We study first-order optimization algorithms for computing the barycenter of Gaussian distributions with respect to the optimal transport metric. Although the objective is geodesically non-convex, Riemannian GD empirically converges rapidly, in fact faster than off-the-shelf methods such as Euclidean GD and SDP solvers. This stands in stark contrast to the best-known theoretical results for Riemannian GD, which depend exponentially on the dimension. In this work, we prove new geodesic convexity results which provide stronger control of the iterates, yielding a dimension-free convergence rate. Our techniques also enable the analysis of two related notions of averaging, the entropically-regularized barycenter and the geometric median, providing the first convergence guarantees for Riemannian GD for these problems.

Robust Principal Component Analysis (RPCA) is a widely used method for recovering low-rank structure from data matrices corrupted by significant and sparse outliers. These corruptions may arise from occlusions, malicious tampering, or other causes for anomalies, and the joint identification of such corruptions with low-rank background is critical for process monitoring and diagnosis. However, existing RPCA methods and their extensions largely do not account for the underlying probabilistic distribution for the data matrices, which in many applications are known and can be highly non-Gaussian. We thus propose a new method called Robust Principal Component Analysis for Exponential Family distributions ($e^{\text{RPCA}}$), which can perform the desired decomposition into low-rank and sparse matrices when such a distribution falls within the exponential family. We present a novel alternating direction method of multiplier optimization algorithm for efficient $e^{\text{RPCA}}$ decomposition. The effectiveness of $e^{\text{RPCA}}$ is then demonstrated in two applications: the first for steel sheet defect detection, and the second for crime activity monitoring in the Atlanta metropolitan area.

Support vector machine (SVM) is one of the most studied paradigms in the realm of machine learning for classification and regression problems. It relies on vectorized input data. However, a significant portion of the real-world data exists in matrix format, which is given as input to SVM by reshaping the matrices into vectors. The process of reshaping disrupts the spatial correlations inherent in the matrix data. Also, converting matrices into vectors results in input data with a high dimensionality, which introduces significant computational complexity. To overcome these issues in classifying matrix input data, support matrix machine (SMM) is proposed. It represents one of the emerging methodologies tailored for handling matrix input data. The SMM method preserves the structural information of the matrix data by using the spectral elastic net property which is a combination of the nuclear norm and Frobenius norm. This article provides the first in-depth analysis of the development of the SMM model, which can be used as a thorough summary by both novices and experts. We discuss numerous SMM variants, such as robust, sparse, class imbalance, and multi-class classification models. We also analyze the applications of the SMM model and conclude the article by outlining potential future research avenues and possibilities that may motivate academics to advance the SMM algorithm.

Wasserstein distance (WD) and the associated optimal transport plan have been proven useful in many applications where probability measures are at stake. In this paper, we propose a new proxy of the squared WD, coined min-SWGG, that is based on the transport map induced by an optimal one-dimensional projection of the two input distributions. We draw connections between min-SWGG and Wasserstein generalized geodesics in which the pivot measure is supported on a line. We notably provide a new closed form for the exact Wasserstein distance in the particular case of one of the distributions supported on a line allowing us to derive a fast computational scheme that is amenable to gradient descent optimization. We show that min-SWGG is an upper bound of WD and that it has a complexity similar to as Sliced-Wasserstein, with the additional feature of providing an associated transport plan. We also investigate some theoretical properties such as metricity, weak convergence, computational and topological properties. Empirical evidences support the benefits of min-SWGG in various contexts, from gradient flows, shape matching and image colorization, among others.

In intelligent transportation systems, traffic data imputation, estimating the missing value from partially observed data is an inevitable and challenging task. Previous studies have not fully considered traffic data's multidimensionality and spatiotemporal correlations, but they are vital to traffic data recovery, especially for high-level missing scenarios. To address this problem, we propose a novel spatiotemporal regularized Tucker decomposition method. First, the traffic matrix is converted into a third-order tensor. Then, based on Tucker decomposition, the tensor is approximated by multiplying non-negative factor matrices with a sparse core tensor. Notably, we do not need to set the tensor rank or determine it through matrix nuclear-norm minimization or tensor rank minimization. The low rankness is characterized by the $l_1$-norm of the core tensor, while the manifold regularization and temporal constraint are employed to capture spatiotemporal correlations and further improve imputation performance. We use an alternating proximal gradient method with guaranteed convergence to address the proposed model. Numerical experiments show that our proposal outperforms matrix-based and tensor-based baselines on real-world spatiotemporal traffic datasets in various missing scenarios.

Gromov-Wasserstein (GW) distance is a powerful tool for comparing and aligning probability distributions supported on different metric spaces. Recently, GW has become the main modeling technique for aligning heterogeneous data for a wide range of graph learning tasks. However, the GW distance is known to be highly sensitive to outliers, which can result in large inaccuracies if the outliers are given the same weight as other samples in the objective function. To mitigate this issue, we introduce a new and robust version of the GW distance called RGW. RGW features optimistically perturbed marginal constraints within a Kullback-Leibler divergence-based ambiguity set. To make the benefits of RGW more accessible in practice, we develop a computationally efficient and theoretically provable procedure using Bregman proximal alternating linearized minimization algorithm. Through extensive experimentation, we validate our theoretical results and demonstrate the effectiveness of RGW on real-world graph learning tasks, such as subgraph matching and partial shape correspondence.

We focus on gradient descent and its variants, as they are widely adopted and scale well when the model dimensionality d is large (Bottou et al., 2018). The optimization problem (OPT) finds many applications: in solving linear systems, logistic regression, support vector machines, and other areas of machine learning (Boyd and Vandenberghe, 2004). Equally important, methods designed for (stochastic) convex optimization also influence the intuition for and design of methods for nonconvex optimization-for example, momentum (Polyak, 1964), AdaGrad (Duchi et al., 2010), and Adam (Kingma and Ba, 2015) were all first analyzed in the convex optimization framework. As models become larger and more complex, the cost and environmental impact of training have rapidly grown as well (Sharir et al., 2020; Patterson et al., 2021). Therefore, it is vital that we develop more efficient and effective methods of solving machine learning optimization tasks. One of the chief challenges in applying gradient-based methods is that they often require tuning one or more stepsize parameters (Goodfellow et al., 2016), and the choice of stepsize can significantly influence a method's convergence speed as well as the quality of the obtained solutions, especially in deep learning (Wilson et al., 2017). The cost and impact of hyperparameter tuning on the optimization process have led to significant research activity in designing parameter-free and adaptive optimization methods in recent years, see e.g.

Federated learning (FL) has recently emerged as a privacy-preserving approach for machine learning in domains that rely on user interactions, particularly recommender systems (RS) and online learning to rank (OLTR). While there has been substantial research on the privacy of traditional FL, little attention has been paid to studying the privacy properties of these interaction-based FL (IFL) systems. In this work, we show that IFL can introduce unique challenges concerning user privacy, particularly when the central server has knowledge and control over the items that users interact with. Specifically, we demonstrate the threat of reconstructing user interactions by presenting RAIFLE, a general optimization-based reconstruction attack framework customized for IFL. RAIFLE employs Active Data Manipulation (ADM), a novel attack technique unique to IFL, where the server actively manipulates the training features of the items to induce adversarial behaviors in the local Figure 1: Schematic diagram of Interaction-based Federated FL updates. We show that RAIFLE is more impactful than existing Learning (IFL). Users interact with items sent by the server FL privacy attacks in the IFL context, and describe how it can and train the FL model using the items and their interactions undermine privacy defenses like secure aggregation and private information with the items. Users may apply privacy defense techniques retrieval. Based on our findings, we propose and discuss such as differential privacy to their updated models before countermeasure guidelines to mitigate our attack in the context of sending local updates to the server.

Machine Learning (ML) systems are becoming instrumental in the public sector, with applications spanning areas like criminal justice, social welfare, financial fraud detection, and public health. While these systems offer great potential benefits to institutional decision-making processes, such as improved efficiency and reliability, they still face the challenge of aligning intricate and nuanced policy objectives with the precise formalization requirements necessitated by ML models. In this paper, we aim to bridge the gap between ML and public sector decision-making by presenting a comprehensive overview of key technical challenges where disjunctions between policy goals and ML models commonly arise. We concentrate on pivotal points of the ML pipeline that connect the model to its operational environment, delving into the significance of representative training data and highlighting the importance of a model setup that facilitates effective decision-making. Additionally, we link these challenges with emerging methodological advancements, encompassing causal ML, domain adaptation, uncertainty quantification, and multi-objective optimization, illustrating the path forward for harmonizing ML and public sector objectives.

Distributed online learning has been proven extremely effective in solving large-scale machine learning problems over streaming data. However, information sharing between learners in distributed learning also raises concerns about the potential leakage of individual learners' sensitive data. To mitigate this risk, differential privacy, which is widely regarded as the "gold standard" for privacy protection, has been widely employed in many existing results on distributed online learning. However, these results often face a fundamental tradeoff between learning accuracy and privacy. In this paper, we propose a locally differentially private gradient tracking based distributed online learning algorithm that successfully circumvents this tradeoff. We prove that the proposed algorithm converges in mean square to the exact optimal solution while ensuring rigorous local differential privacy, with the cumulative privacy budget guaranteed to be finite even when the number of iterations tends to infinity. The algorithm is applicable even when the communication graph among learners is directed. To the best of our knowledge, this is the first result that simultaneously ensures learning accuracy and rigorous local differential privacy in distributed online learning over directed graphs. We evaluate our algorithm's performance by using multiple benchmark machine-learning applications, including logistic regression of the "Mushrooms" dataset and CNN-based image classification of the "MNIST" and "CIFAR-10" datasets, respectively. The experimental results confirm that the proposed algorithm outperforms existing counterparts in both training and testing accuracies.