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Mixtures of Experts (MoE) are known for their ability to learn complex conditional distributions with multiple modes. However, despite their potential, these models are challenging to train and often tend to produce poor performance, explaining their limited popularity. Our hypothesis is that this under-performance is a result of the commonly utilized maximum likelihood (ML) optimization, which leads to mode averaging and a higher likelihood of getting stuck in local maxima. We propose a novel curriculum-based approach to learning mixture models in which each component of the MoE is able to select its own subset of the training data for learning. This approach allows for independent optimization of each component, resulting in a more modular architecture that enables the addition and deletion of components on the fly, leading to an optimization less susceptible to local optima. The curricula can ignore data-points from modes not represented by the MoE, reducing the mode-averaging problem. To achieve a good data coverage, we couple the optimization of the curricula with a joint entropy objective and optimize a lower bound of this objective. We evaluate our curriculum-based approach on a variety of multimodal behavior learning tasks and demonstrate its superiority over competing methods for learning MoE models and conditional generative models.

Representing a manifold of very high-dimensional data with generative models has been shown to be computationally efficient in practice. However, this requires that the data manifold admits a global parameterization. In order to represent manifolds of arbitrary topology, we propose to learn a mixture model of variational autoencoders. Here, every encoder-decoder pair represents one chart of a manifold. We propose a loss function for maximum likelihood estimation of the model weights and choose an architecture that provides us the analytical expression of the charts and of their inverses. Once the manifold is learned, we use it for solving inverse problems by minimizing a data fidelity term restricted to the learned manifold. To solve the arising minimization problem we propose a Riemannian gradient descent algorithm on the learned manifold. We demonstrate the performance of our method for low-dimensional toy examples as well as for deblurring and electrical impedance tomography on certain image manifolds.

Expectation-Maximization (EM) algorithm is a widely used iterative algorithm for computing (local) maximum likelihood estimate (MLE). It can be used in an extensive range of problems, including the clustering of data based on the Gaussian mixture model (GMM). Numerical instability and convergence problems may arise in situations where the sample size is not much larger than the data dimensionality. In such low sample support (LSS) settings, the covariance matrix update in the EM-GMM algorithm may become singular or poorly conditioned, causing the algorithm to crash. On the other hand, in many signal processing problems, a priori information can be available indicating certain structures for different cluster covariance matrices. In this paper, we present a regularized EM algorithm for GMM-s that can make efficient use of such prior knowledge as well as cope with LSS situations. The method aims to maximize a penalized GMM likelihood where regularized estimation may be used to ensure positive definiteness of covariance matrix updates and shrink the estimators towards some structured target covariance matrices. We show that the theoretical guarantees of convergence hold, leading to better performing EM algorithm for structured covariance matrix models or with low sample settings.

Although Physics-Informed Neural Networks (PINNs) have been successfully applied in a wide variety of science and engineering fields, they can fail to accurately predict the underlying solution in slightly challenging convection-diffusion-reaction problems. In this paper, we investigate the reason of this failure from a domain distribution perspective, and identify that learning multi-scale fields simultaneously makes the network unable to advance its training and easily get stuck in poor local minima. We show that the widespread experience of sampling more collocation points in high-loss layer regions hardly help optimize and may even worsen the results. These findings motivate the development of a novel curriculum learning method that encourages neural networks to prioritize learning on easier non-layer regions while downplaying learning on harder layer regions. The proposed method helps PINNs automatically adjust the learning emphasis and thereby facilitate the optimization procedure. Numerical results on typical benchmark equations show that the proposed curriculum learning approach mitigates the failure modes of PINNs and can produce accurate results for very sharp boundary and interior layers. Our work reveals that for equations whose solutions have large scale differences, paying less attention to high-loss regions can be an effective strategy for learning them accurately.

Deep neural networks (DNNs) have greatly impacted numerous fields over the past decade. Yet despite exhibiting superb performance over many problems, their black-box nature still poses a significant challenge with respect to explainability. Indeed, explainable artificial intelligence (XAI) is crucial in several fields, wherein the answer alone -- sans a reasoning of how said answer was derived -- is of little value. This paper uncovers a troubling property of explanation methods for image-based DNNs: by making small visual changes to the input image -- hardly influencing the network's output -- we demonstrate how explanations may be arbitrarily manipulated through the use of evolution strategies. Our novel algorithm, AttaXAI, a model-agnostic, adversarial attack on XAI algorithms, only requires access to the output logits of a classifier and to the explanation map; these weak assumptions render our approach highly useful where real-world models and data are concerned. We compare our method's performance on two benchmark datasets -- CIFAR100 and ImageNet -- using four different pretrained deep-learning models: VGG16-CIFAR100, VGG16-ImageNet, MobileNet-CIFAR100, and Inception-v3-ImageNet. We find that the XAI methods can be manipulated without the use of gradients or other model internals. Our novel algorithm is successfully able to manipulate an image in a manner imperceptible to the human eye, such that the XAI method outputs a specific explanation map. To our knowledge, this is the first such method in a black-box setting, and we believe it has significant value where explainability is desired, required, or legally mandatory.

Motivated by applications in unmanned aerial based ground penetrating radar for detecting buried landmines, we consider the problem of imaging small point like scatterers situated in a lossy medium below a random rough surface. Both the random rough surface and the absorption in the lossy medium significantly impede the target detection and imaging process. Using principal component analysis we effectively remove the reflection from the air-soil interface. We then use a modification of the classical synthetic aperture radar imaging functional to image the targets. This imaging method introduces a user-defined parameter, $\delta$, which scales the resolution by $\sqrt{\delta}$ allowing for target localization with sub wavelength accuracy. Numerical results in two dimensions illustrate the robustness of the approach for imaging multiple targets. However, the depth at which targets are detectable is limited due to the absorption in the lossy medium.

Time complexity in rewriting is naturally understood as the number of steps needed to reduce terms to normal forms. Establishing complexity bounds to this measure is a well-known problem in the rewriting community. A vast majority of techniques to find such bounds consist of modifying termination proofs in order to recover complexity information. This has been done for instance with semantic interpretations, recursive path orders, and dependency pairs. In this paper, we follow the same program by tailoring tuple interpretations to deal with innermost complexity analysis. A tuple interpretation interprets terms as tuples holding upper bounds to the cost of reduction and size of normal forms. In contrast with the full rewriting setting, the strongly monotonic requirement for cost components is dropped when reductions are innermost. This weakened requirement on cost tuples allows us to prove the innermost version of the compatibility result: if all rules in a term rewriting system can be strictly oriented, then the innermost rewrite relation is well-founded. We establish the necessary conditions for which tuple interpretations guarantee polynomial bounds to the runtime of compatible systems and describe a search procedure for such interpretations.

It is a common paradigm in object detection frameworks to treat all samples equally and target at maximizing the performance on average. In this work, we revisit this paradigm through a careful study on how different samples contribute to the overall performance measured in terms of mAP. Our study suggests that the samples in each mini-batch are neither independent nor equally important, and therefore a better classifier on average does not necessarily mean higher mAP. Motivated by this study, we propose the notion of Prime Samples, those that play a key role in driving the detection performance. We further develop a simple yet effective sampling and learning strategy called PrIme Sample Attention (PISA) that directs the focus of the training process towards such samples. Our experiments demonstrate that it is often more effective to focus on prime samples than hard samples when training a detector. Particularly, On the MSCOCO dataset, PISA outperforms the random sampling baseline and hard mining schemes, e.g. OHEM and Focal Loss, consistently by more than 1% on both single-stage and two-stage detectors, with a strong backbone ResNeXt-101.

Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.

Traditional methods for link prediction can be categorized into three main types: graph structure feature-based, latent feature-based, and explicit feature-based. Graph structure feature methods leverage some handcrafted node proximity scores, e.g., common neighbors, to estimate the likelihood of links. Latent feature methods rely on factorizing networks' matrix representations to learn an embedding for each node. Explicit feature methods train a machine learning model on two nodes' explicit attributes. Each of the three types of methods has its unique merits. In this paper, we propose SEAL (learning from Subgraphs, Embeddings, and Attributes for Link prediction), a new framework for link prediction which combines the power of all the three types into a single graph neural network (GNN). GNN is a new type of neural network which directly accepts graphs as input and outputs their labels. In SEAL, the input to the GNN is a local subgraph around each target link. We prove theoretically that our local subgraphs also reserve a great deal of high-order graph structure features related to link existence. Another key feature is that our GNN can naturally incorporate latent features and explicit features. It is achieved by concatenating node embeddings (latent features) and node attributes (explicit features) in the node information matrix for each subgraph, thus combining the three types of features to enhance GNN learning. Through extensive experiments, SEAL shows unprecedentedly strong performance against a wide range of baseline methods, including various link prediction heuristics and network embedding methods.