As the demand to integrate Artificial Intelligence into high-stakes environments continues to grow, explaining the reasoning behind neural-network predictions has shifted from a theoretical curiosity to a strict operational requirement. Our work is motivated by the explanations of autoregressive neural predictions on dynamic physical fields, as in weather forecasting. Gradient-based feature attribution methods are widely used to explain the predictions on such data, in particular due to their scalability to high-dimensional inputs. It is also interesting to remark that gradient-based techniques such as SmoothGrad are now standard on images to robustify the explanations using pointwise averages of the attribution maps obtained from several noised inputs. Our goal is to efficiently adapt this aggregation strategy to dynamic physical fields. To do so, our first contribution is to identify a fundamental failure mode when averaging perturbed attribution maps on dynamic physical fields: stochastic input perturbations do not induce stationary amplitude noise in attribution maps, but instead cause a geometric displacement of the attributions. Consequently, pointwise averaging blurs these spatially misaligned features. To tackle this issue, we introduce WassersteinGrad, which extracts a geometric consensus of perturbed attribution maps by computing their entropic Wasserstein barycenter. The results, obtained on regional weather data and a meteorologist-validated neural model, demonstrate promising explainability properties of WassersteinGrad over gradient-based baselines across both single-step and autoregressive forecasting settings.

The Gaussian mixture model is widely used in unsupervised learning, owing to its simplicity and interpretability. However, a fundamental limitation of the classical Gaussian mixture model is that it forces each observation to belong to exactly one component. In many practical applications, such as genetics, social network analysis, and text mining, an observation may naturally belong to multiple components or exhibit partial membership in several latent components. To overcome this limitation, we propose the mixed membership sub-Gaussian model, which extends the classical Gaussian mixture framework by allowing each observation to belong to multiple components. This model inherits the interpretability of the classical Gaussian mixture model while offering greater flexibility for capturing complex overlapping structures. We develop an efficient spectral algorithm to estimate the mixed membership of each individual observation, and under mild separation conditions on the component centres, we prove that the estimation error of the per-individual membership vector can be made arbitrarily small with high probability. To our knowledge, this is the first work to provide a computationally efficient estimator with such a vanishing-error guarantee for a mixed-membership extension of the Gaussian mixture model. Extensive experimental studies demonstrate that our method outperforms existing approaches that ignore mixed memberships.

Predicting customers' long-term revenue from sparse and irregular transaction data is central to marketing resource allocation in non-contractual settings, yet existing approaches face a trade-off. Traditional probabilistic customer base models deliver robust long-horizon forecasts by imposing strong structural assumptions, while flexible machine-learning models often require substantial training data and careful tuning. We propose a variational-autoencoder-based model that preserves the process-based likelihood of established attrition-transaction-spend models conditional on customer heterogeneity, but replaces the restrictive parametric mixing distribution with a flexible latent representation learned by encoder-decoder networks. The resulting approach (i) provides a single model for customer attrition, transactions and spending, (ii) remains reliable when contextual covariates are unavailable, and (iii) flexibly incorporates rich covariates and nonlinear effects when they are available. This design balances structural stability with the flexibility needed to capture complex purchase dynamics. Across multiple real-world datasets and prediction horizons, the proposed model improves upon the latest benchmarks. Businesses benefit directly, as a better assessment of customers' future revenues improves the efficiency of campaign targeting. For research, this work provides guidance on how to embed domain-specific models into the variational autoencoder framework, enabling flexible representation learning while retaining an econometrically meaningful process structure.

Canada's Carney has enjoyed a long political honeymoon. Mark Carney arrived on Canada's political scene last year as an Ivy League and Oxford educated economist and a former central banker for two countries. He had an impressive resume and ambitions to be prime minister but had never run for public office until replacing Justin Trudeau as Liberal leader. There was concern his lack of political experience would be a liability, but under his leadership, the Liberals won a minority government, which in a year had solidified into a narrow majority following the defection of five opposition members of parliament to his party. Carney tore up the rulebook, jumping from political neophyte to leading a G7 nation, and he is enjoying a lengthy honeymoon both in Canada and around the world as a globetrotting prime minister.

Kevin Durant is expected to sit out Game 4 with a sprained left ankle and bone bruise, as the Rockets face elimination against the Lakers on Sunday night.

China's economy was just starting to open up in the late 1980s when a determined high school dropout made his way to Beijing with 600 pairs of shoes. Ding Shizhong had them made in a relative's factory and now he was going to sell them. The money he earned paid for his first workshop where he began making footwear for other companies. The 17-year-old was one of China's many newly minted entrepreneurs as capitalism took off under the watchful eye of its Communist Party rulers. But, as it turns out, Ding had much bigger plans.

We chose the Google Search corpus [Luo et al., 2021] for our question-answering system as it provides good coverage of the knowledge needed and is publicly available. However, as noted by the authors of RA-VQA, additional knowledge bases may be required to answer some questions correctly. Future work may address the issue by improving the quality and expanding the coverage of knowledge. We do not perceive any immediate ethical concerns associated with the misuse of our proposed system. There is a possibility that the trained KB-VQA system might generate inappropriate or biased content as a result of the training data biases during LLM and LMM pre-training and fine-tuning.