We consider the algorithmic problem of finding the optimal weights and biases for a two-layer fully connected neural network to fit a given set of data points. This problem is known as empirical risk minimization in the machine learning community. We show that the problem is $\exists\mathbb{R}$-complete. This complexity class can be defined as the set of algorithmic problems that are polynomial-time equivalent to finding real roots of a polynomial with integer coefficients. Furthermore, we show that arbitrary algebraic numbers are required as weights to be able to train some instances to optimality, even if all data points are rational. Our results hold even if the following restrictions are all added simultaneously. $\bullet$ There are exactly two output neurons. $\bullet$ There are exactly two input neurons. $\bullet$ The data has only 13 different labels. $\bullet$ The number of hidden neurons is a constant fraction of the number of data points. $\bullet$ The target training error is zero. $\bullet$ The ReLU activation function is used. This shows that even very simple networks are difficult to train. The result explains why typical methods for $\mathsf{NP}$-complete problems, like mixed-integer programming or SAT-solving, cannot train neural networks to global optimality, unless $\mathsf{NP}=\exists\mathbb{R}$. We strengthen a recent result by Abrahamsen, Kleist and Miltzow [NeurIPS 2021].

我们有助于更好地理解由具有Relu激活和给定架构的神经网络表示的功能。使用来自混合整数优化，多面体理论和热带几何的技术，我们为普遍近似定理提供了数学逆向，这表明单个隐藏层足以用于学习任务。特别是，我们调查完全可增值功能是否完全可以通过添加更多层（没有限制大小）来严格增加。由于它为神经假设类别代表的函数类提供给算法和统计方面，这个问题对算法和统计方面具有潜在的影响。然而，据我们所知，这个问题尚未在神经网络文学中调查。我们还在这些神经假设类别中代表功能所需的神经网络的大小上存在上限。

了解训练具有整流线性单元（RELUS）的训练简单神经网络的计算复杂性最近是一项深入研究的主题。缩小差距和文献的补充结果，我们提供了有关训练两层relu网络的参数复杂性相对于各种损失函数的几个结果。经过对其他参数的简要讨论，我们着重分析培训数据对计算复杂性的尺寸$ d $的影响。我们根据w [1]的参数$ d $提供运行时间的下限，并证明已知的蛮力策略基本上是最佳的（假设指数时间假设）。与以前的工作相比，我们的结果适用于广泛（ER）范围的损失功能，包括[0，\ infty] $中的所有$ p \ for $ \ ell^p $ -loss。特别是，我们将已知的多项式时间算法扩展到常数$ d $，并将凸损失函数扩展到更一般的损耗函数，在这些情况下，我们的运行时间下限也匹配。

本文研究了人工神经网络（NNS）与整流线性单元的表现力。为了将它们作为实际计算的模型，我们介绍了最大仿射算术计划的概念，并显示了它们与NNS之间的等效性有关自然复杂度措施。然后我们使用此结果表明，使用多项式NNS可以解决两个基本组合优化问题，这相当于非常特殊的强多项式时间算法。首先，我们显示，对于带有N $节点的任何无向图形，有一个NN大小$ \ Mathcal {O}（n ^ 3）$，它将边缘权重用为输入，计算最小生成树的值图表。其次，我们显示，对于任何带有$ N $节点和$ M $弧的任何定向图，都有一个尺寸$ \ mathcal {o}（m ^ 2n ^ 2）$，它将电弧容量作为输入和计算最大流量。这些结果尤其尤其暗示，相应的参数优化问题的解决方案可以在多项式空间中编码所有边缘权重或电弧容量的方法，并在多项式时间中进行评估，并且由NN提供这种编码。

Deep-learning of artificial neural networks (ANNs) is creating highly functional tools that are, unfortunately, as hard to interpret as their natural counterparts. While it is possible to identify functional modules in natural brains using technologies such as fMRI, we do not have at our disposal similarly robust methods for artificial neural networks. Ideally, understanding which parts of an artificial neural network perform what function might help us to address a number of vexing problems in ANN research, such as catastrophic forgetting and overfitting. Furthermore, revealing a network's modularity could improve our trust in them by making these black boxes more transparent. Here we introduce a new information-theoretic concept that proves useful in understanding and analyzing a network's functional modularity: the relay information $I_R$. The relay information measures how much information groups of neurons that participate in a particular function (modules) relay from inputs to outputs. Combined with a greedy search algorithm, relay information can be used to {\em identify} computational modules in neural networks. We also show that the functionality of modules correlates with the amount of relay information they carry.

Cashews are grown by over 3 million smallholders in more than 40 countries worldwide as a principal source of income. As the third largest cashew producer in Africa, Benin has nearly 200,000 smallholder cashew growers contributing 15% of the country's national export earnings. However, a lack of information on where and how cashew trees grow across the country hinders decision-making that could support increased cashew production and poverty alleviation. By leveraging 2.4-m Planet Basemaps and 0.5-m aerial imagery, newly developed deep learning algorithms, and large-scale ground truth datasets, we successfully produced the first national map of cashew in Benin and characterized the expansion of cashew plantations between 2015 and 2021. In particular, we developed a SpatioTemporal Classification with Attention (STCA) model to map the distribution of cashew plantations, which can fully capture texture information from discriminative time steps during a growing season. We further developed a Clustering Augmented Self-supervised Temporal Classification (CASTC) model to distinguish high-density versus low-density cashew plantations by automatic feature extraction and optimized clustering. Results show that the STCA model has an overall accuracy of 80% and the CASTC model achieved an overall accuracy of 77.9%. We found that the cashew area in Benin has doubled from 2015 to 2021 with 60% of new plantation development coming from cropland or fallow land, while encroachment of cashew plantations into protected areas has increased by 70%. Only half of cashew plantations were high-density in 2021, suggesting high potential for intensification. Our study illustrates the power of combining high-resolution remote sensing imagery and state-of-the-art deep learning algorithms to better understand tree crops in the heterogeneous smallholder landscape.

Local patterns play an important role in statistical physics as well as in image processing. Two-dimensional ordinal patterns were studied by Ribeiro et al. who determined permutation entropy and complexity in order to classify paintings and images of liquid crystals. Here we find that the 2 by 2 patterns of neighboring pixels come in three types. The statistics of these types, expressed by two parameters, contains the relevant information to describe and distinguish textures. The parameters are most stable and informative for isotropic structures.

It is well known that conservative mechanical systems exhibit local oscillatory behaviours due to their elastic and gravitational potentials, which completely characterise these periodic motions together with the inertial properties of the system. The classification of these periodic behaviours and their geometric characterisation are in an on-going secular debate, which recently led to the so-called eigenmanifold theory. The eigenmanifold characterises nonlinear oscillations as a generalisation of linear eigenspaces. With the motivation of performing periodic tasks efficiently, we use tools coming from this theory to construct an optimization problem aimed at inducing desired closed-loop oscillations through a state feedback law. We solve the constructed optimization problem via gradient-descent methods involving neural networks. Extensive simulations show the validity of the approach.

Artificial intelligence(AI) systems based on deep neural networks (DNNs) and machine learning (ML) algorithms are increasingly used to solve critical problems in bioinformatics, biomedical informatics, and precision medicine. However, complex DNN or ML models that are unavoidably opaque and perceived as black-box methods, may not be able to explain why and how they make certain decisions. Such black-box models are difficult to comprehend not only for targeted users and decision-makers but also for AI developers. Besides, in sensitive areas like healthcare, explainability and accountability are not only desirable properties of AI but also legal requirements -- especially when AI may have significant impacts on human lives. Explainable artificial intelligence (XAI) is an emerging field that aims to mitigate the opaqueness of black-box models and make it possible to interpret how AI systems make their decisions with transparency. An interpretable ML model can explain how it makes predictions and which factors affect the model's outcomes. The majority of state-of-the-art interpretable ML methods have been developed in a domain-agnostic way and originate from computer vision, automated reasoning, or even statistics. Many of these methods cannot be directly applied to bioinformatics problems, without prior customization, extension, and domain adoption. In this paper, we discuss the importance of explainability with a focus on bioinformatics. We analyse and comprehensively overview of model-specific and model-agnostic interpretable ML methods and tools. Via several case studies covering bioimaging, cancer genomics, and biomedical text mining, we show how bioinformatics research could benefit from XAI methods and how they could help improve decision fairness.

Generic Object Tracking (GOT) is the problem of tracking target objects, specified by bounding boxes in the first frame of a video. While the task has received much attention in the last decades, researchers have almost exclusively focused on the single object setting. Multi-object GOT benefits from a wider applicability, rendering it more attractive in real-world applications. We attribute the lack of research interest into this problem to the absence of suitable benchmarks. In this work, we introduce a new large-scale GOT benchmark, LaGOT, containing multiple annotated target objects per sequence. Our benchmark allows researchers to tackle key remaining challenges in GOT, aiming to increase robustness and reduce computation through joint tracking of multiple objects simultaneously. Furthermore, we propose a Transformer-based GOT tracker TaMOS capable of joint processing of multiple objects through shared computation. TaMOs achieves a 4x faster run-time in case of 10 concurrent objects compared to tracking each object independently and outperforms existing single object trackers on our new benchmark. Finally, TaMOs achieves highly competitive results on single-object GOT datasets, setting a new state-of-the-art on TrackingNet with a success rate AUC of 84.4%. Our benchmark, code, and trained models will be made publicly available.