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We investigate both theoretically and numerically the consistency between the nonlinear discretization in full order models (FOMs) and reduced order models (ROMs) for incompressible flows. To this end, we consider two cases: (i) FOM-ROM consistency, i.e., when we use the same nonlinearity discretization in the FOM and ROM; and (ii) FOM-ROM inconsistency, i.e., when we use different nonlinearity discretizations in the FOM and ROM. Analytically, we prove that while the FOM-ROM consistency yields optimal error bounds, FOM-ROM inconsistency yields additional terms dependent on the FOM divergence error, which prevent the ROM from recovering the FOM as the number of modes increases. Computationally, we consider channel flow around a cylinder and Kelvin-Helmholtz instability, and show that FOM-ROM consistency yields significantly more accurate results than the FOM-ROM inconsistency.

Computational fluctuating hydrodynamics aims at understanding the impact of thermal fluctuations on fluid motions at small scales through numerical exploration. These fluctuations are modeled as stochastic flux terms and incorporated into the classical Navier-Stokes equations, which need to be solved numerically. In this paper, we present a novel projection-based method for solving the incompressible fluctuating hydrodynamics equations. By analyzing the equilibrium structure factor spectrum of the velocity field, we investigate how the inherent splitting errors affect the numerical solution of the stochastic partial differential equations in the presence of non-periodic boundary conditions, and how iterative corrections can reduce these errors. Our computational examples demonstrate both the capability of our approach to reproduce correctly stochastic properties of fluids at small scales as well as its potential use in the simulations of multi-physics problems.

In this work, we investigate the asymptotic spectral density of the random feature matrix $M = Y Y^\ast$ with $Y = f(WX)$ generated by a single-hidden-layer neural network, where $W$ and $X$ are random rectangular matrices with i.i.d. centred entries and $f$ is a non-linear smooth function which is applied entry-wise. We prove that the Stieltjes transform of the limiting spectral distribution approximately satisfies a quartic self-consistent equation, which is exactly the equation obtained by [Pennington, Worah] and [Benigni, P\'ech\'e] with the moment method. We extend the previous results to the case of additive bias $Y=f(WX+B)$ with $B$ being an independent rank-one Gaussian random matrix, closer modelling the neural network infrastructures encountered in practice. Our key finding is that in the case of additive bias it is impossible to choose an activation function preserving the layer-to-layer singular value distribution, in sharp contrast to the bias-free case where a simple integral constraint is sufficient to achieve isospectrality. To obtain the asymptotics for the empirical spectral density we follow the resolvent method from random matrix theory via the cumulant expansion. We find that this approach is more robust and less combinatorial than the moment method and expect that it will apply also for models where the combinatorics of the former become intractable. The resolvent method has been widely employed, but compared to previous works, it is applied here to non-linear random matrices.

Machine learning models have achieved human-level performance on various tasks. This success comes at a high cost of computation and storage overhead, which makes machine learning algorithms difficult to deploy on edge devices. Typically, one has to partially sacrifice accuracy in favor of an increased performance quantified in terms of reduced memory usage and energy consumption. Current methods compress the networks by reducing the precision of the parameters or by eliminating redundant ones. In this paper, we propose a new insight into network compression through the Bayesian framework. We show that Bayesian neural networks automatically discover redundancy in model parameters, thus enabling self-compression, which is linked to the propagation of uncertainty through the layers of the network. Our experimental results show that the network architecture can be successfully compressed by deleting parameters identified by the network itself while retaining the same level of accuracy.

Community detection is the problem of identifying community structure in graphs. Often the graph is modeled as a sample from the Stochastic Block Model, in which each vertex belongs to a community. The probability that two vertices are connected by an edge depends on the communities of those vertices. In this paper, we consider a model of {\em censored} community detection with two communities, where most of the data is missing as the status of only a small fraction of the potential edges is revealed. In this model, vertices in the same community are connected with probability $p$ while vertices in opposite communities are connected with probability $q$. The connectivity status of a given pair of vertices $\{u,v\}$ is revealed with probability $\alpha$, independently across all pairs, where $\alpha = \frac{t \log(n)}{n}$. We establish the information-theoretic threshold $t_c(p,q)$, such that no algorithm succeeds in recovering the communities exactly when $t < t_c(p,q)$. We show that when $t > t_c(p,q)$, a simple spectral algorithm based on a weighted, signed adjacency matrix succeeds in recovering the communities exactly. While spectral algorithms are shown to have near-optimal performance in the symmetric case, we show that they may fail in the asymmetric case where the connection probabilities inside the two communities are allowed to be different. In particular, we show the existence of a parameter regime where a simple two-phase algorithm succeeds but any algorithm based on the top two eigenvectors of the weighted, signed adjacency matrix fails.

Series expansions have been a cornerstone of applied mathematics and engineering for centuries. In this paper, we revisit the Taylor series expansion from a modern Machine Learning perspective. Specifically, we introduce the Fast Continuous Convolutional Taylor Transform (FC2T2), a variant of the Fast Multipole Method (FMM), that allows for the efficient approximation of low dimensional convolutional operators in continuous space. We build upon the FMM which is an approximate algorithm that reduces the computational complexity of N-body problems from O(NM) to O(N+M) and finds application in e.g. particle simulations. As an intermediary step, the FMM produces a series expansion for every cell on a grid and we introduce algorithms that act directly upon this representation. These algorithms analytically but approximately compute the quantities required for the forward and backward pass of the backpropagation algorithm and can therefore be employed as (implicit) layers in Neural Networks. Specifically, we introduce a root-implicit layer that outputs surface normals and object distances as well as an integral-implicit layer that outputs a rendering of a radiance field given a 3D pose. In the context of Machine Learning, $N$ and $M$ can be understood as the number of model parameters and model evaluations respectively which entails that, for applications that require repeated function evaluations which are prevalent in Computer Vision and Graphics, unlike regular Neural Networks, the techniques introduce in this paper scale gracefully with parameters. For some applications, this results in a 200x reduction in FLOPs compared to state-of-the-art approaches at a reasonable or non-existent loss in accuracy.

A key advantage of isogeometric discretizations is their accurate and well-behaved eigenfrequencies and eigenmodes. For degree two and higher, however, optical branches of spurious outlier frequencies and modes may appear due to boundaries or reduced continuity at patch interfaces. In this paper, we introduce a variational approach based on perturbed eigenvalue analysis that eliminates outlier frequencies without negatively affecting the accuracy in the remainder of the spectrum and modes. We then propose a pragmatic iterative procedure that estimates the perturbation parameters in such a way that the outlier frequencies are effectively reduced. We demonstrate that our approach allows for a much larger critical time-step size in explicit dynamics calculations. In addition, we show that the critical time-step size obtained with the proposed approach does not depend on the polynomial degree of spline basis functions.

The problem of Approximate Nearest Neighbor (ANN) search is fundamental in computer science and has benefited from significant progress in the past couple of decades. However, most work has been devoted to pointsets whereas complex shapes have not been sufficiently treated. Here, we focus on distance functions between discretized curves in Euclidean space: they appear in a wide range of applications, from road segments to time-series in general dimension. For $\ell_p$-products of Euclidean metrics, for any $p$, we design simple and efficient data structures for ANN, based on randomized projections, which are of independent interest. They serve to solve proximity problems under a notion of distance between discretized curves, which generalizes both discrete Fr\'echet and Dynamic Time Warping distances. These are the most popular and practical approaches to comparing such curves. We offer the first data structures and query algorithms for ANN with arbitrarily good approximation factor, at the expense of increasing space usage and preprocessing time over existing methods. Query time complexity is comparable or significantly improved by our algorithms, our algorithm is especially efficient when the length of the curves is bounded.

The ever-growing interest witnessed in the acquisition and development of unmanned aerial vehicles (UAVs), commonly known as drones in the past few years, has brought generation of a very promising and effective technology. Because of their characteristic of small size and fast deployment, UAVs have shown their effectiveness in collecting data over unreachable areas and restricted coverage zones. Moreover, their flexible-defined capacity enables them to collect information with a very high level of detail, leading to high resolution images. UAVs mainly served in military scenario. However, in the last decade, they have being broadly adopted in civilian applications as well. The task of aerial surveillance and situation awareness is usually completed by integrating intelligence, surveillance, observation, and navigation systems, all interacting in the same operational framework. To build this capability, UAV's are well suited tools that can be equipped with a wide variety of sensors, such as cameras or radars. Deep learning has been widely recognized as a prominent approach in different computer vision applications. Specifically, one-stage object detector and two-stage object detector are regarded as the most important two groups of Convolutional Neural Network based object detection methods. One-stage object detector could usually outperform two-stage object detector in speed; however, it normally trails in detection accuracy, compared with two-stage object detectors. In this study, focal loss based RetinaNet, which works as one-stage object detector, is utilized to be able to well match the speed of regular one-stage detectors and also defeat two-stage detectors in accuracy, for UAV based object detection. State-of-the-art performance result has been showed on the UAV captured image dataset-Stanford Drone Dataset (SDD).

Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones.