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Motivated by the increasing availability of data of functional nature, we develop a general probabilistic and statistical framework for extremes of regularly varying random elements $X$ in $L^2[0,1]$. We place ourselves in a Peaks-Over-Threshold framework where a functional extreme is defined as an observation $X$ whose $L^2$-norm $\|X\|$ is comparatively large. Our goal is to propose a dimension reduction framework resulting into finite dimensional projections for such extreme observations. Our contribution is double. First, we investigate the notion of Regular Variation for random quantities valued in a general separable Hilbert space, for which we propose a novel concrete characterization involving solely stochastic convergence of real-valued random variables. Second, we propose a notion of functional Principal Component Analysis (PCA) accounting for the principal `directions' of functional extremes. We investigate the statistical properties of the empirical covariance operator of the angular component of extreme functions, by upper-bounding the Hilbert-Schmidt norm of the estimation error for finite sample sizes. Numerical experiments with simulated and real data illustrate this work.

We discretize a risk-neutral optimal control problem governed by a linear elliptic partial differential equation with random inputs using a Monte Carlo sample-based approximation and a finite element discretization, yielding finite dimensional control problems. We establish an exponential tail bound for the distance between the finite dimensional problems' solutions and the risk-neutral problem's solution. The tail bound implies that solutions to the risk-neutral optimal control problem can be reliably estimated with the solutions to the finite dimensional control problems. Numerical simulations illustrate our theoretical findings.

Interior point methods are widely used for different types of mathematical optimization problems. Many implementations of interior point methods in use today rely on direct linear solvers to solve systems of equations in each iteration. The need to solve ever larger optimization problems more efficiently and the rise of hardware accelerators for general purpose computing has led to a large interest in using iterative linear solvers instead, with the major issue being inevitable ill-conditioning of the linear systems arising as the optimization progresses. We investigate the use of Krylov solvers for interior point methods in solving optimization problems from radiation therapy. We implement a prototype interior point method using a so called doubly augmented formulation of the Karush-Kuhn-Tucker (KKT) linear system of equations, originally proposed by Forsgren and Gill, and evaluate its performance on real optimization problems from radiation therapy. Crucially, our implementation uses a preconditioned conjugate gradient method with Jacobi preconditioning internally. Our measurements of the conditioning of the linear systems indicate that the Jacobi preconditioner improves the conditioning of the systems to a degree that they can be solved iteratively, but there is room for further improvement in that regard. Furthermore, profiling of our prototype code shows that it is suitable for GPU acceleration, which may further improve its performance in practice. Overall, our results indicate that our method can find solutions of acceptable accuracy in reasonable time, even with a simple Jacobi preconditioner.

Classical recommender systems often assume that historical data are stationary and fail to account for the dynamic nature of user preferences, limiting their ability to provide reliable recommendations in time-sensitive settings. This assumption is particularly problematic in finance, where financial products exhibit continuous changes in valuations, leading to frequent shifts in client interests. These evolving interests, summarized in the past client-product interactions, see their utility fade over time with a degree that might differ from one client to another. To address this challenge, we propose a time-dependent collaborative filtering algorithm that can adaptively discount distant client-product interactions using personalized decay functions. Our approach is designed to handle the non-stationarity of financial data and produce reliable recommendations by modeling the dynamic collaborative signals between clients and products. We evaluate our method using a proprietary dataset from BNP Paribas and demonstrate significant improvements over state-of-the-art benchmarks from relevant literature. Our findings emphasize the importance of incorporating time explicitly in the model to enhance the accuracy of financial product recommendation.

Few existing studies focus on the source separation problem with unknown numbers of signals, and how to evaluate the performances of the systems is not yet clear. We propose a solution with a fixed number of output channels to address these two problems, enabling it to avoid the dimensional disaster caused by the permutation problem induced by the alignment of outputs to targets. Specifically, we propose a two-step algorithm based on autoencoders and a new performance evaluation method for situations with mute channels. Experiments conducted on simulated mixtures of radiated ship noise show that the proposed solution can achieve similar separation performance to that attained with a known number of signals. The proposed algorithm achieved competitive performance as two algorithms developed for known numbers of signals, which is highly explainable and extensible and get the state of the art under this framework.

We present SEIF, a methodology that combines static analysis with symbolic execution to verify and explicate information flow paths in a hardware design. SEIF begins with a statically built model of the information flow through a design and uses guided symbolic execution to recognize and eliminate non-flows with high precision or to find corresponding paths through the design state for true flows. We evaluate SEIF on two open-source CPUs, an AES core, and the AKER access control module. SEIF can exhaustively explore 10-12 clock cycles deep in 4-6 seconds on average, and can automatically account for 86-90% of the paths in the statically built model. Additionally, SEIF can be used to find multiple violating paths for security properties, providing a new angle for security verification.

Our research focuses on solving the zero-shot text classification problem in NLP, with a particular emphasis on innovative self-training strategies. To achieve this objective, we propose a novel self-training strategy that uses labels rather than text for training, significantly reducing the model's training time. Specifically, we use categories from Wikipedia as our training set and leverage the SBERT pre-trained model to establish positive correlations between pairs of categories within the same text, facilitating associative training. For new test datasets, we have improved the original self-training approach, eliminating the need for prior training and testing data from each target dataset. Instead, we adopt Wikipedia as a unified training dataset to better approximate the zero-shot scenario. This modification allows for rapid fine-tuning and inference across different datasets, greatly reducing the time required for self-training. Our experimental results demonstrate that this method can adapt the model to the target dataset within minutes. Compared to other BERT-based transformer models, our approach significantly reduces the amount of training data by training only on labels, not the actual text, and greatly improves training efficiency by utilizing a unified training set. Additionally, our method achieves state-of-the-art results on both the Yahoo Topic and AG News datasets.

Momentum is known to accelerate the convergence of gradient descent in strongly convex settings without stochastic gradient noise. In stochastic optimization, such as training neural networks, folklore suggests that momentum may help deep learning optimization by reducing the variance of the stochastic gradient update, but previous theoretical analyses do not find momentum to offer any provable acceleration. Theoretical results in this paper clarify the role of momentum in stochastic settings where the learning rate is small and gradient noise is the dominant source of instability, suggesting that SGD with and without momentum behave similarly in the short and long time horizons. Experiments show that momentum indeed has limited benefits for both optimization and generalization in practical training regimes where the optimal learning rate is not very large, including small- to medium-batch training from scratch on ImageNet and fine-tuning language models on downstream tasks.

Incompleteness is a common problem for existing knowledge graphs (KGs), and the completion of KG which aims to predict links between entities is challenging. Most existing KG completion methods only consider the direct relation between nodes and ignore the relation paths which contain useful information for link prediction. Recently, a few methods take relation paths into consideration but pay less attention to the order of relations in paths which is important for reasoning. In addition, these path-based models always ignore nonlinear contributions of path features for link prediction. To solve these problems, we propose a novel KG completion method named OPTransE. Instead of embedding both entities of a relation into the same latent space as in previous methods, we project the head entity and the tail entity of each relation into different spaces to guarantee the order of relations in the path. Meanwhile, we adopt a pooling strategy to extract nonlinear and complex features of different paths to further improve the performance of link prediction. Experimental results on two benchmark datasets show that the proposed model OPTransE performs better than state-of-the-art methods.

We examine the problem of question answering over knowledge graphs, focusing on simple questions that can be answered by the lookup of a single fact. Adopting a straightforward decomposition of the problem into entity detection, entity linking, relation prediction, and evidence combination, we explore simple yet strong baselines. On the popular SimpleQuestions dataset, we find that basic LSTMs and GRUs plus a few heuristics yield accuracies that approach the state of the art, and techniques that do not use neural networks also perform reasonably well. These results show that gains from sophisticated deep learning techniques proposed in the literature are quite modest and that some previous models exhibit unnecessary complexity.