For the theoretical exposition, we first establish the following Lemmas. Lemma A.1 proves that the derivative of the function φis bounded in the `2-norm when the domain is restricted to the support of P. Lemma A.1. Lemma A.3 proves that the function fΘ, as a function of Θ, is Lipschitz with respect to the k k norm. Joint first authors contributed equally Corresponding author 35th Conference on Neural Information Processing Systems (NeurIPS 2021). Thus, from equation (1), h φ(PC(θ)) φ(θ),x PC(θ)i 0. (2) We now observe that, dφ(x,θ) dφ(x,PC(θ)) dφ(PC(θ),θ) = h φ(PC(θ)) φ(θ),x PC(θ)i 0. Hence the result.

Recent advances in center-based clustering continue to improve upon the drawbacks of Lloyd's celebrated $k$-means algorithm over $60$ years after its introduction. Various methods seek to address poor local minima, sensitivity to outliers, and data that are not well-suited to Euclidean measures of fit, but many are supported largely empirically. Moreover, combining such approaches in a piecemeal manner can result in ad hoc methods, and the limited theoretical results supporting each individual contribution may no longer hold. Toward addressing these issues in a principled way, this paper proposes a cohesive robust framework for center-based clustering under a general class of dissimilarity measures. In particular, we present a rigorous theoretical treatment within a Median-of-Means (MoM) estimation framework, showing that it subsumes several popular $k$-means variants. In addition to unifying existing methods, we derive uniform concentration bounds that complete their analyses, and bridge these results to the MoM framework via Dudley's chaining arguments. Importantly, we neither require any assumptions on the distribution of the outlying observations nor on the relative number of observations $n$ to features $p$. We establish strong consistency and an error rate of $O(n^{-1/2})$ under mild conditions, surpassing the best-known results in the literature. The methods are empirically validated thoroughly on real and synthetic datasets.

Semi-supervised graph domain adaptation, as a branch of graph transfer learning, aims to annotate unlabeled target graph nodes by utilizing transferable knowledge learned from a label-scarce source graph. However, most existing studies primarily concentrate on aligning feature distributions directly to extract domain-invariant features, while ignoring the utilization of the intrinsic structure information in graphs. Inspired by the significance of data structure information in enhancing models' generalization performance, this paper aims to investigate how to leverage the structure information to assist graph transfer learning. To this end, we propose an innovative framework called TFGDA. Specially, TFGDA employs a structure alignment strategy named STSA to encode graphs' topological structure information into the latent space, greatly facilitating the learning of transferable features.

Recent advances in center-based clustering continue to improve upon the drawbacks of Lloyd's celebrated k -means algorithm over 60 years after its introduction. Various methods seek to address poor local minima, sensitivity to outliers, and data that are not well-suited to Euclidean measures of fit, but many are supported largely empirically. Moreover, combining such approaches in a piecemeal manner can result in ad hoc methods, and the limited theoretical results supporting each individual contribution may no longer hold. Toward addressing these issues in a principled way, this paper proposes a cohesive robust framework for center-based clustering under a general class of dissimilarity measures. In particular, we present a rigorous theoretical treatment within a Median-of-Means (MoM) estimation framework, showing that it subsumes several popular k -means variants.

This paper addresses the limitations of traditional vector quantization (clustering) algorithms, particularly K-Means and its variant K-Means++, and explores the Stochastic Quantization (SQ) algorithm as a scalable alternative for high-dimensional unsupervised and semi-supervised learning problems. Some traditional clustering algorithms suffer from inefficient memory utilization during computation, necessitating the loading of all data samples into memory, which becomes impractical for large-scale datasets. While variants such as Mini-Batch K-Means partially mitigate this issue by reducing memory usage, they lack robust theoretical convergence guarantees due to the non-convex nature of clustering problems. In contrast, the Stochastic Quantization algorithm provides strong theoretical convergence guarantees, making it a robust alternative for clustering tasks. We demonstrate the computational efficiency and rapid convergence of the algorithm on an image classification problem with partially labeled data, comparing model accuracy across various ratios of labeled to unlabeled data. To address the challenge of high dimensionality, we trained Triplet Network to encode images into low-dimensional representations in a latent space, which serve as a basis for comparing the efficiency of both the Stochastic Quantization algorithm and traditional quantization algorithms. Furthermore, we enhance the algorithm's convergence speed by introducing modifications with an adaptive learning rate.

In this paper, we regard clustering as ensembles of k-ary affinity relations and clusters correspond to subsets of objects with maximal average affinity relations. The average affinity relation of a cluster is relaxed and well approximated by a constrained homogenous function. We present an efficient procedure to solve this optimization problem, and show that the underlying clusters can be robustly revealed by using priors systematically constructed from the data. Our method can automatically select some points to form clusters, leaving other points un-grouped; thus it is inherently robust to large numbers of outliers, which has seriously limited the applicability of classical methods. Our method also provides a unified solution to clustering from k-ary affinity relations with k 2, that is, it applies to both graph-based and hypergraph-based clustering problems.

In this paper, we regard clustering as ensembles of k-ary affinity relations and clusters correspond to subsets of objects with maximal average affinity relations. The average affinity relation of a cluster is relaxed and well approximated by a constrained homogenous function. We present an efficient procedure to solve this optimization problem, and show that the underlying clusters can be robustly revealed by using priors systematically constructed from the data. Our method can automatically select some points to form clusters, leaving other points un-grouped; thus it is inherently robust to large numbers of outliers, which has seriously limited the applicability of classical methods. Our method also provides a unified solution to clustering from k-ary affinity relations with k 2, that is, it applies to both graph-based and hypergraph-based clustering problems.