Quantum many-body systems involving bosonic modes or gauge fields have infinite-dimensional local Hilbert spaces which must be truncated to perform simulations of real-time dynamics on classical or quantum computers. To analyze the truncation error, we develop methods for bounding the rate of growth of local quantum numbers such as the occupation number of a mode at a lattice site, or the electric field at a lattice link. Our approach applies to various models of bosons interacting with spins or fermions, and also to both abelian and non-abelian gauge theories. We show that if states in these models are truncated by imposing an upper limit $\Lambda$ on each local quantum number, and if the initial state has low local quantum numbers, then an error at most $\epsilon$ can be achieved by choosing $\Lambda$ to scale polylogarithmically with $\epsilon^{-1}$, an exponential improvement over previous bounds based on energy conservation. For the Hubbard-Holstein model, we numerically compute a bound on $\Lambda$ that achieves accuracy $\epsilon$, obtaining significantly improved estimates in various parameter regimes. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution. Building on that result, we formulate quantum algorithms for dynamical simulation of lattice gauge theories and of models with bosonic modes; the gate complexity depends almost linearly on spacetime volume in the former case, and almost quadratically on time in the latter case. We establish a lower bound showing that there are systems involving bosons for which this quadratic scaling with time cannot be improved. By applying our result on the truncation error in time evolution, we also prove that spectrally isolated energy eigenstates can be approximated with accuracy $\epsilon$ by truncating local quantum numbers at $\Lambda=\textrm{polylog}(\epsilon^{-1})$.
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Multiple antenna arrays play a key role in wireless networks for communications but also localization and sensing. The use of large antenna arrays pushes towards a propagation regime in which the wavefront is no longer plane but spherical. This allows to infer the position and orientation of an arbitrary source from the received signal without the need of using multiple anchor nodes. To understand the fundamental limits of large antenna arrays for localization, this paper fusions wave propagation theory with estimation theory, and computes the Cram{\'e}r-Rao Bound (CRB) for the estimation of the three Cartesian coordinates of the source on the basis of the electromagnetic vector field, observed over a rectangular surface area. To simplify the analysis, we assume that the source is a dipole, whose center is located on the line perpendicular to the surface center, with an orientation a priori known. Numerical and asymptotic results are given to quantify the CRBs, and to gain insights into the effect of various system parameters on the ultimate estimation accuracy. It turns out that surfaces of practical size may guarantee a centimeter-level accuracy in the mmWave bands.
In this work, we present a new high order Discontinuous Galerkin time integration scheme for second-order (in time) differential systems that typically arise from the space discretization of the elastodynamics equation. By rewriting the original equation as a system of first order differential equations we introduce the method and show that the resulting discrete formulation is well-posed, stable and retains super-optimal rate of convergence with respect to the discretization parameters, namely the time step and the polynomial approximation degree. A set of two- and three-dimensional numerical experiments confirm the theoretical bounds. Finally, the method is applied to real geophysical applications.
A mega-constellation of low-altitude earth orbit (LEO) satellites (SATs) are envisaged to provide a global coverage SAT network in beyond fifth-generation (5G) cellular systems. LEO SAT networks exhibit extremely long link distances of many users under time-varying SAT network topology. This makes existing multiple access protocols, such as random access channel (RACH) based cellular protocol designed for fixed terrestrial network topology, ill-suited. To overcome this issue, in this paper, we propose a novel grant-free random access solution for LEO SAT networks, dubbed emergent random access channel protocol (eRACH). In stark contrast to existing model-based and standardized protocols, eRACH is a model-free approach that emerges through interaction with the non-stationary network environment, using multi-agent deep reinforcement learning (MADRL). Furthermore, by exploiting known SAT orbiting patterns, eRACH does not require central coordination or additional communication across users, while training convergence is stabilized through the regular orbiting patterns. Compared to RACH, we show from various simulations that our proposed eRACH yields 54.6% higher average network throughput with around two times lower average access delay while achieving 0.989 Jain's fairness index.
We analyze the orthogonal greedy algorithm when applied to dictionaries $\mathbb{D}$ whose convex hull has small entropy. We show that if the metric entropy of the convex hull of $\mathbb{D}$ decays at a rate of $O(n^{-\frac{1}{2}-\alpha})$ for $\alpha > 0$, then the orthogonal greedy algorithm converges at the same rate on the variation space of $\mathbb{D}$. This improves upon the well-known $O(n^{-\frac{1}{2}})$ convergence rate of the orthogonal greedy algorithm in many cases, most notably for dictionaries corresponding to shallow neural networks. These results hold under no additional assumptions on the dictionary beyond the decay rate of the entropy of its convex hull. In addition, they are robust to noise in the target function and can be extended to convergence rates on the interpolation spaces of the variation norm. Finally, we show that these improved rates are sharp and prove a negative result showing that the iterates generated by the orthogonal greedy algorithm cannot in general be bounded in the variation norm of $\mathbb{D}$.
This work presents Reliable-NIDS (R-NIDS), a novel methodology for Machine Learning (ML) based Network Intrusion Detection Systems (NIDSs) that allows ML models to work on integrated datasets, empowering the learning process with diverse information from different datasets. Therefore, R-NIDS targets the design of more robust models, that generalize better than traditional approaches. We also propose a new dataset, called UNK21. It is built from three of the most well-known network datasets (UGR'16, USNW-NB15 and NLS-KDD), each one gathered from its own network environment, with different features and classes, by using a data aggregation approach present in R-NIDS. Following R-NIDS, in this work we propose to build two well-known ML models (a linear and a non-linear one) based on the information of three of the most common datasets in the literature for NIDS evaluation, those integrated in UNK21. The results that the proposed methodology offers show how these two ML models trained as a NIDS solution could benefit from this approach, being able to generalize better when training on the newly proposed UNK21 dataset. Furthermore, these results are carefully analyzed with statistical tools that provide high confidence on our conclusions.
Despite their overwhelming capacity to overfit, deep neural networks trained by specific optimization algorithms tend to generalize well to unseen data. Recently, researchers explained it by investigating the implicit regularization effect of optimization algorithms. A remarkable progress is the work (Lyu&Li, 2019), which proves gradient descent (GD) maximizes the margin of homogeneous deep neural networks. Except GD, adaptive algorithms such as AdaGrad, RMSProp and Adam are popular owing to their rapid training process. However, theoretical guarantee for the generalization of adaptive optimization algorithms is still lacking. In this paper, we study the implicit regularization of adaptive optimization algorithms when they are optimizing the logistic loss on homogeneous deep neural networks. We prove that adaptive algorithms that adopt exponential moving average strategy in conditioner (such as Adam and RMSProp) can maximize the margin of the neural network, while AdaGrad that directly sums historical squared gradients in conditioner can not. It indicates superiority on generalization of exponential moving average strategy in the design of the conditioner. Technically, we provide a unified framework to analyze convergent direction of adaptive optimization algorithms by constructing novel adaptive gradient flow and surrogate margin. Our experiments can well support the theoretical findings on convergent direction of adaptive optimization algorithms.
Tumor detection in biomedical imaging is a time-consuming process for medical professionals and is not without errors. Thus in recent decades, researchers have developed algorithmic techniques for image processing using a wide variety of mathematical methods, such as statistical modeling, variational techniques, and machine learning. In this paper, we propose a semi-automatic method for liver segmentation of 2D CT scans into three labels denoting healthy, vessel, or tumor tissue based on graph cuts. First, we create a feature vector for each pixel in a novel way that consists of the 59 intensity values in the time series data and propose a simplified perimeter cost term in the energy functional. We normalize the data and perimeter terms in the functional to expedite the graph cut without having to optimize the scaling parameter $\lambda$. In place of a training process, predetermined tissue means are computed based on sample regions identified by expert radiologists. The proposed method also has the advantage of being relatively simple to implement computationally. It was evaluated against the ground truth on a clinical CT dataset of 10 tumors and yielded segmentations with a mean Dice similarity coefficient (DSC) of .77 and mean volume overlap error (VOE) of 36.7%. The average processing time was 1.25 minutes per slice.
Modern CNN-based object detectors rely on bounding box regression and non-maximum suppression to localize objects. While the probabilities for class labels naturally reflect classification confidence, localization confidence is absent. This makes properly localized bounding boxes degenerate during iterative regression or even suppressed during NMS. In the paper we propose IoU-Net learning to predict the IoU between each detected bounding box and the matched ground-truth. The network acquires this confidence of localization, which improves the NMS procedure by preserving accurately localized bounding boxes. Furthermore, an optimization-based bounding box refinement method is proposed, where the predicted IoU is formulated as the objective. Extensive experiments on the MS-COCO dataset show the effectiveness of IoU-Net, as well as its compatibility with and adaptivity to several state-of-the-art object detectors.
Machine Learning is a widely-used method for prediction generation. These predictions are more accurate when the model is trained on a larger dataset. On the other hand, the data is usually divided amongst different entities. For privacy reasons, the training can be done locally and then the model can be safely aggregated amongst the participants. However, if there are only two participants in \textit{Collaborative Learning}, the safe aggregation loses its power since the output of the training already contains much information about the participants. To resolve this issue, they must employ privacy-preserving mechanisms, which inevitably affect the accuracy of the model. In this paper, we model the training process as a two-player game where each player aims to achieve a higher accuracy while preserving its privacy. We introduce the notion of \textit{Price of Privacy}, a novel approach to measure the effect of privacy protection on the accuracy of the model. We develop a theoretical model for different player types, and we either find or prove the existence of a Nash Equilibrium with some assumptions. Moreover, we confirm these assumptions via a Recommendation Systems use case: for a specific learning algorithm, we apply three privacy-preserving mechanisms on two real-world datasets. Finally, as a complementary work for the designed game, we interpolate the relationship between privacy and accuracy for this use case and present three other methods to approximate it in a real-world scenario.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
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