The suggested procedure involves integrating the proximity function null P as a mechanism to guide exploration on the space of walks. While the use of null P that we have explained has focused on classification and metric analysis, its geometric interpretation and ability to quantify similarity between walks suggest a broader applicability, particularly in settings where the reward landscape is sparse or the graph structure is too large for exhaustive exploration. We propose an improvement to the exploration strategy used in reinforcement learning algorithms that incrementally construct walks within graph-based environments. Traditionally, these algorithms alternate between exploitation (choosing the next node to maximize an estimated reward) and exploration (randomly selecting a new node). The novelty lies in replacing random exploration with a proximity-guided strategy using a function null P . Instead of sampling uniformly, the agent compares potential path extensions to a reference set of high-reward walks, prioritizing those that are most similar in structure. This approach introduces a more informed, data-driven method for exploration, focusing on areas of the graph that resemble previously successful trajectories.

We study elliptical distributions in locally convex vector spaces, and determine conditions when they can or cannot be used to satisfy differential privacy (DP). A requisite condition for a sanitized statistical summary to satisfy DP is that the corresponding privacy mechanism must induce equivalent probability measures for all possible input databases. We show that elliptical distributions with the same dispersion operator, C, are equivalent if the difference of their means lies in the Cameron-Martin space of C . In the case of releasing finite-dimensional summaries using elliptical perturbations, we show that the privacy parameter null can be computed in terms of a one-dimensional maximization problem. We apply this result to consider multivariate Laplace, t, Gaussian, and K -norm noise. Surprisingly, we show that the multivariate Laplace noise does not achieve null -DP in any dimension greater than one. Finally, we show that when the dimension of the space is infinite, no elliptical distribution can be used to give null -DP; only (null,δ)-DP is possible.

The decision maker must select an action based on the context and all past observations.

We introduce a novel framework that utilizes the internal weight activations of modern Large Language Models (LLMs) to construct a metric space of languages. Unlike traditional approaches based on hand-crafted linguistic features, our method automatically derives high-dimensional vector representations by computing weight importance scores via an adapted pruning algorithm. Our approach captures intrinsic language characteristics that reflect linguistic phenomena. We validate our approach across diverse datasets and multilingual LLMs, covering 106 languages. The results align well with established linguistic families while also revealing unexpected inter-language connections that may indicate historical contact or language evolution. The source code, computed language latent vectors, and visualization tool are made publicly available at https://github.com/mshamrai/deep-language-geometry.

The Gromov-Wasserstein (GW) variant of optimal transport, designed to compare probability densities defined over distinct metric spaces, has emerged as an important tool for the analysis of data with complex structure, such as ensembles of point clouds or networks. To overcome certain limitations, such as the restriction to comparisons of measures of equal mass and sensitivity to outliers, several unbalanced or partial transport relaxations of the GW distance have been introduced in the recent literature. This paper is concerned with the Conic Gromov-Wasserstein (CGW) distance introduced by S ejourn e, Vialard, and Peyr e [35]. We provide a novel formulation in terms of semi-couplings, and extend the framework beyond the metric measure space setting, to compare more general network and hypernetwork structures. With this new formulation, we establish several fundamental properties of the CGW metric, including its scaling behavior under dilation, variational convergence in the limit of volume growth constraints, and comparison bounds with established optimal transport metrics. We further derive quantitative bounds that characterize the robustness of the CGW metric to perturbations in the underlying measures. The hypernetwork formulation of CGW admits a simple and provably convergent block coordinate ascent algorithm for its estimation, and we demonstrate the computational tractability and scalability of our approach through experiments on synthetic and real-world high-dimensional and structured datasets.

Multi-modal Knowledge Graph Completion (MMKGC) aims to uncover hidden world knowledge in multimodal knowledge graphs by leveraging both multimodal and structural entity information. However, the inherent imbalance in multimodal knowledge graphs, where modality distributions vary across entities, poses challenges in utilizing additional modality data for robust entity representation. Existing MMKGC methods typically rely on attention or gate-based fusion mechanisms but overlook complementarity contained in multi-modal data. In this paper, we propose a novel framework named Mixture of Complementary Modality Experts (MoCME), which consists of a Complementarity-guided Modality Knowledge Fusion (CMKF) module and an Entropy-guided Negative Sampling (EGNS) mechanism. Additionally, we introduce an Entropy-guided Negative Sampling mechanism to dynamically prioritize informative and uncertain negative samples to enhance training effectiveness and model robustness. Extensive experiments on five benchmark datasets demonstrate that our MoCME achieves state-of-the-art performance, surpassing existing approaches. Introduction Knowledge graphs (KGs) [1, 2, 3, 4, 5] model real-world knowledge through structured representations in the form of triples--comprising a head entity, a relation, and a tail entity--which are typically constructed manually based on existing databases. However, the inherent incompleteness of KGs [6, 7], coupled with the high cost of annotating factual triples, has given rise to the task of Knowledge Graph Completion (KGC), which aims to predict and infer missing but plausible triples within an existing knowledge graph. Conventional KGC methods [1, 2, 3, 4] predominantly rely on Knowledge Graph Embedding (KGE) techniques, where entities and relations are embedded into continuous vector spaces to learn structural representations that model the relational patterns of triples and assess their plausibility .

Knowledge graph representation learning approaches provide a mapping between symbolic knowledge in the form of triples in a knowledge graph (KG) and their feature vectors. Knowledge graph embedding (KGE) models often represent relations in a KG as geometric transformations. Most state-of-the-art (SOTA) KGE models are derived from elementary geometric transformations (EGTs), such as translation, scaling, rotation, and reflection, or their combinations. These geometric transformations enable the models to effectively preserve specific structural and relational patterns of the KG. However, the current use of EGTs by KGEs remains insufficient without considering relation-specific transformations. Although recent models attempted to address this problem by ensembling SOTA baseline models in different ways, only a single or composite version of geometric transformations are used by such baselines to represent all the relations. In this paper, we propose a framework that evaluates how well each relation fits with different geometric transformations. Based on this ranking, the model can: (1) assign the best-matching transformation to each relation, or (2) use majority voting to choose one transformation type to apply across all relations. That is, the model learns a single relation-specific EGT in low dimensional vector space through an attention mechanism. Furthermore, we use the correlation between relations and EGTs, which are learned in a low dimension, for relation embeddings in a high dimensional vector space. The effectiveness of our models is demonstrated through comprehensive evaluations on three benchmark KGs as well as a real-world financial KG, witnessing a performance comparable to leading models

Supervised machine learning pipelines trained on features derived from persistent homology have been experimentally observed to ignore much of the information contained in a persistence diagram. Computing persistence diagrams is often the most computationally demanding step in such a pipeline, however. To explore this, we introduce several methods to generate topological feature vectors from unreduced boundary matrices. We compared the performance of pipelines trained on vectorizations of unreduced PDs to vectorizations of fully-reduced PDs across several data and task types. Our results indicate that models trained on PDs built from unreduced diagrams can perform on par and even outperform those trained on fully-reduced diagrams on some tasks. This observation suggests that machine learning pipelines which incorporate topology-based features may benefit in terms of computational cost and performance by utilizing information contained in unreduced boundary matrices.

Cross-domain recommendation (CDR) aims to address the persistent cold-start problem in Recommender Systems. Current CDR research concentrates on transferring cold-start users' information from the auxiliary domain to the target domain. However, these systems face two main issues: the underutilization of multimodal data, which hinders effective cross-domain alignment, and the neglect of side users who interact solely within the target domain, leading to inadequate learning of the target domain's vector space distribution. To address these issues, we propose a model leveraging Multimodal data and Side users for diffusion Cross-domain recommendation (MuSiC). We first employ a multimodal large language model to extract item multimodal features and leverage a large language model to uncover user features using prompt learning without fine-tuning. Secondly, we propose the cross-domain diffusion module to learn the generation of feature vectors in the target domain. This approach involves learning feature distribution from side users and understanding the patterns in cross-domain transformation through overlapping users. Subsequently, the trained diffusion module is used to generate feature vectors for cold-start users in the target domain, enabling the completion of cross-domain recommendation tasks. Finally, our experimental evaluation of the Amazon dataset confirms that MuSiC achieves state-of-the-art performance, significantly outperforming all selected baselines. Our code is available: https://anonymous.4open.science/r/MuSiC-310A/.

Causal inference is central to statistics and scientific discovery, enabling researchers to identify cause-and-effect relationships beyond associations. While traditionally studied within Euclidean spaces, contemporary applications increasingly involve complex, non-Euclidean data structures that reside in abstract metric spaces, known as random objects, such as images, shapes, networks, and distributions. This paper introduces a novel framework for causal inference with continuous treatments applied to non-Euclidean data. To address the challenges posed by the lack of linear structures, we leverage Hilbert space embeddings of the metric spaces to facilitate Fréchet mean estimation and causal effect mapping. Motivated by a study on the impact of exposure to fine particulate matter on age-at-death distributions across U.S. counties, we propose a nonparametric, doubly-debiased causal inference approach for outcomes as random objects with continuous treatments. Our framework can accommodate moderately high-dimensional vector-valued confounders and derive efficient influence functions for estimation to ensure both robustness and interpretability. We establish rigorous asymptotic properties of the cross-fitted estimators and employ conformal inference techniques for counterfactual outcome prediction. Validated through numerical experiments and applied to real-world environmental data, our framework extends causal inference methodologies to complex data structures, broadening its applicability across scientific disciplines.