Fr\'echet regression is becoming a mainstay in modern data analysis for analyzing non-traditional data types belonging to general metric spaces. This novel regression method is especially useful in the analysis of complex health data such as continuous monitoring and imaging data. Fr\'echet regression utilizes the pairwise distances between the random objects, which makes the choice of metric crucial in the estimation. In this paper, existing dimension reduction methods for Fr\'echet regression are reviewed, and the effect of metric choice on the estimation of the dimension reduction subspace is explored for the regression between random responses and Euclidean predictors. Extensive numerical studies illustrate how different metrics affect the central and central mean space estimators. Two real applications involving analysis of brain connectivity networks of subjects with and without Parkinson's disease and an analysis of the distributions of glycaemia based on continuous glucose monitoring data are provided, to demonstrate how metric choice can influence findings in real applications.

With the explosive growth of Internet data, users are facing the problem of information overload, which makes it a challenge to efficiently obtain the required resources. Recommendation systems have emerged in this context. By filtering massive amounts of information, they provide users with content that meets their needs, playing a key role in scenarios such as advertising recommendation and product recommendation. However, traditional click-through rate prediction and TOP-K recommendation mechanisms are gradually unable to meet the recommendations needs in modern life scenarios due to high computational complexity, large memory consumption, long feature selection time, and insufficient feature interaction. This paper proposes a recommendations system model based on a separation embedding cross-network. The model uses an embedding neural network layer to transform sparse feature vectors into dense embedding vectors, and can independently perform feature cross operations on different dimensions, thereby improving the accuracy and depth of feature mining. Experimental results show that the model shows stronger adaptability and higher prediction accuracy in processing complex data sets, effectively solving the problems existing in existing models.

Heterogeneous graphs are present in various domains, such as social networks, recommendation systems, and biological networks. Unlike homogeneous graphs, heterogeneous graphs consist of multiple types of nodes and edges, each representing different entities and relationships. Generating realistic heterogeneous graphs that capture the complex interactions among diverse entities is a difficult task due to several reasons. The generator has to model both the node type distribution along with the feature distribution for each node type. In this paper, we look into solving challenges in heterogeneous graph generation, by employing a two phase hierarchical structure, wherein the first phase creates a skeleton graph with node types using a prior diffusion based model and in the second phase, we use an encoder and a sampler structure as generator to assign node type specific features to the nodes. A discriminator is used to guide training of the generator and feature vectors are sampled from a node feature pool. We conduct extensive experiments with subsets of IMDB and DBLP datasets to show the effectiveness of our method and also the need for various architecture components.

An ideal winner in such an election is a Condorcet winner, a candidate that "beats" any other candidate in a pairwise voting contest. Formally, i C is a Condorcet winner if, for every candidate j C\{i}, more than half of the voters prefer i over j. To avoid ties, we assume the number n of voters is odd. Unfortunately, it is easy to construct elections where a Condorcet winner does not exist; indeed typically there is no Condorcet winner. Given this, Elkind, Lang and Saffidine [12] proposed a relaxation where, rather than a single winning candidate, we desire a set of winning candidates that collectively beats any other candidate. Specifically, a set of candidates S C is a Condorcet winning set if, for every candidate j C \S, more than half of the voters prefer some (voter-dependent) candidate in S to j. Elkind et al. [12] showed that a Condorcet winning set always exists provided we allow the winning set to be large enough.

Consider a nonparametric contextual multi-arm bandit problem where each arm a \in [K] is associated to a nonparametric reward function f_a: [0,1] \to \mathbb{R} mapping from contexts to the expected reward. Suppose that there is a large set of arms, yet there is a simple but unknown structure amongst the arm reward functions, e.g. We present a novel algorithm which learns data-driven similarities amongst the arms, in order to implement adaptive partitioning of the context-arm space for more efficient learning. We provide regret bounds along with simulations that highlight the algorithm's dependence on the local geometry of the reward functions.

We study reinforcement learning in continuous state and action spaces endowed with a metric. We provide a refined analysis of the algorithm of Sinclair, Banerjee, and Yu (2019) and show that its regret scales with the zooming dimension of the instance. This parameter, which originates in the bandit literature, captures the size of the subsets of near optimal actions and is always smaller than the covering dimension used in previous analyses. As such, our results are the first provably adaptive guarantees for reinforcement learning in metric spaces.

From optimal transport to robust dimensionality reduction, many machine learning applicationscan be cast into the min-max optimization problems over Riemannian manifolds. Though manymin-max algorithms have been analyzed in the Euclidean setting, it has been elusive how theseresults translate to the Riemannian case. Zhang et al. (2022) have recently identified that geodesic convexconcave Riemannian problems admit always Sion's saddle point solutions. Immediately, an importantquestion that arises is if a performance gap between the Riemannian and the optimal Euclidean spaceconvex concave algorithms is necessary. Our work is the first to answer the question in the negative:We prove that the Riemannian corrected extragradient (RCEG) method achieves last-iterate at alinear convergence rate at the geodesically strongly convex concave case, matching the euclidean one.Our results also extend to the stochastic or non-smooth case where RCEG & Riemanian gradientascent descent (RGDA) achieve respectively near-optimal convergence rates up to factors dependingon curvature of the manifold.

Learning useful representations is a key ingredient to the success of modern machine learning. Currently, representation learning mostly relies on embedding data into Euclidean space. However, recent work has shown that data in some domains is better modeled by non-euclidean metric spaces, and inappropriate geometry can result in inferior performance. In this paper, we aim to eliminate the inductive bias imposed by the embedding space geometry. Namely, we propose to map data into more general non-vector metric spaces: a weighted graph with a shortest path distance.

As clearly indicated in the title, this paper submission is an extension of the PointNet work of [19], to appear at CVPR 2017. The goal is to classify and segment (3D) point clouds. Novel contributions over [19] are the use of a hierarchical network, leveraging neighbourhoods at different scales, and a mechanism to deal with varying sampling densities, effectively generating receptive fields that vary in a data dependent manner. All this leads to state-of-the-art results. PointNet seems an important extension over PointNet, in that it allows to properly exploit local spatial information.

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