We apply one-dimensional convolutional neural networks to the Frobenius traces of elliptic curves over $\mathbb{Q}$ and evaluate and interpret their predictive capacity. In keeping with similar experiments by Kazalicki--Vlah, Bujanović--Kazalicki--Novak, and Pozdnyakov, we observe high accuracy predictions for the analytic rank across a range of conductors. We interpret the prediction using saliency curves and explore the interesting interplay between murmurations and Mestre--Nagao sums, the details of which vary with the conductor and the (predicted) rank.

We present a complete theoretical characterization of Latent Posterior Factors (LPF), a principled framework for aggregating multiple heterogeneous evidence items in probabilistic prediction tasks. Multi-evidence reasoning arises pervasively in high-stakes domains including healthcare diagnosis, financial risk assessment, legal case analysis, and regulatory compliance, yet existing approaches either lack formal guarantees or fail to handle multi-evidence scenarios architecturally. LPF encodes each evidence item into a Gaussian latent posterior via a variational autoencoder, converting posteriors to soft factors through Monte Carlo marginalization, and aggregating factors via exact Sum-Product Network inference (LPF-SPN) or a learned neural aggregator (LPF-Learned). We prove seven formal guarantees spanning the key desiderata for trustworthy AI: Calibration Preservation (ECE <= epsilon + C/sqrt(K_eff)); Monte Carlo Error decaying as O(1/sqrt(M)); a non-vacuous PAC-Bayes bound with train-test gap of 0.0085 at N=4200; operation within 1.12x of the information-theoretic lower bound; graceful degradation as O(epsilon*delta*sqrt(K)) under corruption, maintaining 88% performance with half of evidence adversarially replaced; O(1/sqrt(K)) calibration decay with R^2=0.849; and exact epistemic-aleatoric uncertainty decomposition with error below 0.002%. All theorems are empirically validated on controlled datasets spanning up to 4,200 training examples. Our theoretical framework establishes LPF as a foundation for trustworthy multi-evidence AI in safety-critical applications.

Decision-making problems often feature uncertainty stemming from heterogeneous and context-dependent human preferences. To address this, we propose a sequential learning-and-optimization pipeline to learn preference distributions and leverage them to solve downstream problems, for example risk-averse formulations. We focus on human choice settings that can be formulated as (integer) linear programs. In such settings, existing inverse optimization and choice modelling methods infer preferences from observed choices but typically produce point estimates or fail to capture contextual shifts, making them unsuitable for risk-averse decision-making. Using a bounded-variance score function gradient estimator, we train a predictive model mapping contextual features to a rich class of parameterizable distributions. This approach yields a maximum likelihood estimate. The model generates scenarios for unseen contexts in the subsequent optimization phase. In a synthetic ridesharing environment, our approach reduces average post-decision surprise by up to 114$\times$ compared to a risk-neutral approach with perfect predictions and up to 25$\times$ compared to leading risk-averse baselines.

Neural networks are central to modern artificial intelligence, yet their training remains highly sensitive to data contamination. Standard neural classifiers are trained by minimizing the categorical cross-entropy loss, corresponding to maximum likelihood estimation under a multinomial model. While statistically efficient under ideal conditions, this approach is highly vulnerable to contaminated observations including label noises corrupting supervision in the output space, and adversarial perturbations inducing worst-case deviations in the input space. In this paper, we propose a unified and statistically grounded framework for robust neural classification that addresses both forms of contamination within a single learning objective. We formulate neural network training as a minimum-divergence estimation problem and introduce rSDNet, a robust learning algorithm based on the general class of $S$-divergences. The resulting training objective inherits robustness properties from classical statistical estimation, automatically down-weighting aberrant observations through model probabilities. We establish essential population-level properties of rSDNet, including Fisher consistency, classification calibration implying Bayes optimality, and robustness guarantees under uniform label noise and infinitesimal feature contamination. Experiments on three benchmark image classification datasets show that rSDNet improves robustness to label corruption and adversarial attacks while maintaining competitive accuracy on clean data, Our results highlight minimum-divergence learning as a principled and effective framework for robust neural classification under heterogeneous data contamination.

Mirror Descent (MD) is a scalable first-order method widely used in large-scale optimization, with applications in image processing, policy optimization, and neural network training. This paper generalizes MD to optimization on Riemannian manifolds. In particular, we develop a Riemannian Mirror Descent (RMD) framework via reparameterization and further propose a stochastic variant of RMD. We also establish non-asymptotic convergence guarantees for both RMD and stochastic RMD. As an application to the Stiefel manifold, our RMD framework reduces to the Curvilinear Gradient Descent (CGD) method proposed in [26]. Moreover, when specializing the stochastic RMD framework to the Stiefel setting, we obtain a stochastic extension of CGD, which effectively addresses large-scale manifold optimization problems.

In sensitive contexts, providers of machine learning algorithms are increasingly required to give explanations for their algorithms' decisions. However, explanation receivers might not trust the provider, who potentially could output misleading or manipulated explanations. In this work, we investigate an auditing framework in which a third-party auditor or a collective of users attempts to sanity-check explanations: they can query model decisions and the corresponding local explanations, pool all the information received, and then check for basic consistency properties. We prove upper and lower bounds on the amount of queries that are needed for an auditor to succeed within this framework. Our results show that successful auditing requires a potentially exorbitant number of queries -- particularly in high dimensional cases. Our analysis also reveals that a key property is the ``locality'' of the provided explanations --- a quantity that so far has not been paid much attention to in the explainability literature. Looking forward, our results suggest that for complex high-dimensional settings, merely providing a pointwise prediction and explanation could be insufficient, as there is no way for the users to verify that the provided explanations are not completely made-up.

THE LOST WEDDING PHOTOS: See JFK Jr and Carolyn Bessette at their secret nuptials... and read every intimate detail of ultra-private ceremony Tulsi Gabbard lets Iran nuke bombshell slip as Senate hearing spirals for Trump's embattled spy chief Candace Owens's sickening low-blow at Karoline Leavitt as Iran war sparks wild attacks Lunatic Megyn Kelly is FINALLY ruined! Her appalling X-rated smear of my friend proves it... but now I know her truly disturbing plan: JOSH HAMMER Inside the epidemic of midlife women who are repulsed by their husbands, the age and'vital statistics' that make men most at risk - and the telltale signs YOUR marriage is about to die: Special report by SADIE NICHOLAS Meghan gives glimpse of'mama's little helpers' Archie and Lilibet in'behind the scenes' video of her latest As Ever launch Shameful hypocrisy of NASCAR star Daniel Suarez's nepo-baby wife: 'Victim' mask slips as she ignites new Las Vegas drama... and dark family past rears its ugly head Princess Kate dons her favourite tiara and the late Queen's earrings as she arrives at King's banquet for the Nigerian President in country's first state visit in almost 40 years Everything JFK Jr told friends about his love affair with'sexual dynamo' Madonna... her unprintable pillow talk... and his perverse incest request that she couldn't go through with Site of'Jesus' crucifixion' forced to shut for Holy Week in unprecedented move tied to biblical prophecies of the Antichrist Ugly new Nicole Kidman and Keith Urban divorce fight ERUPTS: Her friends share humiliating details of'midlife crisis'... and reveal brutal REAL reason daughter Sunday Rose'snubbed' him Outrage after Seattle museum vandal destroys $250,000 of famous Dale Chihuly glass at city's museum dedicated to him Amanda Bynes, 39, 'is now a size 4 after losing 35lb' thanks to weight-loss medication... after hitting 180lb Chilling unclassified threat report reveals the'most likely' terror attack scenario on US soil Three's Company bombshell Jenilee Harrison who was also on Dallas and The Love Boat still looks great at 67, see her now The discovery of a Roman artifact in the Americas has sparked a debate about who truly discovered the New World. While Christopher Columbus is hailed as the first in 1492, archaeologists uncovered a small terracotta head of a bearded man carved with distinctive European features tucked inside a Mexican tomb. The artifact, known as the Tecaxic-Calixtlahuaca Head, was discovered in 1933 inside a sealed pre-Hispanic burial beneath multiple intact layers, indicating it had not been disturbed after its placement. Experts say its facial features, beard style and craftsmanship bear a striking resemblance to objects from the ancient Mediterranean rather than indigenous Mesoamerican traditions.

Distributed learning is essential for training large-scale deep models.Asynchronous SGD (ASGD) and its variants are commonly used distributed learning methods, particularly in scenarios where the computing capabilities of workers in the cluster are heterogeneous.Momentum has been acknowledged for its benefits in both optimization and generalization in deep model training. However, existing works have found that naively incorporating momentum into ASGD can impede the convergence.In this paper, we propose a novel method called ordered momentum (OrMo) for ASGD. In OrMo, momentum is incorporated into ASGD by organizing the gradients in order based on their iteration indexes. We theoretically prove the convergence of OrMo with both constant and delay-adaptive learning rates for non-convex problems. To the best of our knowledge, this is the first work to establish the convergence analysis of ASGD with momentum without dependence on the maximum delay. Empirical results demonstrate that OrMo can achieve better convergence performance compared with ASGD and other asynchronous methods with momentum.

We consider learning a sparse model from linear measurements taken by a network of agents. Different from existing decentralized methods designed based on the LASSO regression with explicit $\ell_1$ norm regularization, we exploit the implicit regularization of decentralized optimization method applied to an over-parameterized nonconvex least squares formulation without penalization. Our first result shows that despite nonconvexity, if the network connectivity is good, the well-known decentralized gradient descent algorithm (DGD) with small initialization and early stopping can compute the statistically optimal solution. Sufficient conditions on the initialization scale, choice of step size, network connectivity, and stopping time are further provided to achieve convergence. Our result recovers the convergence rate of gradient descent in the centralized setting, showing its tightness. Based on the analysis of DGD, we further propose a communication-efficient version, termed T-DGD, by truncating the iterates before transmission. In the high signal-to-noise ratio (SNR) regime, we show that T-DGD achieves comparable statistical accuracy to DGD, while the communication cost is logarithmic in the number of parameters. Numerical results are provided to validate the effectiveness of DGD and T-DGD for sparse learning through implicit regularization.

Recent years have seen a significant progress in the general-purpose problem solving abilities of large vision and language models (LVLMs), such as ChatGPT, Gemini, etc.; some of these breakthroughs even seem to enable AI models to outperform human abilities in varied tasks that demand higher-order cognitive skills. Are the current large AI models indeed capable of generalized problem solving as humans do? A systematic analysis of AI capabilities for joint vision and text reasoning, however, is missing in the current scientific literature. In this paper, we make an effort towards filling this gap, by evaluating state-of-the-art LVLMs on their mathematical and algorithmic reasoning abilities using visuo-linguistic problems from children's Olympiads. Specifically, we consider problems from the Mathematical Kangaroo (MK) Olympiad, which is a popular international competition targeted at children from grades 1-12, that tests children's deeper mathematical abilities using puzzles that are appropriately gauged to their age and skills. Using the puzzles from MK, we created a dataset, dubbed SMART-840, consisting of 840 problems from years 2020-2024. With our dataset, we analyze LVLMs power on mathematical reasoning; their responses on our puzzles offer a direct way to compare against that of children. Our results show that modern LVLMs do demonstrate increasingly powerful reasoning skills in solving problems for higher grades, but lack the foundations to correctly answer problems designed for younger children. Further analysis shows that there is no significant correlation between the reasoning capabilities of AI models and that of young children, and their capabilities appear to be based on a different type of reasoning than the cumulative knowledge that underlies children's mathematical skills.