我们考虑在非负轨道中包含的半格式集中的多项式优化问题（POP）（紧凑型集合上的每个POP都可以通过对Origin的简单翻译来以这种格式放置）。通过将每个变量平行，可以将这样的POP转换为等效的POP。使用偶数对称性和因子宽度的概念，我们根据Dickinson-Povh提出了基于P \'Olya的Potitivstellensatz的扩展，提出了半决赛弛豫的层次结构。作为其显着特征和关键特征，可以任意选择每个结果的半芬特弛豫的最大矩阵大小，此外，我们证明了新层次结构返回的值的序列收敛到原始POP的最佳值，以$ o的速率$ o。 （\ varepsilon^{ - c}）$如果半gebraic集具有非空内饰。当应用于（i）多层神经网络的鲁棒性认证和（ii）计算积极的最大奇异值时，我们的方法基于p \'olya的Potitivstellensatz提供了更好的界限，并且比标准瞬间层次结构更快地运行了几百倍。

在一组多项式不等式上最小化多项式的问题是NP-顽固的非凸问题。得益于实际代数几何形状的强大结果，人们可以将此问题转换为有限维凸问题的嵌套序列。在关联的层次结构的每个步骤中，都需要解决固定尺寸的半决赛程序，该程序又可以使用有效的数值工具来解决。但是，实际上，没有免费的午餐，这种优化方法通常包括严重的可伸缩性问题。幸运的是，对于许多应用程序，我们可以查看眼睛中的问题，并利用描述问题的成本和约束所产生的固有数据结构，例如稀疏性或对称性。本书提出了一些研究工作，以应对重要的计算含义来应对这一科学挑战，并提供了替代优化方案的开发，这些方案至少在某些已确定的问题类别中，可以很好地扩展计算复杂性。本书中提出的算法框架主要利用输入数据的稀疏结构来解决大规模的多项式优化问题。我们提出了松弛的稀疏性探索层次结构，以解决不受约束或受约束的问题。与密集的层次结构相比，它们在实践中提供了更快的溶液近似值，但也具有相同的理论收敛保证。我们的框架不仅限于静态多项式优化，因此我们揭示了由动态系统分析产生的近似值的近似值。我们还向涉及非交易变量的问题（例如，任意大小或量子物理运算符的矩阵）提供了各种扩展。

Selecting the number of topics in LDA models is considered to be a difficult task, for which alternative approaches have been proposed. The performance of the recently developed singular Bayesian information criterion (sBIC) is evaluated and compared to the performance of alternative model selection criteria. The sBIC is a generalization of the standard BIC that can be implemented to singular statistical models. The comparison is based on Monte Carlo simulations and carried out for several alternative settings, varying with respect to the number of topics, the number of documents and the size of documents in the corpora. Performance is measured using different criteria which take into account the correct number of topics, but also whether the relevant topics from the DGPs are identified. Practical recommendations for LDA model selection in applications are derived.

Applying deep learning concepts from image detection and graph theory has greatly advanced protein-ligand binding affinity prediction, a challenge with enormous ramifications for both drug discovery and protein engineering. We build upon these advances by designing a novel deep learning architecture consisting of a 3-dimensional convolutional neural network utilizing channel-wise attention and two graph convolutional networks utilizing attention-based aggregation of node features. HAC-Net (Hybrid Attention-Based Convolutional Neural Network) obtains state-of-the-art results on the PDBbind v.2016 core set, the most widely recognized benchmark in the field. We extensively assess the generalizability of our model using multiple train-test splits, each of which maximizes differences between either protein structures, protein sequences, or ligand extended-connectivity fingerprints. Furthermore, we perform 10-fold cross-validation with a similarity cutoff between SMILES strings of ligands in the training and test sets, and also evaluate the performance of HAC-Net on lower-quality data. We envision that this model can be extended to a broad range of supervised learning problems related to structure-based biomolecular property prediction. All of our software is available as open source at https://github.com/gregory-kyro/HAC-Net/.

Counterfactual explanation is a common class of methods to make local explanations of machine learning decisions. For a given instance, these methods aim to find the smallest modification of feature values that changes the predicted decision made by a machine learning model. One of the challenges of counterfactual explanation is the efficient generation of realistic counterfactuals. To address this challenge, we propose VCNet-Variational Counter Net-a model architecture that combines a predictor and a counterfactual generator that are jointly trained, for regression or classification tasks. VCNet is able to both generate predictions, and to generate counterfactual explanations without having to solve another minimisation problem. Our contribution is the generation of counterfactuals that are close to the distribution of the predicted class. This is done by learning a variational autoencoder conditionally to the output of the predictor in a join-training fashion. We present an empirical evaluation on tabular datasets and across several interpretability metrics. The results are competitive with the state-of-the-art method.

Despite their impressive performance on diverse tasks, large language models (LMs) still struggle with tasks requiring rich world knowledge, implying the limitations of relying solely on their parameters to encode a wealth of world knowledge. This paper aims to understand LMs' strengths and limitations in memorizing factual knowledge, by conducting large-scale knowledge probing experiments of 10 models and 4 augmentation methods on PopQA, our new open-domain QA dataset with 14k questions. We find that LMs struggle with less popular factual knowledge, and that scaling fails to appreciably improve memorization of factual knowledge in the tail. We then show that retrieval-augmented LMs largely outperform orders of magnitude larger LMs, while unassisted LMs remain competitive in questions about high-popularity entities. Based on those findings, we devise a simple, yet effective, method for powerful and efficient retrieval-augmented LMs, which retrieves non-parametric memories only when necessary. Experimental results show that this significantly improves models' performance while reducing the inference costs.

We introduce the Conditional Independence Regression CovariancE (CIRCE), a measure of conditional independence for multivariate continuous-valued variables. CIRCE applies as a regularizer in settings where we wish to learn neural features $\varphi(X)$ of data $X$ to estimate a target $Y$, while being conditionally independent of a distractor $Z$ given $Y$. Both $Z$ and $Y$ are assumed to be continuous-valued but relatively low dimensional, whereas $X$ and its features may be complex and high dimensional. Relevant settings include domain-invariant learning, fairness, and causal learning. The procedure requires just a single ridge regression from $Y$ to kernelized features of $Z$, which can be done in advance. It is then only necessary to enforce independence of $\varphi(X)$ from residuals of this regression, which is possible with attractive estimation properties and consistency guarantees. By contrast, earlier measures of conditional feature dependence require multiple regressions for each step of feature learning, resulting in more severe bias and variance, and greater computational cost. When sufficiently rich features are used, we establish that CIRCE is zero if and only if $\varphi(X) \perp \!\!\! \perp Z \mid Y$. In experiments, we show superior performance to previous methods on challenging benchmarks, including learning conditionally invariant image features.

This paper describes Waymo's Collision Avoidance Testing (CAT) methodology: a scenario-based testing method that evaluates the safety of the Waymo Driver Automated Driving Systems' (ADS) intended functionality in conflict situations initiated by other road users that require urgent evasive maneuvers. Because SAE Level 4 ADS are responsible for the dynamic driving task (DDT), when engaged, without immediate human intervention, evaluating a Level 4 ADS using scenario-based testing is difficult due to the potentially infinite number of operational scenarios in which hazardous situations may unfold. To that end, in this paper we first describe the safety test objectives for the CAT methodology, including the collision and serious injury metrics and the reference behavior model representing a non-impaired eyes on conflict human driver used to form an acceptance criterion. Afterward, we introduce the process for identifying potentially hazardous situations from a combination of human data, ADS testing data, and expert knowledge about the product design and associated Operational Design Domain (ODD). The test allocation and execution strategy is presented next, which exclusively utilize simulations constructed from sensor data collected on a test track, real-world driving, or from simulated sensor data. The paper concludes with the presentation of results from applying CAT to the fully autonomous ride-hailing service that Waymo operates in San Francisco, California and Phoenix, Arizona. The iterative nature of scenario identification, combined with over ten years of experience of on-road testing, results in a scenario database that converges to a representative set of responder role scenarios for a given ODD. Using Waymo's virtual test platform, which is calibrated to data collected as part of many years of ADS development, the CAT methodology provides a robust and scalable safety evaluation.

Training a Neural Radiance Field (NeRF) without pre-computed camera poses is challenging. Recent advances in this direction demonstrate the possibility of jointly optimising a NeRF and camera poses in forward-facing scenes. However, these methods still face difficulties during dramatic camera movement. We tackle this challenging problem by incorporating undistorted monocular depth priors. These priors are generated by correcting scale and shift parameters during training, with which we are then able to constrain the relative poses between consecutive frames. This constraint is achieved using our proposed novel loss functions. Experiments on real-world indoor and outdoor scenes show that our method can handle challenging camera trajectories and outperforms existing methods in terms of novel view rendering quality and pose estimation accuracy.

The integration of data and knowledge from several sources is known as data fusion. When data is available in a distributed fashion or when different sensors are used to infer a quantity of interest, data fusion becomes essential. In Bayesian settings, a priori information of the unknown quantities is available and, possibly, shared among the distributed estimators. When the local estimates are fused, such prior might be overused unless it is accounted for. This paper explores the effects of shared priors in Bayesian data fusion contexts, providing fusion rules and analysis to understand the performance of such fusion as a function of the number of collaborative agents and the uncertainty of the priors. Analytical results are corroborated through experiments in a variety of estimation and classification problems.