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Understanding quantum channels and the strange behavior of their capacities is a key objective of quantum information theory. Here we study a remarkably simple, low-dimensional, single-parameter family of quantum channels with exotic quantum information-theoretic features. As the simplest example from this family, we focus on a qutrit-to-qutrit channel that is intuitively obtained by hybridizing together a simple degradable channel and a completely useless qubit channel. Such hybridizing makes this channel's capacities behave in a variety of interesting ways. For instance, the private and classical capacity of this channel coincide and can be explicitly calculated, even though the channel does not belong to any class for which the underlying information quantities are known to be additive. Moreover, the quantum capacity of the channel can be computed explicitly, given a clear and compelling conjecture is true. This "spin alignment conjecture," which may be of independent interest, is proved in certain special cases and additional numerical evidence for its validity is provided. Finally, we generalize the qutrit channel in two ways, and the resulting channels and their capacities display similarly rich behavior. In the companion paper [Phys. Rev. Lett. 130, 200801 (2023); arXiv:2202.08377], we further show that the qutrit channel demonstrates superadditivity when transmitting quantum information jointly with a variety of assisting channels, in a manner unknown before.

A common forecasting setting in real world applications considers a set of possibly heterogeneous time series of the same domain. Due to different properties of each time series such as length, obtaining forecasts for each individual time series in a straight-forward way is challenging. This paper proposes a general framework utilizing a similarity measure in Dynamic Time Warping to find similar time series to build neighborhoods in a k-Nearest Neighbor fashion, and improve forecasts of possibly simple models by averaging. Several ways of performing the averaging are suggested, and theoretical arguments underline the usefulness of averaging for forecasting. Additionally, diagnostics tools are proposed allowing a deep understanding of the procedure.

Ensuring the security of networked systems is a significant problem, considering the susceptibility of modern infrastructures and technologies to adversarial interference. A central component of this problem is how defensive resources should be allocated to mitigate the severity of potential attacks on the system. In this paper, we consider this in the context of a General Lotto game, where a defender and attacker deploys resources on the nodes of a network, and the objective is to secure as many links as possible. The defender secures a link only if it out-competes the attacker on both of its associated nodes. For bipartite networks, we completely characterize equilibrium payoffs and strategies for both the defender and attacker. Surprisingly, the resulting payoffs are the same for any bipartite graph. On arbitrary network structures, we provide lower and upper bounds on the defender's max-min value. Notably, the equilibrium payoff from bipartite networks serves as the lower bound. These results suggest that more connected networks are easier to defend against attacks. We confirm these findings with simulations that compute deterministic allocation strategies on large random networks. This also highlights the importance of randomization in the equilibrium strategies.

In a recent article the authors showed that the radiative Transfer equations with multiple frequencies and scattering can be formulated as a nonlinear integral system. In the present article, the formulation is extended to handle reflective boundary conditions. The fixed point method to solve the system is shown to be monotone. The discretization is done with a $P^1$ Finite Element Method. The convolution integrals are precomputed at every vertices of the mesh and stored in compressed hierarchical matrices, using Partially Pivoted Adaptive Cross-Approximation. Then the fixed point iterations involve only matrix vector products. The method is $O(N\sqrt[3]{N}\ln N)$, with respect to the number of vertices, when everything is smooth. A numerical implementation is proposed and tested on two examples. As there are some analogies with ray tracing the programming is complex.

Offline reinforcement learning (RL) seeks to derive an effective control policy from previously collected data. To circumvent errors due to inadequate data coverage, behavior-regularized methods optimize the control policy while concurrently minimizing deviation from the data collection policy. Nevertheless, these methods often exhibit subpar practical performance, particularly when the offline dataset is collected by sub-optimal policies. In this paper, we propose a novel algorithm employing in-sample policy iteration that substantially enhances behavior-regularized methods in offline RL. The core insight is that by continuously refining the policy used for behavior regularization, in-sample policy iteration gradually improves itself while implicitly avoids querying out-of-sample actions to avert catastrophic learning failures. Our theoretical analysis verifies its ability to learn the in-sample optimal policy, exclusively utilizing actions well-covered by the dataset. Moreover, we propose competitive policy improvement, a technique applying two competitive policies, both of which are trained by iteratively improving over the best competitor. We show that this simple yet potent technique significantly enhances learning efficiency when function approximation is applied. Lastly, experimental results on the D4RL benchmark indicate that our algorithm outperforms previous state-of-the-art methods in most tasks.

Nonconvex-nonconcave minimax optimization has received intense attention over the last decade due to its broad applications in machine learning. Unfortunately, most existing algorithms cannot be guaranteed to converge globally and even suffer from limit cycles. To address this issue, we propose a novel single-loop algorithm called doubly smoothed gradient descent ascent method (DSGDA), which naturally balances the primal and dual updates. The proposed DSGDA can get rid of limit cycles in various challenging nonconvex-nonconcave examples in the literature, including Forsaken, Bilinearly-coupled minimax, Sixth-order polynomial, and PolarGame. We further show that under an one-sided Kurdyka-\L{}ojasiewicz condition with exponent $\theta\in(0,1)$ (resp. convex primal/concave dual function), DSGDA can find a game-stationary point with an iteration complexity of $\mathcal{O}(\epsilon^{-2\max\{2\theta,1\}})$ (resp. $\mathcal{O}(\epsilon^{-4})$). These match the best results for single-loop algorithms that solve nonconvex-concave or convex-nonconcave minimax problems, or problems satisfying the rather restrictive one-sided Polyak-\L{}ojasiewicz condition. Our work demonstrates, for the first time, the possibility of having a simple and unified single-loop algorithm for solving nonconvex-nonconcave, nonconvex-concave, and convex-nonconcave minimax problems.

Data in Knowledge Graphs often represents part of the current state of the real world. Thus, to stay up-to-date the graph data needs to be updated frequently. To utilize information from Knowledge Graphs, many state-of-the-art machine learning approaches use embedding techniques. These techniques typically compute an embedding, i.e., vector representations of the nodes as input for the main machine learning algorithm. If a graph update occurs later on -- specifically when nodes are added or removed -- the training has to be done all over again. This is undesirable, because of the time it takes and also because downstream models which were trained with these embeddings have to be retrained if they change significantly. In this paper, we investigate embedding updates that do not require full retraining and evaluate them in combination with various embedding models on real dynamic Knowledge Graphs covering multiple use cases. We study approaches that place newly appearing nodes optimally according to local information, but notice that this does not work well. However, we find that if we continue the training of the old embedding, interleaved with epochs during which we only optimize for the added and removed parts, we obtain good results in terms of typical metrics used in link prediction. This performance is obtained much faster than with a complete retraining and hence makes it possible to maintain embeddings for dynamic Knowledge Graphs.

Modern neural network training relies heavily on data augmentation for improved generalization. After the initial success of label-preserving augmentations, there has been a recent surge of interest in label-perturbing approaches, which combine features and labels across training samples to smooth the learned decision surface. In this paper, we propose a new augmentation method that leverages the first and second moments extracted and re-injected by feature normalization. We replace the moments of the learned features of one training image by those of another, and also interpolate the target labels. As our approach is fast, operates entirely in feature space, and mixes different signals than prior methods, one can effectively combine it with existing augmentation methods. We demonstrate its efficacy across benchmark data sets in computer vision, speech, and natural language processing, where it consistently improves the generalization performance of highly competitive baseline networks.

Clustering is one of the most fundamental and wide-spread techniques in exploratory data analysis. Yet, the basic approach to clustering has not really changed: a practitioner hand-picks a task-specific clustering loss to optimize and fit the given data to reveal the underlying cluster structure. Some types of losses---such as k-means, or its non-linear version: kernelized k-means (centroid based), and DBSCAN (density based)---are popular choices due to their good empirical performance on a range of applications. Although every so often the clustering output using these standard losses fails to reveal the underlying structure, and the practitioner has to custom-design their own variation. In this work we take an intrinsically different approach to clustering: rather than fitting a dataset to a specific clustering loss, we train a recurrent model that learns how to cluster. The model uses as training pairs examples of datasets (as input) and its corresponding cluster identities (as output). By providing multiple types of training datasets as inputs, our model has the ability to generalize well on unseen datasets (new clustering tasks). Our experiments reveal that by training on simple synthetically generated datasets or on existing real datasets, we can achieve better clustering performance on unseen real-world datasets when compared with standard benchmark clustering techniques. Our meta clustering model works well even for small datasets where the usual deep learning models tend to perform worse.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.