The efficiency of Markov Chain Monte Carlo (MCMC) depends on how the underlying geometry of the problem is taken into account. For distributions with strongly varying curvature, Riemannian metrics help in efficient exploration of the target distribution. Unfortunately, they have significant computational overhead due to e.g. repeated inversion of the metric tensor, and current geometric MCMC methods using the Fisher information matrix to induce the manifold are in practice slow. We propose a new alternative Riemannian metric for MCMC, by embedding the target distribution into a higher-dimensional Euclidean space as a Monge patch and using the induced metric determined by direct geometric reasoning. Our metric only requires first-order gradient information and has fast inverse and determinants, and allows reducing the computational complexity of individual iterations from cubic to quadratic in the problem dimensionality. We demonstrate how Lagrangian Monte Carlo in this metric efficiently explores the target distributions.
相關內容
The perfectly matched layer (PML) formulation is a prominent way of handling radiation problems in unbounded domain and has gained interest due to its simple implementation in finite element codes. However, its simplicity can be advanced further using the isogeometric framework. This work presents a spline based PML formulation which avoids additional coordinate transformation as the formulation is based on the same space in which the numerical solution is sought. The procedure can be automated for any convex artificial boundary. This removes restrictions on the domain construction using PML and can therefore reduce computational cost and improve mesh quality. The usage of spline basis functions with higher continuity also improves the accuracy of the numerical solution.
We consider minimizing a smooth and strongly convex objective function using a stochastic Newton method. At each iteration, the algorithm is given an oracle access to a stochastic estimate of the Hessian matrix. The oracle model includes popular algorithms such as the Subsampled Newton and Newton Sketch, which can efficiently construct stochastic Hessian estimates for many tasks. Despite using second-order information, these existing methods do not exhibit superlinear convergence, unless the stochastic noise is gradually reduced to zero during the iteration, which would lead to a computational blow-up in the per-iteration cost. We address this limitation with Hessian averaging: instead of using the most recent Hessian estimate, our algorithm maintains an average of all past estimates. This reduces the stochastic noise while avoiding the computational blow-up. We show that this scheme enjoys local $Q$-superlinear convergence with a non-asymptotic rate of $(\Upsilon\sqrt{\log (t)/t}\,)^{t}$, where $\Upsilon$ is proportional to the level of stochastic noise in the Hessian oracle. A potential drawback of this (uniform averaging) approach is that the averaged estimates contain Hessian information from the global phase of the iteration, i.e., before the iterates converge to a local neighborhood. This leads to a distortion that may substantially delay the superlinear convergence until long after the local neighborhood is reached. To address this drawback, we study a number of weighted averaging schemes that assign larger weights to recent Hessians, so that the superlinear convergence arises sooner, albeit with a slightly slower rate. Remarkably, we show that there exists a universal weighted averaging scheme that transitions to local convergence at an optimal stage, and still enjoys a superlinear convergence~rate nearly (up to a logarithmic factor) matching that of uniform Hessian averaging.
We prove a central limit theorem for the entropic transportation cost between subgaussian probability measures, centered at the population cost. This is the first result which allows for asymptotically valid inference for entropic optimal transport between measures which are not necessarily discrete. In the compactly supported case, we complement these results with new, faster, convergence rates for the expected entropic transportation cost between empirical measures. Our proof is based on strengthening convergence results for dual solutions to the entropic optimal transport problem.
We employ kernel-based approaches that use samples from a probability distribution to approximate a Kolmogorov operator on a manifold. The self-tuning variable-bandwidth kernel method [Berry & Harlim, Appl. Comput. Harmon. Anal., 40(1):68--96, 2016] computes a large, sparse matrix that approximates the differential operator. Here, we use the eigendecomposition of the discretization to (i) invert the operator, solving a differential equation, and (ii) represent gradient vector fields on the manifold. These methods only require samples from the underlying distribution and, therefore, can be applied in high dimensions or on geometrically complex manifolds when spatial discretizations are not available. We also employ an efficient $k$-$d$ tree algorithm to compute the sparse kernel matrix, which is a computational bottleneck.
This work focuses on reducing the computational cost of repeated video encodes by using a lower resolution clip as a proxy. Features extracted from the low resolution clip are used to learn an optimal lagrange multiplier for rate control on the original resolution clip. In addition to reducing the computational cost and encode time by using lower resolution clips, we also investigate the use of older, but faster codecs such as H.264 to create proxies. This work shows that the computational load is reduced by 22 times using 144p proxies. Our tests are based on the YouTube UGC dataset, hence our results are based on a practical instance of the adaptive bitrate encoding problem. Further improvements are possible, by optimising the placement and sparsity of operating points required for the rate distortion curves.
The majority of internet traffic is video content. This drives the demand for video compression in order to deliver high quality video at low target bitrates. This paper investigates the impact of adjusting the rate distortion equation on compression performance. An constant of proportionality, k, is used to modify the Lagrange multiplier used in H.265 (HEVC). Direct optimisation methods are deployed to maximise BD-Rate improvement for a particular clip. This leads to up to 21% BD-Rate improvement for an individual clip. Furthermore we use a more realistic corpus of material provided by YouTube. The results show that direct optimisation using BD-rate as the objective function can lead to further gains in bitrate savings that are not available with previous approaches.
Population dynamics is the study of temporal and spatial variation in the size of populations of organisms and is a major part of population ecology. One of the main difficulties in analyzing population dynamics is that we can only obtain observation data with coarse time intervals from fixed-point observations due to experimental costs or other constraints. Recently, modeling population dynamics by using continuous normalizing flows (CNFs) and dynamic optimal transport has been proposed to infer the expected trajectory of samples from a fixed-point observed population. While the sample behavior in CNF is deterministic, the actual sample in biological systems moves in an essentially random yet directional manner. Moreover, when a sample moves from point A to point B in dynamical systems, its trajectory is such that the corresponding action has the smallest possible value, known as the principle of least action. To satisfy these requirements of the sample trajectories, we formulate the Lagrangian Schr\"odinger bridge (LSB) problem and propose to solve it approximately using neural SDE with regularization. We also develop a model architecture that enables faster computation. Our experiments show that our solution to the LSB problem can approximate the dynamics at the population level and that using the prior knowledge introduced by the Lagrangian enables us to estimate the trajectories of individual samples with stochastic behavior.
The problem of continuous inverse optimal control (over finite time horizon) is to learn the unknown cost function over the sequence of continuous control variables from expert demonstrations. In this article, we study this fundamental problem in the framework of energy-based model, where the observed expert trajectories are assumed to be random samples from a probability density function defined as the exponential of the negative cost function up to a normalizing constant. The parameters of the cost function are learned by maximum likelihood via an "analysis by synthesis" scheme, which iterates (1) synthesis step: sample the synthesized trajectories from the current probability density using the Langevin dynamics via back-propagation through time, and (2) analysis step: update the model parameters based on the statistical difference between the synthesized trajectories and the observed trajectories. Given the fact that an efficient optimization algorithm is usually available for an optimal control problem, we also consider a convenient approximation of the above learning method, where we replace the sampling in the synthesis step by optimization. Moreover, to make the sampling or optimization more efficient, we propose to train the energy-based model simultaneously with a top-down trajectory generator via cooperative learning, where the trajectory generator is used to fast initialize the synthesis step of the energy-based model. We demonstrate the proposed methods on autonomous driving tasks, and show that they can learn suitable cost functions for optimal control.
This paper addresses the numerical solution of nonlinear eigenvector problems such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational physics and chemistry. These problems characterize critical points of energy minimization problems on the infinite-dimensional Stiefel manifold. To efficiently compute minimizers, we propose a novel Riemannian gradient descent method induced by an energy-adaptive metric. Quantified convergence of the methods is established under suitable assumptions on the underlying problem. A non-monotone line search and the inexact evaluation of Riemannian gradients substantially improve the overall efficiency of the method. Numerical experiments illustrate the performance of the method and demonstrates its competitiveness with well-established schemes.
Works on quantum computing and cryptanalysis has increased significantly in the past few years. Various constructions of quantum arithmetic circuits, as one of the essential components in the field, has also been proposed. However, there has only been a few studies on finite field inversion despite its essential use in realizing quantum algorithms, such as in Shor's algorithm for Elliptic Curve Discrete Logarith Problem (ECDLP). In this study, we propose to reduce the depth of the existing quantum Fermat's Little Theorem (FLT)-based inversion circuit for binary finite field. In particular, we propose follow a complete waterfall approach to translate the Itoh-Tsujii's variant of FLT to the corresponding quantum circuit and remove the inverse squaring operations employed in the previous work by Banegas et al., lowering the number of CNOT gates (CNOT count), which contributes to reduced overall depth and gate count. Furthermore, compare the cost by firstly constructing our method and previous work's in Qiskit quantum computer simulator and perform the resource analysis. Our approach can serve as an alternative for a time-efficient implementation.
小貼士
登錄享
相關主題
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191