This paper presents an algorithm and its implementation in the software package NCSOStools for finding sums of hermitian squares and commutators decompositions for polynomials in noncommuting variables. The algorithm is based on noncommutative analogs of the classical Gram matrix method and the Newton polytope method, which allows us to use semidefinite programming. Throughout the paper several...

We describe regularization tools for training large-scale artiicial feed-forward neural networks. In a companion paper (in this issue) we give the basic ideas and some theoretical results regarding the Gauss-Newton method compared to other methods such as the Levenberg-Marquardt method applied on small and medium size problems. We propose algorithms that explicitly use a sequence of Tikhonov re...

In this paper, a monotone linear complementarity reformulation of the nonnegative least squares problems in inequality sense is introduced. Then Mehrotra’s predictor-corrector interior point algorithm is applied to solve the resulting monotone linear complementarity problem. Our numerical experiments on several randomly generated test problems show that Mehrotra’s algorithm overperforms the wid...

We introduce a partial proximal point algorithm for solving nuclear norm regularized and semidefinite matrix least squares problems with linear equality constraints. For the inner subproblems, we show that the positive definiteness of the generalized Hessian of the objective function for the inner subproblems is equivalent to the constraint nondegeneracy of the corresponding primal problem, whi...

This paper presents a novel 2D shape deformation algorithm based on nonlinear least squares optimization. The algorithm aims to preserve two local shape properties: Laplacian coordinates of the boundary curve and local area of the shape interior, which are together represented in a non-quadric energy function. An iterative Gauss-Newton method is used to minimize this nonlinear energy function. ...

We present an iterative approach to solve separable nonlinear least squares problems arising in the estimation of wavelength-dependent point spread function (PSF) parameters for hyperspectral imaging. A variable projection Gauss-Newton method is used to solve the nonlinear least squares problem. An analysis shows that the Jacobian can be potentially very ill-conditioned. To deal with this ill-c...

A least-squares approach to computing inverse dynamics is proposed. The method utilizes equations of motion for a multi-segment body, incorporating terms for ground reaction forces and torques. The resulting system is overdetermined at each point in time, because kinematic and force measurements outnumber unknown torques, and may be solved using weighted least squares to yield estimates of the ...

We consider the problem of fitting a parametric curve to a given point cloud (e.g., measurement data). Least-squares approximation, i.e., minimization of the l2 norm of residuals (the Euclidean distances to the data points), is the most common approach. This is due to its computational simplicity [1]. However, in the case of data that is affected by noise or contains outliers, this is not alway...

In this paper a second-order method for solving large-scale strongly convex `1-regularized problems is developed. The proposed method is a NewtonCG (Conjugate Gradients) algorithm with backtracking line-search embedded in a doubly-continuation scheme. Worst-case iteration complexity of the proposed Newton-CG is established. Based on the analysis of Newton-CG, worstcase iteration complexity of t...

A sum-of-squares is a polynomial that can be expressed as a sum of squares of other polynomials. Determining if a sum-of-squares decomposition exists for a given polynomial is equivalent to a linear matrix inequality feasibility problem. The computation required to solve the feasibility problem depends on the number of monomials used in the decomposition. The Newton polytope is a method to prun...