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Many approaches have been proposed for early classification of time series in light of itssignificance in a wide range of applications including healthcare, transportation and fi-nance. Until now, the early classification problem has been dealt with by considering onlyirrevocable decisions. This paper introduces a new problem calledearly and revocabletimeseries classification, where the decision maker can revoke its earlier decisions based on thenew available measurements. In order to formalize and tackle this problem, we propose anew cost-based framework and derive two new approaches from it. The first approach doesnot consider explicitly the cost of changing decision, while the second one does. Exten-sive experiments are conducted to evaluate these approaches on a large benchmark of realdatasets. The empirical results obtained convincingly show (i) that the ability of revok-ing decisions significantly improves performance over the irrevocable regime, and (ii) thattaking into account the cost of changing decision brings even better results in general.Keywords:revocable decisions, cost estimation, online decision making

In this article, we study approximation properties of the variation spaces corresponding to shallow neural networks with a variety of activation functions. We introduce two main tools for estimating the metric entropy, approximation rates, and $n$-widths of these spaces. First, we introduce the notion of a smoothly parameterized dictionary and give upper bounds on the non-linear approximation rates, metric entropy and $n$-widths of their absolute convex hull. The upper bounds depend upon the order of smoothness of the parameterization. This result is applied to dictionaries of ridge functions corresponding to shallow neural networks, and they improve upon existing results in many cases. Next, we provide a method for lower bounding the metric entropy and $n$-widths of variation spaces which contain certain classes of ridge functions. This result gives sharp lower bounds on the $L^2$-approximation rates, metric entropy, and $n$-widths for variation spaces corresponding to neural networks with a range of important activation functions, including ReLU$^k$, sigmoidal activation functions with bounded variation, and the B-spline activation functions.

Detecting the reflection symmetry plane of an object represented by a 3D point cloud is a fundamental problem in 3D computer vision and geometry processing due to its various applications such as compression, object detection, robotic grasping, 3D surface reconstruction, etc. There exist several efficient approaches for solving this problem for clean 3D point clouds. However, this problem becomes difficult to solve in the presence of outliers and missing parts due to occlusions while scanning the objects through 3D scanners. The existing methods try to overcome these challenges mostly by voting-based techniques but fail in challenging settings. In this work, we propose a statistical estimator for the plane of reflection symmetry that is robust to outliers and missing parts. We pose the problem of finding the optimal estimator as an optimization problem on a 2-sphere that quickly converges to the global solution. We further propose a 3D point descriptor that is invariant to 3D reflection symmetry using the spectral properties of the geodesic distance matrix constructed from the neighbors of a point. This helps us in decoupling the chicken-and-egg problem of finding optimal symmetry plane and correspondences between the reflective symmetric points. We show that the proposed approach achieves the state-of-the-art performance on the benchmarks dataset.

Splitting-based time integration approaches such as fractional steps, alternating direction implicit, operator splitting, and locally one-dimensional methods partition the system of interest into components and solve individual components implicitly in a cost-effective way. This work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge-Kutta (GARK) methods. Specifically, we develop implicit-implicit (IMIM) GARK schemes, provide the order conditions and stability analysis for this class, and explain their application to partitioned systems of ordinary differential equations. We show that classical splitting methods belong to the IMIM GARK family, and therefore can be studied in this unified framework. New IMIM-GARK splitting methods are developed and tested using parabolic systems.

We establish improved uniform error bounds for the time-splitting methods for the long-time dynamics of the Schr\"odinger equation with small potential and the nonlinear Schr\"odinger equation (NLSE) with weak nonlinearity. For the Schr\"odinger equation with small potential characterized by a dimensionless parameter $\varepsilon \in (0, 1]$ representing the amplitude of the potential, we employ the unitary flow property of the (second-order) time-splitting Fourier pseudospectral (TSFP) method in $L^2$-norm to prove a uniform error bound at $C(T)(h^m +\tau^2)$ up to the long time $T_\varepsilon= T/\varepsilon$ for any $T>0$ and uniformly for $0<\varepsilon\le1$, while $h$ is the mesh size, $\tau$ is the time step, $m \ge 2$ depends on the regularity of the exact solution, and $C(T) =C_0+C_1T$ grows at most linearly with respect to $T$ with $C_0$ and $C_1$ two positive constants independent of $T$, $\varepsilon$, $h$ and $\tau$. Then by introducing a new technique of {\sl regularity compensation oscillation} (RCO) in which the high frequency modes are controlled by regularity and the low frequency modes are analyzed by phase cancellation and energy method, an improved uniform error bound at $O(h^{m-1} + \varepsilon \tau^2)$ is established in $H^1$-norm for the long-time dynamics up to the time at $O(1/\varepsilon)$ of the Schr\"odinger equation with $O(\varepsilon)$-potential with $m \geq 3$, which is uniformly for $\varepsilon\in(0,1]$. Moreover, the RCO technique is extended to prove an improved uniform error bound at $O(h^{m-1} + \varepsilon^2\tau^2)$ in $H^1$-norm for the long-time dynamics up to the time at $O(1/\varepsilon^2)$ of the cubic NLSE with $O(\varepsilon^2)$-nonlinearity strength, uniformly for $\varepsilon \in (0, 1]$. Extensions to the first-order and fourth-order time-splitting methods are discussed.

Capturing solution near the singular point of any nonlinear SBVPs is challenging because coefficients involved in the differential equation blow up near singularities. In this article, we aim to construct a general method based on orthogonal polynomials as wavelets. We discuss multiresolution analysis for wavelets generated by orthogonal polynomials, e.g., Hermite, Legendre, Chebyshev, Laguerre, and Gegenbauer. Then we use these wavelets for solving nonlinear SBVPs. These wavelets can deal with singularities easily and efficiently. To deal with the nonlinearity, we use both Newton's quasilinearization and the Newton-Raphson method. To show the importance and accuracy of the proposed methods, we solve the Lane-Emden type of problems and compare the computed solutions with the known solutions. As the resolution is increased the computed solutions converge to exact solutions or known solutions. We observe that the proposed technique performs well on a class of Lane-Emden type BVPs. As the paper deals with singularity, non-linearity significantly and different wavelets are used to compare the results.

When solving the American options with or without dividends, numerical methods often obtain lower convergence rates if further treatment is not implemented even using high-order schemes. In this article, we present a fast and explicit fourth-order compact scheme for solving the free boundary options. In particular, the early exercise features with the asset option and option sensitivity are computed based on a coupled of nonlinear PDEs with fixed boundaries for which a high order analytical approximation is obtained. Furthermore, we implement a new treatment at the left boundary by introducing a third-order Robin boundary condition. Rather than computing the optimal exercise boundary from the analytical approximation, we simply obtain it from the asset option based on the linear relationship at the left boundary. As such, a high order convergence rate can be achieved. We validate by examples that the improvement at the left boundary yields a fourth-order convergence rate without further implementation of mesh refinement, Rannacher time-stepping, and/or smoothing of the initial condition. Furthermore, we extensively compare, the performance of our present method with several 5(4) Runge-Kutta pairs and observe that Dormand and Prince and Bogacki and Shampine 5(4) pairs are faster and provide more accurate numerical solutions. Based on numerical results and comparison with other existing methods, we can validate that the present method is very fast and provides more accurate solutions with very coarse grids.

Throughput is a main performance objective in communication networks. This paper considers a fundamental maximum throughput routing problem -- the all-or-nothing multicommodity flow (ANF) problem -- in arbitrary directed graphs and in the practically relevant but challenging setting where demands can be (much) larger than the edge capacities. Hence, in addition to assigning requests to valid flows for each routed commodity, an admission control mechanism is required which prevents overloading the network when routing commodities. We make several contributions. On the theoretical side we obtain substantially improved bi-criteria approximation algorithms for this NP-hard problem. We present two non-trivial linear programming relaxations and show how to convert their fractional solutions into integer solutions via randomized rounding. One is an exponential-size formulation (solvable in polynomial time using a separation oracle) that considers a ``packing'' view and allows a more flexible approach, while the other is a generalization of the compact LP formulation of Liu et al. (INFOCOM'19) that allows for easy solving via standard LP solvers. We obtain a polynomial-time randomized algorithm that yields an arbitrarily good approximation on the weighted throughput while violating the edge capacity constraints by only a small multiplicative factor. We also describe a deterministic rounding algorithm by derandomization, using the method of pessimistic estimators. We complement our theoretical results with a proof of concept empirical evaluation.

We consider unconstrained optimization problems with nonsmooth and convex objective function in the form of mathematical expectation. The proposed method approximates the objective function with a sample average function by using different sample size in each iteration. The sample size is chosen in an adaptive manner based on the Inexact Restoration. The method uses line search and assumes descent directions with respect to the current approximate function. We prove the almost sure convergence under the standard assumptions. The convergence rate is also considered and the worst-case complexity of $\mathcal{O} (\varepsilon^{-2})$ is proved. Numerical results for two types of problems, machine learning hinge loss and stochastic linear complementarity problems, show the efficiency of the proposed scheme.

The piecewise constant Mumford-Shah (PCMS) model and the Rudin-Osher-Fatemi (ROF) model are two of the most famous variational models in image segmentation and image restoration, respectively. They have ubiquitous applications in image processing. In this paper, we explore the linkage between these two important models. We prove that for two-phase segmentation problem the optimal solution of the PCMS model can be obtained by thresholding the minimizer of the ROF model. This linkage is still valid for multiphase segmentation under mild assumptions. Thus it opens a new segmentation paradigm: image segmentation can be done via image restoration plus thresholding. This new paradigm, which circumvents the innate non-convex property of the PCMS model, therefore improves the segmentation performance in both efficiency (much faster than state-of-the-art methods based on PCMS model, particularly when the phase number is high) and effectiveness (producing segmentation results with better quality) due to the flexibility of the ROF model in tackling degraded images, such as noisy images, blurry images or images with information loss. As a by-product of the new paradigm, we derive a novel segmentation method, coined thresholded-ROF (T-ROF) method, to illustrate the virtue of manipulating image segmentation through image restoration techniques. The convergence of the T-ROF method under certain conditions is proved, and elaborate experimental results and comparisons are presented.