The Procrustes-Wasserstein problem consists in matching two high-dimensional point clouds in an unsupervised setting, and has many applications in natural language processing and computer vision. We consider a planted model with two datasets $X,Y$ that consist of $n$ datapoints in $\mathbb{R}^d$, where $Y$ is a noisy version of $X$, up to an orthogonal transformation and a relabeling of the data points. This setting is related to the graph alignment problem in geometric models. In this work, we focus on the euclidean transport cost between the point clouds as a measure of performance for the alignment. We first establish information-theoretic results, in the high ($d \gg \log n$) and low ($d \ll \log n$) dimensional regimes. We then study computational aspects and propose the Ping-Pong algorithm, alternatively estimating the orthogonal transformation and the relabeling, initialized via a Franke-Wolfe convex relaxation. We give sufficient conditions for the method to retrieve the planted signal after one single step. We provide experimental results to compare the proposed approach with the state-of-the-art method of Grave et al. (2019).

Since the weak convergence for stochastic processes does not account for the growth of information over time which is represented by the underlying filtration, a slightly erroneous stochastic model in weak topology may cause huge loss in multi-periods decision making problems. To address such discontinuities Aldous introduced the extended weak convergence, which can fully characterise all essential properties, including the filtration, of stochastic processes; however was considered to be hard to find efficient numerical implementations. In this paper, we introduce a novel metric called High Rank PCF Distance (HRPCFD) for extended weak convergence based on the high rank path development method from rough path theory, which also defines the characteristic function for measure-valued processes. We then show that such HRPCFD admits many favourable analytic properties which allows us to design an efficient algorithm for training HRPCFD from data and construct the HRPCF-GAN by using HRPCFD as the discriminator for conditional time series generation. Our numerical experiments on both hypothesis testing and generative modelling validate the out-performance of our approach compared with several state-of-the-art methods, highlighting its potential in broad applications of synthetic time series generation and in addressing classic financial and economic challenges, such as optimal stopping or utility maximisation problems.

Assigning tasks to service providers is a frequent procedure across various applications. Often the tasks arrive dynamically while the service providers remain static. Preventing task rejection caused by service provider overload is of utmost significance. To ensure a positive experience in relevant applications for both service providers and tasks, fairness must be considered. To address the issue, we model the problem as an online matching within a bipartite graph and tackle two minimax problems: one focuses on minimizing the highest waiting time of a task, while the other aims to minimize the highest workload of a service provider. We show that the second problem can be expressed as a linear program and thus solved efficiently while maintaining a reasonable approximation to the objective of the first problem. We developed novel methods that utilize the two minimax problems. We conducted extensive simulation experiments using real data and demonstrated that our novel heuristics, based on the linear program, performed remarkably well.

For constrained, not necessarily monotone submodular maximization, all known approximation algorithms with ratio greater than $1/e$ require continuous ideas, such as queries to the multilinear extension of a submodular function and its gradient, which are typically expensive to simulate with the original set function. For combinatorial algorithms, the best known approximation ratios for both size and matroid constraint are obtained by a simple randomized greedy algorithm of Buchbinder et al. [9]: $1/e \approx 0.367$ for size constraint and $0.281$ for the matroid constraint in $\mathcal O (kn)$ queries, where $k$ is the rank of the matroid. In this work, we develop the first combinatorial algorithms to break the $1/e$ barrier: we obtain approximation ratio of $0.385$ in $\mathcal O (kn)$ queries to the submodular set function for size constraint, and $0.305$ for a general matroid constraint. These are achieved by guiding the randomized greedy algorithm with a fast local search algorithm. Further, we develop deterministic versions of these algorithms, maintaining the same ratio and asymptotic time complexity. Finally, we develop a deterministic, nearly linear time algorithm with ratio $0.377$.

This paper presents the first systematic study of evalution of Deep Neural Network (DNN) designed and trained to predict the evolution of a stochastic dynamical system, using wildfire prediction as a case study. We show that traditional evaluation methods based on threshold based classification metrics and error-based scoring rules assess a DNN's ability to replicate the observed ground truth (GT), but do not measure the fidelity of the DNN's learning of the underlying stochastic process. To address this gap, we propose a new system property: Statistic-GT, representing the GT of the stochastic process, and an evaluation metric that exclusively assesses fidelity to Statistic-GT. Utilizing a synthetic dataset, we introduce a stochastic framework to characterize this property and establish criteria for a metric to be a valid measure of the proposed property. We formally show that Expected Calibration Error (ECE) tests the necessary condition for fidelity to Statistic-GT. We perform empirical experiments, differentiating ECE's behavior from conventional metrics and demonstrate that ECE exclusively measures fidelity to the stochastic process. Extending our analysis to real-world wildfire data, we highlight the limitations of traditional evaluation methods and discuss the utility of evaluating fidelity to the stochastic process alongside existing metrics.

We propose an actor-critic algorithm for a family of complex problems arising in algebraic statistics and discrete optimization. The core task is to produce a sample from a finite subset of the non-negative integer lattice defined by a high-dimensional polytope. We translate the problem into a Markov decision process and devise an actor-critic reinforcement learning (RL) algorithm to learn a set of good moves that can be used for sampling. We prove that the actor-critic algorithm converges to an approximately optimal sampling policy. To tackle complexity issues that typically arise in these sampling problems, and to allow the RL to function at scale, our solution strategy takes three steps: decomposing the starting point of the sample, using RL on each induced subproblem, and reconstructing to obtain a sample in the original polytope. In this setup, the proof of convergence applies to each subproblem in the decomposition. We test the method in two regimes. In statistical applications, a high-dimensional polytope arises as the support set for the reference distribution in a model/data fit test for a broad family of statistical models for categorical data. We demonstrate how RL can be used for model fit testing problems for data sets for which traditional MCMC samplers converge too slowly due to problem size and sparsity structure. To test the robustness of the algorithm and explore its generalization properties, we apply it to synthetically generated data of various sizes and sparsity levels.

We study conformal prediction in the one-shot federated learning setting. The main goal is to compute marginally and training-conditionally valid prediction sets, at the server-level, in only one round of communication between the agents and the server. Using the quantile-of-quantiles family of estimators and split conformal prediction, we introduce a collection of computationally-efficient and distribution-free algorithms that satisfy the aforementioned requirements. Our approaches come from theoretical results related to order statistics and the analysis of the Beta-Beta distribution. We also prove upper bounds on the coverage of all proposed algorithms when the nonconformity scores are almost surely distinct. For algorithms with training-conditional guarantees, these bounds are of the same order of magnitude as those of the centralized case. Remarkably, this implies that the one-shot federated learning setting entails no significant loss compared to the centralized case. Our experiments confirm that our algorithms return prediction sets with coverage and length similar to those obtained in a centralized setting.

Curvature serves as a potent and descriptive invariant, with its efficacy validated both theoretically and practically within graph theory. We employ a definition of generalized Ricci curvature proposed by Ollivier, which Lin and Yau later adapted to graph theory, known as Ollivier-Ricci curvature (ORC). ORC measures curvature using the Wasserstein distance, thereby integrating geometric concepts with probability theory and optimal transport. Jost and Liu previously discussed the lower bound of ORC by showing the upper bound of the Wasserstein distance. We extend the applicability of these bounds to discrete spaces with metrics on integers, specifically hypergraphs. Compared to prior work on ORC in hypergraphs by Coupette, Dalleiger, and Rieck, which faced computational challenges, our method introduces a simplified approach with linear computational complexity, making it particularly suitable for analyzing large-scale networks. Through extensive simulations and application to synthetic and real-world datasets, we demonstrate the significant improvements our method offers in evaluating ORC.

We propose to learn the time-varying stochastic computational resource usage of software as a graph structured Schr\"odinger bridge problem. In general, learning the computational resource usage from data is challenging because resources such as the number of CPU instructions and the number of last level cache requests are both time-varying and statistically correlated. Our proposed method enables learning the joint time-varying stochasticity in computational resource usage from the measured profile snapshots in a nonparametric manner. The method can be used to predict the most-likely time-varying distribution of computational resource availability at a desired time. We provide detailed algorithms for stochastic learning in both single and multi-core cases, discuss the convergence guarantees, computational complexities, and demonstrate their practical use in two case studies: a single-core nonlinear model predictive controller, and a synthetic multi-core software.

Almost all classification methods such as XGBoost [1], Random Forest [2], Logistic Regression [3] are able to produce probability estimates. Output thresholding is a process to tune the decision threshold which is later used to assign class predictions based on a model's probability estimates for instances during inference [4]. For binary classification tasks, instances with probability estimates higher than or equal to the threshold are assigned positives class, otherwise as negative which is depicted in Table 1. Adjusting the threshold is particularly important for imbalanced classification problems where the train datasets have a smaller number of samples in the minority classes compared to the other classes. Output thresholding is one of the methods to address class imbalance problem [5]. Since the distribution of classes is skewed and probability estimates often favor the majority class, using a default classification threshold of 0.5 may not be the most effective approach for such problems [6]. Therefore it is essential to perform a search for the threshold to use during inference. Output thresholding is also considered to address class imbalance problem for convolutional neural networks [7].