An Efficient Quadrature Sequence and Sparsifying Methodology for Mean-Field Variational Inference

This work proposes a quasirandom sequence of quadratures for high-dimensional mean-field variational inference and a related sparsifying methodology. Each iterate of the sequence contains two evaluations points that combine to correctly integrate all univariate quadratic functions, as well as univariate cubics if the mean-field factors are symmetric. More importantly, averaging results over short subsequences achieves periodic exactness on a much larger space of multivariate polynomials of quadratic total degree. This framework is devised by first considering stochastic blocked mean-field quadratures, which may be useful in other contexts. By replacing pseudorandom sequences with quasirandom sequences, over half of all multivariate quadratic basis functions integrate exactly with only 4 function evaluations, and the exactness dimension increases for longer subsequences. Analysis shows how these efficient integrals characterize the dominant log-posterior contributions to mean-field variational approximations, including diagonal Hessian approximations, to support a robust sparsifying methodology in deep learning algorithms. A numerical demonstration of this approach on a simple Convolutional Neural Network for MNIST retains high test accuracy, 96.9%, while training over 98.9% of parameters to zero in only 10 epochs, bearing potential to reduce both storage and energy requirements for deep learning models

Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines

Calabr{\`o} et al. [10] changed the paradigm of the mass and stiffness computation from the traditional element-wise assembly to a row-wise concept, showing that the latter one offers integration that may be orders of magnitude faster. Considering a B-spline basis function as a non-negative measure, each mass matrix row is integrated by its own quadrature rule with respect to that measure. Each rule is easy to compute as it leads to a linear system of equations, however, the quadrature rules are of the Newton-Cotes type, that is, they require a number of quadrature points that is equal to the dimension of the spline space. In this work, we propose weighted quadrature rules of Gaussian type which require the minimum number of quadrature points while guaranteeing exactness of integration with respect to the weight function. The weighted Gaussian rules arise as solutions of non-linear systems of equations. We derive rules for the mass and stiffness matrices for uniform $C^1$ quadratic and $C^2$ cubic isogeometric discretizations. In each parameter direction, our rules require locally only $p+1$ quadrature points, $p$ being the polynomial degree. While the nodes cannot be reused for various weight functions as in [10], the computational cost of the mass and stiffness matrix assembly is comparable.RYC-2017-2264

Gaussian quadrature rules for C1 quintic splines with uniform knot vectors

We provide explicit quadrature rules for spaces of C1 quintic splines with uniform knot sequences over finite domains. The quadrature nodes and weights are derived via an explicit recursion that avoids numerical solvers. Each rule is optimal, that is, requires the minimal number of nodes, for a given function space. For each of n subintervals, generically, only two nodes are required which reduces the evaluation cost by 2/3 when compared to the classical Gaussian quadrature for polynomials over each knot span. Numerical experiments show fast convergence, as n grows, to the “two-third” quadrature rule of Hughes et al. (2010) for infinite domains

Multi-objective variational curves

Riemannian cubics in tension are critical points of the linear combination of two objective functionals, namely the squared norms of the velocity and acceleration of a curve on a Riemannian manifold. We view this variational problem of finding a curve as a multi-objective optimization problem and construct the Pareto fronts for some given instances where the manifold is a sphere and where the manifold is a torus. The Pareto front for the curves on the torus turns out to be particularly interesting: the front is disconnected and it reveals two distinct Riemannian cubics with the same boundary data, which is the first known nontrivial instance of this kind. We also discuss some convexity conditions involving the Pareto fronts for curves on general Riemannian manifolds

Gaussian quadrature for C1C^1 cubic Clough-Tocher macro-triangles

A numerical integration rule for multivariate cubic polynomials over n-dimensional simplices was designed by Hammer and Stroud [14]. The quadrature rule requires n + 2 quadrature points: the barycentre of the simplex and n + 1 points that lie on the connecting lines between the barycentre and the vertices of the simplex. In the planar case, this particular rule belongs to a two-parameter family of quadrature rules that admit exact integration of bivariate polynomials of total degree three over triangles. We prove that this rule is exact for a larger space, namely the C1 cubic Clough-Tocher spline space over macro-triangles if and only if the split-point is the barycentre. This results into a factor of three reduction in the number of quadrature points needed to integrate the Clough-Tocher spline space exactly