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Quantum dynamics can be simulated on a quantum computer by exponentiating elementary terms from the Hamiltonian in a sequential manner. However, such an implementation of Trotter steps has gate complexity depending on the total Hamiltonian term number, comparing unfavorably to algorithms using more advanced techniques. We develop methods to perform faster Trotter steps with complexity sublinear in the number of terms. We achieve this for a class of Hamiltonians whose interaction strength decays with distance according to power law. Our methods include one based on a recursive block encoding and one based on an average-cost simulation, overcoming the normalization-factor barrier of these advanced quantum simulation techniques. We also realize faster Trotter steps when certain blocks of Hamiltonian coefficients have low rank. Combining with a tighter error analysis, we show that it suffices to use $\left(\eta^{1/3}n^{1/3}+\frac{n^{2/3}}{\eta^{2/3}}\right)n^{1+o(1)}$ gates to simulate uniform electron gas with $n$ spin orbitals and $\eta$ electrons in second quantization in real space, asymptotically improving over the best previous work. We obtain an analogous result when the external potential of nuclei is introduced under the Born-Oppenheimer approximation. We prove a circuit lower bound when the Hamiltonian coefficients take a continuum range of values, showing that generic $n$-qubit $2$-local Hamiltonians with commuting terms require at least $\Omega(n^2)$ gates to evolve with accuracy $\epsilon=\Omega(1/poly(n))$ for time $t=\Omega(\epsilon)$. Our proof is based on a gate-efficient reduction from the approximate synthesis of diagonal unitaries within the Hamming weight-$2$ subspace, which may be of independent interest. Our result thus suggests the use of Hamiltonian structural properties as both necessary and sufficient to implement Trotter steps with lower gate complexity.

We consider alternating gradient descent (AGD) with fixed step size $\eta > 0$, applied to the asymmetric matrix factorization objective. We show that, for a rank-$r$ matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$, $T = \left( \left(\frac{\sigma_1(\mathbf{A})}{\sigma_r(\mathbf{A})}\right)^2 \log(1/\epsilon)\right)$ iterations of alternating gradient descent suffice to reach an $\epsilon$-optimal factorization $\| \mathbf{A} - \mathbf{X}_T^{\vphantom{\intercal}} \mathbf{Y}_T^{\intercal} \|_{\rm F}^2 \leq \epsilon \| \mathbf{A} \|_{\rm F}^2$ with high probability starting from an atypical random initialization. The factors have rank $d>r$ so that $\mathbf{X}_T\in\mathbb{R}^{m \times d}$ and $\mathbf{Y}_T \in\mathbb{R}^{n \times d}$. Experiments suggest that our proposed initialization is not merely of theoretical benefit, but rather significantly improves convergence of gradient descent in practice. Our proof is conceptually simple: a uniform PL-inequality and uniform Lipschitz smoothness constant are guaranteed for a sufficient number of iterations, starting from our random initialization. Our proof method should be useful for extending and simplifying convergence analyses for a broader class of nonconvex low-rank factorization problems.

The Laplace-Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle deformation). Using classical potential theory, the Laplace-Beltrami operator can be pre-/post-conditioned with an integral operator whose kernel is translation invariant, resulting in well-conditioned Fredholm integral equations of the second-kind. These equations have the standard~$1/r$ kernel from potential theory, and therefore the equations can be solved rapidly and accurately using a combination of fast multipole methods (FMMs) and high-order quadrature corrections. In this work we detail such a scheme, presenting two alternative integral formulations of the Laplace-Beltrami problem, each of whose solution can be obtained via FMM acceleration. We then present several applications of the solvers, focusing on the computation of what are known as harmonic vector fields, relevant for many applications in electromagnetics. A battery of numerical results are presented for each application, detailing the performance of the solver in various geometries.

Although the complete scope of the sixth generation of mobile technologies (6G) is still unclear, the prominence of the Internet of Things (IoT) and Artificial Intelligence (AI) / Machine Learning (ML) in the networking field is undeniable. In this regard, key technology enablers for the previous generation, 5G, such as software-defined networking and network function virtualization, fall short to accomplish the stringent requirements envisioned for 6G verticals. This PhD thesis goes back to basics, by exploring missing functionality gaps in relation to these technologies, in order to provide the ''glue'' for holistic and fully-fledged networking solutions for 6G, aligned with standards and industry recommendations. Although ambitious, and in a very early stage, this PhD thesis illustrates an initial design for in-band control in Software-Defined Networking (SDN) that could facilitate the interoperability among constrained IoT devices. The current design demonstrates promising results in terms of resource-usage and robustness, which are pivotal features for constrained networks. Next steps include the integration of the approach with a real testbed comprised of constrained IoT devices and the implementation of a federated learning environment at the edge.

The paper describes of the Ryze Tello drone to move autonomously using a basic vision system. The drone's position is determined by identifying AprilTags' position relative to the drone's built-in camera. The accuracy of the drone's position readings and distance calculations was tested under controlled conditions, and errors were analysed. The study showed a decrease in absolute error with decreasing drone distance from the marker, a little change in the relative error for large distances, and a sharp decrease in the relative error for small distances. The method is satisfactory for determining the drone's position relative to a marker.

Graphs are used widely to model complex systems, and detecting anomalies in a graph is an important task in the analysis of complex systems. Graph anomalies are patterns in a graph that do not conform to normal patterns expected of the attributes and/or structures of the graph. In recent years, graph neural networks (GNNs) have been studied extensively and have successfully performed difficult machine learning tasks in node classification, link prediction, and graph classification thanks to the highly expressive capability via message passing in effectively learning graph representations. To solve the graph anomaly detection problem, GNN-based methods leverage information about the graph attributes (or features) and/or structures to learn to score anomalies appropriately. In this survey, we review the recent advances made in detecting graph anomalies using GNN models. Specifically, we summarize GNN-based methods according to the graph type (i.e., static and dynamic), the anomaly type (i.e., node, edge, subgraph, and whole graph), and the network architecture (e.g., graph autoencoder, graph convolutional network). To the best of our knowledge, this survey is the first comprehensive review of graph anomaly detection methods based on GNNs.

Following unprecedented success on the natural language tasks, Transformers have been successfully applied to several computer vision problems, achieving state-of-the-art results and prompting researchers to reconsider the supremacy of convolutional neural networks (CNNs) as {de facto} operators. Capitalizing on these advances in computer vision, the medical imaging field has also witnessed growing interest for Transformers that can capture global context compared to CNNs with local receptive fields. Inspired from this transition, in this survey, we attempt to provide a comprehensive review of the applications of Transformers in medical imaging covering various aspects, ranging from recently proposed architectural designs to unsolved issues. Specifically, we survey the use of Transformers in medical image segmentation, detection, classification, reconstruction, synthesis, registration, clinical report generation, and other tasks. In particular, for each of these applications, we develop taxonomy, identify application-specific challenges as well as provide insights to solve them, and highlight recent trends. Further, we provide a critical discussion of the field's current state as a whole, including the identification of key challenges, open problems, and outlining promising future directions. We hope this survey will ignite further interest in the community and provide researchers with an up-to-date reference regarding applications of Transformer models in medical imaging. Finally, to cope with the rapid development in this field, we intend to regularly update the relevant latest papers and their open-source implementations at \url{//github.com/fahadshamshad/awesome-transformers-in-medical-imaging}.

Training machine learning models in a meaningful order, from the easy samples to the hard ones, using curriculum learning can provide performance improvements over the standard training approach based on random data shuffling, without any additional computational costs. Curriculum learning strategies have been successfully employed in all areas of machine learning, in a wide range of tasks. However, the necessity of finding a way to rank the samples from easy to hard, as well as the right pacing function for introducing more difficult data can limit the usage of the curriculum approaches. In this survey, we show how these limits have been tackled in the literature, and we present different curriculum learning instantiations for various tasks in machine learning. We construct a multi-perspective taxonomy of curriculum learning approaches by hand, considering various classification criteria. We further build a hierarchical tree of curriculum learning methods using an agglomerative clustering algorithm, linking the discovered clusters with our taxonomy. At the end, we provide some interesting directions for future work.

Small data challenges have emerged in many learning problems, since the success of deep neural networks often relies on the availability of a huge amount of labeled data that is expensive to collect. To address it, many efforts have been made on training complex models with small data in an unsupervised and semi-supervised fashion. In this paper, we will review the recent progresses on these two major categories of methods. A wide spectrum of small data models will be categorized in a big picture, where we will show how they interplay with each other to motivate explorations of new ideas. We will review the criteria of learning the transformation equivariant, disentangled, self-supervised and semi-supervised representations, which underpin the foundations of recent developments. Many instantiations of unsupervised and semi-supervised generative models have been developed on the basis of these criteria, greatly expanding the territory of existing autoencoders, generative adversarial nets (GANs) and other deep networks by exploring the distribution of unlabeled data for more powerful representations. While we focus on the unsupervised and semi-supervised methods, we will also provide a broader review of other emerging topics, from unsupervised and semi-supervised domain adaptation to the fundamental roles of transformation equivariance and invariance in training a wide spectrum of deep networks. It is impossible for us to write an exclusive encyclopedia to include all related works. Instead, we aim at exploring the main ideas, principles and methods in this area to reveal where we are heading on the journey towards addressing the small data challenges in this big data era.