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We construct several classes of neural networks with ReLU and BiSU (Binary Step Unit) activations, which exactly emulate the lowest order Finite Element (FE) spaces on regular, simplicial partitions of polygonal and polyhedral domains $\Omega \subset \mathbb{R}^d$, $d=2,3$. For continuous, piecewise linear (CPwL) functions, our constructions generalize previous results in that arbitrary, regular simplicial partitions of $\Omega$ are admitted, also in arbitrary dimension $d\geq 2$. Vector-valued elements emulated include the classical Raviart-Thomas and the first family of N\'{e}d\'{e}lec edge elements on triangles and tetrahedra. Neural Networks emulating these FE spaces are required in the correct approximation of boundary value problems of electromagnetism in nonconvex polyhedra $\Omega \subset \mathbb{R}^3$, thereby constituting an essential ingredient in the application of e.g. the methodology of ``physics-informed NNs'' or ``deep Ritz methods'' to electromagnetic field simulation via deep learning techniques. They satisfy exact (De Rham) sequence properties, and also spawn discrete boundary complexes on $\partial\Omega$ which satisfy exact sequence properties for the surface divergence and curl operators $\mathrm{div}_\Gamma$ and $\mathrm{curl}_\Gamma$, respectively, thereby enabling ``neural boundary elements'' for computational electromagnetism. We indicate generalizations of our constructions to higher-order compatible spaces and other, non-compatible classes of discretizations in particular the Crouzeix-Raviart elements and Hybridized, Higher Order (HHO) methods.

Multi-block CCA constructs linear relationships explaining coherent variations across multiple blocks of data. We view the multi-block CCA problem as finding leading generalized eigenvectors and propose to solve it via a proximal gradient descent algorithm with $\ell_1$ constraint for high dimensional data. In particular, we use a decaying sequence of constraints over proximal iterations, and show that the resulting estimate is rate-optimal under suitable assumptions. Although several previous works have demonstrated such optimality for the $\ell_0$ constrained problem using iterative approaches, the same level of theoretical understanding for the $\ell_1$ constrained formulation is still lacking. We also describe an easy-to-implement deflation procedure to estimate multiple eigenvectors sequentially. We compare our proposals to several existing methods whose implementations are available on R CRAN, and the proposed methods show competitive performances in both simulations and a real data example.

The Gromov-Hausdorff distance $(d_{GH})$ proves to be a useful distance measure between shapes. In order to approximate $d_{GH}$ for compact subsets $X,Y\subset\mathbb{R}^d$, we look into its relationship with $d_{H,iso}$, the infimum Hausdorff distance under Euclidean isometries. As already known for dimension $d\geq 2$, the $d_{H,iso}$ cannot be bounded above by a constant factor times $d_{GH}$. For $d=1$, however, we prove that $d_{H,iso}\leq\frac{5}{4}d_{GH}$. We also show that the bound is tight. In effect, this gives rise to an $O(n\log{n})$-time algorithm to approximate $d_{GH}$ with an approximation factor of $\left(1+\frac{1}{4}\right)$.

The popular LSPE($\lambda$) algorithm for policy evaluation is revisited to derive a concentration bound that gives high probability performance guarantees from some time on.

We lay the foundations of a new theory for algorithms and computational complexity by parameterizing the instances of a computational problem as a moduli scheme. Considering the geometry of the scheme associated to 3-SAT, we separate P and NP.

$k$-means clustering is a fundamental problem in various disciplines. This problem is nonconvex, and standard algorithms are only guaranteed to find a local optimum. Leveraging the structure of local solutions characterized in [1], we propose a general algorithmic framework for escaping undesirable local solutions and recovering the global solution (or the ground truth). This framework consists of alternating between the following two steps iteratively: (i) detect mis-specified clusters in a local solution and (ii) improve the current local solution by non-local operations. We discuss implementation of these steps, and elucidate how the proposed framework unifies variants of $k$-means algorithm in literature from a geometric perspective. In addition, we introduce two natural extensions of the proposed framework, where the initial number of clusters is misspecified. We provide theoretical justification for our approach, which is corroborated with extensive experiments.

There is growing interest in object detection in advanced driver assistance systems and autonomous robots and vehicles. To enable such innovative systems, we need faster object detection. In this work, we investigate the trade-off between accuracy and speed with domain-specific approximations, i.e. category-aware image size scaling and proposals scaling, for two state-of-the-art deep learning-based object detection meta-architectures. We study the effectiveness of applying approximation both statically and dynamically to understand the potential and the applicability of them. By conducting experiments on the ImageNet VID dataset, we show that domain-specific approximation has great potential to improve the speed of the system without deteriorating the accuracy of object detectors, i.e. up to 7.5x speedup for dynamic domain-specific approximation. To this end, we present our insights toward harvesting domain-specific approximation as well as devise a proof-of-concept runtime, AutoFocus, that exploits dynamic domain-specific approximation.

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).

This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.

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.