Computing Multi-Homogeneous Bezout Numbers is Hard

The multi-homogeneous Bezout number is a bound for the number of solutions of a system of multi-homogeneous polynomial equations, in a suitable product of projective spaces. Given an arbitrary, not necessarily multi-homogeneous system, one can ask for the optimal multi-homogenization that would minimize the Bezout number. In this paper, it is proved that the problem of computing, or even estimating the optimal multi-homogeneous Bezout number is actually NP-hard. In terms of approximation theory for combinatorial optimization, the problem of computing the best multi-homogeneous structure does not belong to APX, unless P = NP. Moreover, polynomial time algorithms for estimating the minimal multi-homogeneous Bezout number up to a fixed factor cannot exist even in a randomized setting, unless BPP contains NP

Trifocal Relative Pose from Lines at Points and its Efficient Solution

We present a new minimal problem for relative pose estimation mixing point features with lines incident at points observed in three views and its efficient homotopy continuation solver. We demonstrate the generality of the approach by analyzing and solving an additional problem with mixed point and line correspondences in three views. The minimal problems include correspondences of (i) three points and one line and (ii) three points and two lines through two of the points which is reported and analyzed here for the first time. These are difficult to solve, as they have 216 and - as shown here - 312 solutions, but cover important practical situations when line and point features appear together, e.g., in urban scenes or when observing curves. We demonstrate that even such difficult problems can be solved robustly using a suitable homotopy continuation technique and we provide an implementation optimized for minimal problems that can be integrated into engineering applications. Our simulated and real experiments demonstrate our solvers in the camera geometry computation task in structure from motion. We show that new solvers allow for reconstructing challenging scenes where the standard two-view initialization of structure from motion fails.Comment: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while most authors were in residence at Brown University's Institute for Computational and Experimental Research in Mathematics -- ICERM, in Providence, R

Maximum likelihood geometry in the presence of data zeros

Given a statistical model, the maximum likelihood degree is the number of complex solutions to the likelihood equations for generic data. We consider discrete algebraic statistical models and study the solutions to the likelihood equations when the data contain zeros and are no longer generic. Focusing on sampling and model zeros, we show that, in these cases, the solutions to the likelihood equations are contained in a previously studied variety, the likelihood correspondence. The number of these solutions give a lower bound on the ML degree, and the problem of finding critical points to the likelihood function can be partitioned into smaller and computationally easier problems involving sampling and model zeros. We use this technique to compute a lower bound on the ML degree for $2 \times 2 \times 2 \times 2$ tensors of border rank $\leq 2$ and $3 \times n$ tables of rank $\leq 2$ for $n=11, 12, 13, 14$, the first four values of $n$ for which the ML degree was previously unknown

Optimization with Sparsity-Inducing Penalties

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel selection. It turns out that many of the related estimation problems can be cast as convex optimization problems by regularizing the empirical risk with appropriate non-smooth norms. The goal of this paper is to present from a general perspective optimization tools and techniques dedicated to such sparsity-inducing penalties. We cover proximal methods, block-coordinate descent, reweighted $\ell_2$-penalized techniques, working-set and homotopy methods, as well as non-convex formulations and extensions, and provide an extensive set of experiments to compare various algorithms from a computational point of view

H∞H_\infty optimization of multiple tuned mass dampers for multimodal vibration control

In this paper, a new computational method for the purpose of multimodal vibration mitigation using multiple tuned mass dampers is proposed. Classically, the minimization of the maximum amplitude is carried out using direct $H_\infty$ optimization. However, as shall be shown in the paper, this approach is prone to being trapped in local minima, in view of the nonsmooth character of the problem at hand. This is why this paper presents an original alternative to this approach through norm-homotopy optimization. This approach, combined with an efficient technique to compute the structural response, is shown to outperform direct $H_\infty$ optimization in terms of speed and performance. Essentially, the outcome of the algorithm leads to the concept of all-equal-peak design for which all the controlled peaks are equal in amplitude. This unique design is new with respect to the existing body of knowledge.Comment: This is a new version of a preprint previously named "All-equal-peak design of multiple tuned mass dampers using norm-homotopy optimization