We provide two fundamental results on the population (infinite-sample) likelihood function of Gaussian mixture models with M 3 components. Our first main result shows that the population likelihood function has bad local maxima even in the special case of equally-weighted mixtures of well-separated and spherical Gaussians. We prove that the log-likelihood value of these bad local maxima can be arbitrarily worse than that of any global optimum, thereby resolving an open question of Srebro [2007].

An augmented Lagrangian (AL) can convert a constrained optimization problem into a sequence of simpler (e.g., unconstrained) problems, which are then usually solved with local solvers. Recently, surrogate-based Bayesian optimization (BO) sub-solvers have been successfully deployed in the AL framework for a more global search in the presence of inequality constraints; however, a drawback was that expected improvement (EI) evaluations relied on Monte Carlo. Here we introduce an alternative slack variable AL, and show that in this formulation the EI may be evaluated with library routines. The slack variables furthermore facilitate equality as well as inequality constraints, and mixtures thereof. We show our new slack "ALBO" compares favorably to the original. Its superiority over conventional alternatives is reinforced on several mixed constraint examples.

Influence maximization in social networks has typically been studied in the context of contagion models and irreversible processes. In this paper, we consider an alternate model that treats individual opinions as spins in an Ising system at dynamic equilibrium. We formalize the Ising influence maximization problem, which has a natural physical interpretation as maximizing the magnetization given a budget of external magnetic field. Under the mean-field (MF) approximation, we present a gradient ascent algorithm that uses the susceptibility to efficiently calculate local maxima of the magnetization, and we develop a number of sufficient conditions for when the MF magnetization is concave and our algorithm converges to a global optimum. We apply our algorithm on random and real-world networks, demonstrating, remarkably, that the MF optimal external fields (i.e., the external fields which maximize the MF magnetization) shift from focusing on high-degree individuals at high temperatures to focusing on low-degree individuals at low temperatures. We also establish a number of novel results about the structure of steady-states in the ferromagnetic MF Ising model on general graph topologies, which are of independent interest.

Many real phenomena, including behaviors, involve strategic interactions that can be learned from data. We focus on learning tree structured potential games where equilibria are represented by local maxima of an underlying potential function. We cast the learning problem within a max margin setting and show that the problem is NP-hard even when the strategic interactions form a tree. We develop a variant of dual decomposition to estimate the underlying game and demonstrate with synthetic and real decision/voting data that the game theoretic perspective (carving out local maxima) enables meaningful recovery.

Many practical perception systems exist within larger processes that include interactions with users or additional components capable of evaluating the quality of predicted solutions. In these contexts, it is beneficial to provide these oracle mechanisms with multiple highly likely hypotheses rather than a single prediction. In this work, we pose the task of producing multiple outputs as a learning problem over an ensemble of deep networks - introducing a novel stochastic gradient descent based approach to minimize the loss with respect to an oracle. Our method is simple to implement, agnostic to both architecture and loss function, and parameter-free. Our approach achieves lower oracle error compared to existing methods on a wide range of tasks and deep architectures. We also show qualitatively that the diverse solutions produced often provide interpretable representations of task ambiguity.

Principal components analysis (PCA) is the optimal linear encoder of data. Sparse linear encoders (e.g., sparse PCA) produce more interpretable features that can promote better generalization.

It is, however, naval drones that have made Russia's Black Sea fleet particularly vulnerable. Under relentless attacks, Moscow was forced to withdraw the core of its fleet from Crimea and move them further east, to Novorossiysk. And even there, Russian ships remain within reach of Ukrainian drones.

We propose an operator learning approach to accelerate geometric Markov chain Monte Carlo (MCMC) for solving infinite-dimensional nonlinear Bayesian inverse problems. While geometric MCMC employs high-quality proposals that adapt to posterior local geometry, it requires computing local gradient and Hessian information of the log-likelihood, incurring a high cost when the parameter-to-observable (PtO) map is defined through expensive model simulations. We consider a delayed-acceptance geometric MCMC method driven by a neural operator surrogate of the PtO map, where the proposal is designed to exploit fast surrogate approximations of the log-likelihood and, simultaneously, its gradient and Hessian. To achieve a substantial speedup, the surrogate needs to be accurate in predicting both the observable and its parametric derivative (the derivative of the observable with respect to the parameter). Training such a surrogate via conventional operator learning using input--output samples often demands a prohibitively large number of model simulations. In this work, we present an extension of derivative-informed operator learning [O'Leary-Roseberry et al., J. Comput. Phys., 496 (2024)] using input--output--derivative training samples. Such a learning method leads to derivative-informed neural operator (DINO) surrogates that accurately predict the observable and its parametric derivative at a significantly lower training cost than the conventional method. Cost and error analysis for reduced basis DINO surrogates are provided. Numerical studies on PDE-constrained Bayesian inversion demonstrate that DINO-driven MCMC generates effective posterior samples 3--9 times faster than geometric MCMC and 60--97 times faster than prior geometry-based MCMC. Furthermore, the training cost of DINO surrogates breaks even after collecting merely 10--25 effective posterior samples compared to geometric MCMC.

Large language models are playing an increasingly significant role in molecular research, yet existing models often generate erroneous information, posing challenges to accurate molecular comprehension. Traditional evaluation metrics for generated content fail to assess a model's accuracy in molecular understanding. To rectify the absence of factual evaluation, we present MoleculeQA, a novel question answering (QA) dataset which possesses 62K QA pairs over 23K molecules. Each QA pair, composed of a manual question, a positive option and three negative options, has consistent semantics with a molecular description from authoritative molecular corpus. MoleculeQA is not only the first benchmark for molecular factual bias evaluation but also the largest QA dataset for molecular research. A comprehensive evaluation on MoleculeQA for existing molecular LLMs exposes their deficiencies in specific areas and pinpoints several particularly crucial factors for molecular understanding.

Recent advancements in Intrusion Detection Systems (IDS), integrating Explainable AI (XAI) methodologies, have led to notable improvements in system performance via precise feature selection. However, a thorough understanding of cyber-attacks requires inherently explainable decision-making processes within IDS. In this paper, we present the Interpretable Generalization Mechanism (IG), poised to revolutionize IDS capabilities. IG discerns coherent patterns, making it interpretable in distinguishing between normal and anomalous network traffic. Further, the synthesis of coherent patterns sheds light on intricate intrusion pathways, providing essential insights for cybersecurity forensics. By experiments with real-world datasets NSL-KDD, UNSW-NB15, and UKM-IDS20, IG is accurate even at a low ratio of training-to-test. With 10%-to-90%, IG achieves Precision (PRE)=0.93, Recall (REC)=0.94, and Area Under Curve (AUC)=0.94 in NSL-KDD; PRE=0.98, REC=0.99, and AUC=0.99 in UNSW-NB15; and PRE=0.98, REC=0.98, and AUC=0.99 in UKM-IDS20. Notably, in UNSW-NB15, IG achieves REC=1.0 and at least PRE=0.98 since 40%-to-60%; in UKM-IDS20, IG achieves REC=1.0 and at least PRE=0.88 since 20%-to-80%. Importantly, in UKM-IDS20, IG successfully identifies all three anomalous instances without prior exposure, demonstrating its generalization capabilities. These results and inferences are reproducible. In sum, IG showcases superior generalization by consistently performing well across diverse datasets and training-to-test ratios (from 10%-to-90% to 90%-to-10%), and excels in identifying novel anomalies without prior exposure. Its interpretability is enhanced by coherent evidence that accurately distinguishes both normal and anomalous activities, significantly improving detection accuracy and reducing false alarms, thereby strengthening IDS reliability and trustworthiness.