Gaussian mixture models (GMMs) are widely used in machine learning for tasks such as clustering, classification, image reconstruction, and generative modeling. A key challenge in working with GMMs is defining a computationally efficient and geometrically meaningful metric. The mixture Wasserstein (MW) distance adapts the Wasserstein metric to GMMs and has been applied in various domains, including domain adaptation, dataset comparison, and reinforcement learning. However, its high computational cost -- arising from repeated Wasserstein distance computations involving matrix square root estimations and an expensive linear program -- limits its scalability to high-dimensional and large-scale problems. To address this, we propose multiple novel slicing-based approximations to the MW distance that significantly reduce computational complexity while preserving key optimal transport properties. From a theoretical viewpoint, we establish several weak and strong equivalences between the introduced metrics, and show the relations to the original MW distance and the well-established sliced Wasserstein distance. Furthermore, we validate the effectiveness of our approach through numerical experiments, demonstrating computational efficiency and applications in clustering, perceptual image comparison, and GMM minimization

This paper studies computationally efficient methods and their minimax optimality for high-dimensional clustering and signal recovery under block signal structures. We propose two sets of methods, cross-block feature aggregation PCA (CFA-PCA) and moving average PCA (MA-PCA), designed for sparse and dense block signals, respectively. Both methods adaptively utilize block signal structures, applicable to non-Gaussian data with heterogeneous variances and non-diagonal covariance matrices. Specifically, the CFA method utilizes a block-wise U-statistic to aggregate and select block signals non-parametrically from data with unknown cluster labels. We show that the proposed methods are consistent for both clustering and signal recovery under mild conditions and weaker signal strengths than the existing methods without considering block structures of signals. Furthermore, we derive both statistical and computational minimax lower bounds (SMLB and CMLB) for high-dimensional clustering and signal recovery under block signals, where the CMLBs are restricted to algorithms with polynomial computation complexity. The minimax boundaries partition signals into regions of impossibility and possibility. No algorithm (or no polynomial time algorithm) can achieve consistent clustering or signal recovery if the signals fall into the statistical (or computational) region of impossibility. We show that the proposed CFA-PCA and MA-PCA methods can achieve the CMLBs for the sparse and dense block signal regimes, respectively, indicating the proposed methods are computationally minimax optimal. A tuning parameter selection method is proposed based on post-clustering signal recovery results. Simulation studies are conducted to evaluate the proposed methods. A case study on global temperature change demonstrates their utility in practice.

Analyzing high-dimensional data presents challenges due to the "curse of dimensionality'', making computations intensive. Dimension reduction techniques, categorized as linear or non-linear, simplify such data. Non-linear methods are particularly essential for efficiently visualizing and processing complex data structures in interactive and graphical applications. This research develops an incremental non-linear dimension reduction method using the Geometric Multi-Resolution Analysis (GMRA) framework for streaming data. The proposed method enables real-time data analysis and visualization by incrementally updating the cluster map, PCA basis vectors, and wavelet coefficients. Numerical experiments show that the incremental GMRA accurately represents non-linear manifolds even with small initial samples and aligns closely with batch GMRA, demonstrating efficient updates and maintaining the multiscale structure. The findings highlight the potential of Incremental GMRA for real-time visualization and interactive graphics applications that require adaptive high-dimensional data representations.

The indicator matrix plays an important role in machine learning, but optimizing it is an NP-hard problem. We propose a new relaxation of the indicator matrix and prove that this relaxation forms a manifold, which we call the Relaxed Indicator Matrix Manifold (RIM manifold). Based on Riemannian geometry, we develop a Riemannian toolbox for optimization on the RIM manifold. Specifically, we provide several methods of Retraction, including a fast Retraction method to obtain geodesics. We point out that the RIM manifold is a generalization of the double stochastic manifold, and it is much faster than existing methods on the double stochastic manifold, which has a complexity of \( \mathcal{O}(n^3) \), while RIM manifold optimization is \( \mathcal{O}(n) \) and often yields better results. We conducted extensive experiments, including image denoising, with millions of variables to support our conclusion, and applied the RIM manifold to Ratio Cut, we provide a rigorous convergence proof and achieve clustering results that outperform the state-of-the-art methods. Our Code in \href{https://github.com/Yuan-Jinghui/Riemannian-Optimization-on-Relaxed-Indicator-Matrix-Manifold}{here}.

While there has been tremendous activity in the area of statistical network inference on graphs, hypergraphs have not enjoyed the same attention, on account of their relative complexity and the lack of tractable statistical models. We introduce a hyper-edge-centric model for analyzing hypergraphs, called the interaction hypergraph, which models natural sampling methods for hypergraphs in neuroscience and communication networks, and accommodates interactions involving different numbers of entities. We define latent embeddings for the interactions in such a network, and analyze their estimators. In particular, we show that a spectral estimate of the interaction latent positions can achieve perfect clustering once enough interactions are observed.

Dealers' position in the trading network is shown to have a significant impact on asset prices. 1 However, it remains challenging to account for the structure of trading networks during the estimation of dealer and asset features' impact on asset prices. Structural approaches usually rely on specific network structures to reduce complexity in estimation (e.g. Pint er and Usl u, 2022; Eisfeldt et al., 2023; Cohen et al., 2024), which limits the accuracy and generalizability of the estimation method. Reduced-form approach uses centrality measures to capture dealers' position in the network(e.g. Di Maggio et al., 2017; Hollifield et al., 2017; Li and Sch urhoff, 2019), but recent papers point out linear regressions with centrality measures can lead to biased estimation when the network is sparse (Cai, 2022).

We introduce Hodge Diffusion Maps, a novel manifold learning algorithm designed to analyze and extract topological information from high-dimensional data-sets. This method approximates the exterior derivative acting on differential forms, thereby providing an approximation of the Hodge Laplacian operator. Hodge Diffusion Maps extend existing non-linear dimensionality reduction techniques, including vector diffusion maps, as well as the theories behind diffusion maps and Laplacian Eigenmaps. Our approach captures higher-order topological features of the data-set by projecting it into lower-dimensional Euclidean spaces using the Hodge Laplacian. We develop a theoretical framework to estimate the approximation error of the exterior derivative, based on sample points distributed over a real manifold. Numerical experiments support and validate the proposed methodology.

The integration of Generative Artificial Intelligence (GenAI) within the Internet of Things (IoT) is garnering considerable interest. This growing attention stems from the continuous evolution and widespread adoption they are both having individually, enough to spontaneously reshape numerous sectors, including Healthcare, Manufacturing, and Smart Cities. Hence, their increasing popularity has catalyzed further extensive research for understanding the potential of the duo GenAI-IoT, how they interplay, and to which extent their synergy can innovate the state-of-the-art in their individual scenarios. However, despite the increasing prominence of GenAI for IoT Computing, much of the existing research remains focused on specific, narrowly scoped applications. This fragmented approach highlights the need for a more comprehensive analysis of the potential, challenges, and implications of GenAI integration within the broader IoT ecosystem. This survey exactly aims to address this gap by providing a holistic overview of the opportunities, issues, and considerations arising from the convergence of these mainstream paradigms. Our contribution is realized through a systematic literature review following the PRISMA methodology. A comparison framework is presented, and well-defined research questions are outlined to comprehensively explore the past, present, and future directions of GenAI integration with IoT Computing, offering valuable insights for both experts and newcomers.

We propose smoothed primal-dual algorithms for solving stochastic and smooth nonconvex optimization problems with linear inequality constraints. Our algorithms are single-loop and only require a single stochastic gradient based on one sample at each iteration. A distinguishing feature of our algorithm is that it is based on an inexact gradient descent framework for the Moreau envelope, where the gradient of the Moreau envelope is estimated using one step of a stochastic primal-dual augmented Lagrangian method. To handle inequality constraints and stochasticity, we combine the recently established global error bounds in constrained optimization with a Moreau envelope-based analysis of stochastic proximal algorithms. For obtaining $\varepsilon$-stationary points, we establish the optimal $O(\varepsilon^{-4})$ sample complexity guarantee for our algorithms and provide extensions to stochastic linear constraints. We also show how to improve this complexity to $O(\varepsilon^{-3})$ by using variance reduction and the expected smoothness assumption. Unlike existing methods, the iterations of our algorithms are free of subproblems, large batch sizes or increasing penalty parameters and use dual variable updates to ensure feasibility.

Clustering is a fundamental data processing task used for grouping records based on one or more features. In the vertically partitioned setting, data is distributed among entities, with each holding only a subset of those features. A key challenge in this scenario is that computing distances between records requires access to all distributed features, which may be privacy-sensitive and cannot be directly shared with other parties. The goal is to compute the joint clusters while preserving the privacy of each entity's dataset. Existing solutions using secret sharing or garbled circuits implement privacy-preserving variants of Lloyd's algorithm but incur high communication costs, scaling as O(nkt), where n is the number of data points, k the number of clusters, and t the number of rounds. These methods become impractical for large datasets or several parties, limiting their use to LAN settings only. On the other hand, a different line of solutions rely on differential privacy (DP) to outsource the local features of the parties to a central server. However, they often significantly degrade the utility of the clustering outcome due to excessive noise. In this work, we propose a novel solution based on homomorphic encryption and DP, reducing communication complexity to O(n+kt). In our method, parties securely outsource their features once, allowing a computing party to perform clustering operations under encryption. DP is applied only to the clusters' centroids, ensuring privacy with minimal impact on utility. Our solution clusters 100,000 two-dimensional points into five clusters using only 73MB of communication, compared to 101GB for existing works, and completes in just under 3 minutes on a 100Mbps network, whereas existing works take over 1 day. This makes our solution practical even for WAN deployments, all while maintaining accuracy comparable to plaintext k-means algorithms.