In this paper, we propose a direct parallel-in-time (PinT) algorithm for time-dependent problems with first- or second-order derivative. We use a second-order boundary value method as the time integrator that leads to a tridiagonal time discretization matrix. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix, which yields a direct parallel implementation across all time levels. A crucial issue on this methodology is how the condition number of the eigenvector matrix $V$ grows as $n$ is increased, where $n$ is the number of time levels. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for computing $V$ and $V^{-1}$, by which we prove that $\mathrm{Cond}_2(V)=\mathcal{O}(n^{2})$. This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as $n$ grows and thus, compared to other direct PinT algorithms, a much larger $n$ can be used to yield satisfactory parallelism. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores.
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High-rate product codes (PCs) and staircase codes (SCs) are ubiquitous codes in high-speed optical communication achieving near-capacity performance on the binary symmetric channel. Their success is mostly due to very efficient iterative decoding algorithms that require very little complexity. In this paper, we extend the density evolution (DE) analysis for PCs and SCs to a channel with ternary output and ternary message passing, where the third symbol marks an erasure. We investigate the performance of a standard error-and-erasure decoder and of its simplification using DE. The proposed analysis can be used to find component code configurations and quantizer levels for the channel output. We also show how the use of even-weight BCH subcodes as component codes can improve the decoding performance at high rates. The DE results are verified by Monte-Carlo simulations, which show that additional coding gains of up to 0.6 dB are possible by ternary decoding, at only a small additional increase in complexity compared to traditional binary message passing.
Chernoff bounds are a powerful application of the Markov inequality to produce strong bounds on the tails of probability distributions. They are often used to bound the tail probabilities of sums of Poisson trials, or in regression to produce conservative confidence intervals for the parameters of such trials. The bounds provide expressions for the tail probabilities that can be inverted for a given probability/confidence to provide tail intervals. The inversions involve the solution of transcendental equations and it is often convenient to substitute approximations that can be exactly solved e.g. by the quadratic equation. In this paper we introduce approximations for the Chernoff bounds whose inversion can be exactly solved with a quadratic equation, but which are closer approximations than those adopted previously.
We consider the problem of quantizing samples of finite-rate-of-innovation (FRI) and bandlimited (BL) signals by using an integrate-and-fire time encoding machine (IF-TEM). We propose a uniform design of the quantization levels and show by numerical simulations that quantization using IF-TEM improves the recovery of FRI and BL signals in comparison with classical uniform sampling in the Fourier-domain and Nyquist methods, respectively. In terms of mean square error (MSE), the reduction reaches at least 5 dB for both classes of signals. Our numerical evaluations also demonstrate that the MSE further decreases when the number of pulses comprising the FRI signal increases. A similar observation is demonstrated for BL signals. In particular, we show that, in contrast to the classical method, increasing the frequency of the IF-TEM input decreases the quantization step size, which can reduce the MSE.
We investigate a clustering problem with data from a mixture of Gaussians that share a common but unknown, and potentially ill-conditioned, covariance matrix. We start by considering Gaussian mixtures with two equally-sized components and derive a Max-Cut integer program based on maximum likelihood estimation. We prove its solutions achieve the optimal misclassification rate when the number of samples grows linearly in the dimension, up to a logarithmic factor. However, solving the Max-cut problem appears to be computationally intractable. To overcome this, we develop an efficient spectral algorithm that attains the optimal rate but requires a quadratic sample size. Although this sample complexity is worse than that of the Max-cut problem, we conjecture that no polynomial-time method can perform better. Furthermore, we gather numerical and theoretical evidence that supports the existence of a statistical-computational gap. Finally, we generalize the Max-Cut program to a $k$-means program that handles multi-component mixtures with possibly unequal weights. It enjoys similar optimality guarantees for mixtures of distributions that satisfy a transportation-cost inequality, encompassing Gaussian and strongly log-concave distributions.
We consider a generic and explicit tamed Euler--Maruyama scheme for multidimensional time-inhomogeneous stochastic differential equations with multiplicative Brownian noise. The diffusion coefficient is uniformly elliptic, H\"older continuous and weakly differentiable in the spatial variables while the drift satisfies the Ladyzhenskaya--Prodi--Serrin condition, as considered by Krylov and R\"ockner (2005). In the discrete scheme, the drift is tamed by replacing it by an approximation. A strong rate of convergence of the scheme is provided in terms of the approximation error of the drift in a suitable and possibly very weak topology. A few examples of approximating drifts are discussed in detail. The parameters of the approximating drifts can vary and be fine-tuned to achieve the standard $1/2$-strong convergence rate with a logarithmic factor.
The Poisson equation is critical to get a self-consistent solution in plasma fluid simulations used for Hall effect thrusters and streamers discharges. Solving the 2D Poisson equation with zero Dirichlet boundary conditions using a deep neural network is investigated using multiple-scale architectures, defined in terms of number of branches, depth and receptive field. The latter is found critical to correctly capture large topological structures of the field. The investigation of multiple architectures, losses, and hyperparameters provides an optimum network to solve accurately the steady Poisson problem. Generalization to new resolutions and domain sizes is then proposed using a proper scaling of the network. Finally, found neural network solver, called PlasmaNet, is coupled with an unsteady Euler plasma fluid equations solver. The test case corresponds to electron plasma oscillations which is used to assess the accuracy of the neural network solution in a time-dependent simulation. In this time-evolving problem, a physical loss is necessary to produce a stable simulation. PlasmaNet is then benchmarked on meshes with increasing number of nodes, and compared with an existing solver based on a standard linear system algorithm for the Poisson equation. It outperforms the classical plasma solver, up to speedups 700 times faster on large meshes. PlasmaNet is finally tested on a more complex case of discharge propagation involving chemistry and advection. The guidelines established in previous sections are applied to build the CNN to solve the same Poisson equation but in cylindrical coordinates. Results reveal good CNN predictions with significant speedup. These results pave the way to new computational strategies to predict unsteady problems involving a Poisson equation, including configurations with coupled multiphysics interactions such as in plasma flows.
In this paper, we present a polynomial-time algorithm for the maximum clique problem, which implies P = NP. Our algorithm is based on a continuous game-theoretic representation of this problem and at its heart lies a discrete-time dynamical system. The rule of our dynamical system depends on a parameter such that if this parameter is equal to the maximum-clique size, the iterates of our dynamical system are guaranteed to converge to a maximum clique.
This study investigates the fundamental limits of variable-length compression in which prefix-free constraints are not imposed (i.e., one-to-one codes are studied) and non-vanishing error probabilities are permitted. Due in part to a crucial relation between the variable-length and fixed-length compression problems, our analysis requires a careful and refined analysis of the fundamental limits of fixed-length compression in the setting where the error probabilities are allowed to approach either zero or one polynomially in the blocklength. To obtain the refinements, we employ tools from moderate deviations and strong large deviations. Finally, we provide the third-order asymptotics for the problem of variable-length compression with non-vanishing error probabilities. We show that unlike several other information-theoretic problems in which the third-order asymptotics are known, for the problem of interest here, the third-order term depends on the permissible error probability.
Multi-marginal optimal transport (MOT) is a generalization of optimal transport to multiple marginals. Optimal transport has evolved into an important tool in many machine learning applications, and its multi-marginal extension opens up for addressing new challenges in the field of machine learning. However, the usage of MOT has been largely impeded by its computational complexity which scales exponentially in the number of marginals. Fortunately, in many applications, such as barycenter or interpolation problems, the cost function adheres to structures, which has recently been exploited for developing efficient computational methods. In this work we derive computational bounds for these methods. With $m$ marginal distributions supported on $n$ points, we provide a $ \mathcal{\tilde O}(d(G)m n^2\epsilon^{-2})$ bound for a $\epsilon$-accuracy when the problem is associated with a tree with diameter $d(G)$. For the special case of the Wasserstein barycenter problem, which corresponds to a star-shaped tree, our bound is in alignment with the existing complexity bound for it.
We consider expected risk minimization when the range of the estimator is required to be nonnegative, motivated by the settings of maximum likelihood estimation (MLE) and trajectory optimization. To facilitate nonlinear interpolation, we hypothesize that search is conducted over a Reproducing Kernel Hilbert Space (RKHS). To solve it, we develop first and second-order variants of stochastic mirror descent employing (i) pseudo-gradients and (ii) complexity-reducing projections. Compressive projection in first-order scheme is executed via kernel orthogonal matching pursuit (KOMP), and overcome the fact that the vanilla RKHS parameterization grows unbounded with time. Moreover, pseudo-gradients are needed when stochastic estimates of the gradient of the expected cost are only computable up to some numerical errors, which arise in, e.g., integral approximations. The second-order scheme develops a Hessian inverse approximation via recursively averaged pseudo-gradient outer products. For the first-order scheme, we establish tradeoffs between accuracy of convergence in mean and the projection budget parameter under constant step-size and compression budget are established, as well as non-asymptotic bounds on the model complexity. Analogous convergence results are established for the second-order scheme under an additional eigenvalue decay condition on the Hessian of the optimal RKHS element. Experiments demonstrate favorable performance on inhomogeneous Poisson Process intensity estimation in practice.
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