law, we prove coarea/tube estimates showing that this mass is linear in the slab width, with leading constant given by a surface integral over the routing interface in the binary case. These estimates yield quantitative soft-to-hard risk bounds and, under compactness and uniform margin control, $Γ$-convergence of the soft objectives to the hard-routing objective. The main conclusion is that the zero-temperature limit is controlled by a thin geometric layer around routing interfaces, not by the full input space. We then use this geometric core in two more model-dependent directions. In a teacher--student setting, we prove a conditional landscape-transfer principle showing that, when the profiled hard-routing problem has favorable identifiability and curvature and the relevant derivatives transfer at boundary-layer scale, small-temperature soft routing inherits approximate teacher recovery and strict-saddle behavior away from teacher-equivalent partitions. We also give a reduced two-expert Gaussian calculation that illustrates a local symmetry-breaking mechanism aligned with the teacher separator.