Local low-rank approach to nonlinear matrix completion

This paper deals with a problem of matrix completion in which each column vector of the matrix belongs to a low-dimensional differentiable manifold (LDDM), with the target matrix being high or full rank. To solve this problem, algorithms based on polynomial mapping and matrix-rank minimization (MRM) have been proposed; such methods assume that each column vector of the target matrix is generated as a vector in a low-dimensional linear subspace (LDLS) and mapped to a pth order polynomial and that the rank of a matrix whose column vectors are dth monomial features of target column vectors is deficient. However, a large number of columns and observed values are needed to strictly solve the MRM problem using this method when p is large; therefore, this paper proposes a new method for obtaining the solution by minimizing the rank of the submatrix without transforming the target matrix, so as to obtain high estimation accuracy even when the number of columns is small. This method is based on the assumption that an LDDM can be approximated locally as an LDLS to achieve high completion accuracy without transforming the target matrix. Numerical examples show that the proposed method has a higher accuracy than other low-rank approaches.

Local low-rank approach to nonlinear-matrix completion

This paper deals with a problem of matrix completion in which each column vector of the matrix belongs to a low-dimensional differentiable manifold (LDDM), with the target matrix being high or full rank. To solve this problem, algorithms based on polynomial mapping and matrix-rank minimization (MRM) have been proposed; such methods assume that each column vector of the target matrix is generated as a vector in a low-dimensional linear subspace (LDLS) and mapped to a p-th order polynomial, and that the rank of a matrix whose column vectors are d-th monomial features of target column vectors is defficient. However, a large number of columns and observed values are needed to strictly solve the MRM problem using this method when p is large; therefore, this paper proposes a new method for obtaining the solution by minimizing the rank of the submatrix without transforming the target matrix, so as to obtain high estimation accuracy even when the number of columns is small. This method is based on the assumption that an LDDM can be approximated locally as an LDLS to achieve high completion accuracy without transforming the target matrix. Numerical examples show that the proposed method has a higher accuracy than other low-rank approaches.

Local low-rank approach to nonlinear-matrix completion

This paper deals with a problem of matrix completion in which each column vector of the matrix belongs to a low-dimensional differentiable manifold (LDDM), with the target matrix being high or full rank. To solve this problem, algorithms based on polynomial mapping and matrix-rank minimization (MRM) have been proposed; such methods assume that each column vector of the target matrix is generated as a vector in a low-dimensional linear subspace (LDLS) and mapped to a p-th order polynomial, and that the rank of a matrix whose column vectors are d-th monomial features of target column vectors is deficient. However, a large number of columns and observed values are needed to strictly solve the MRM problem using this method when p is large; therefore, this paper proposes a new method for obtaining the solution by minimizing the rank of the submatrix without transforming the target matrix, so as to obtain high estimation accuracy even when the number of columns is small. This method is based on the assumption that an LDDM can be approximated locally as an LDLS to achieve high completion accuracy without transforming the target matrix. Numerical examples show that the proposed method has a higher accuracy than other low-rank approaches.

On the effect of projections on the Billingsley dimensions

We are interested in the behavior of Billingsley dimensions under projections onto a lower dimensional linear subspace. The results in this paper establish the connections with various dimensions of subsets [Formula: see text] of [Formula: see text] and their projections, and generalize many known results about Hausdorff and packing dimensions of projections of [Formula: see text]. In particular, we improve, through these results, one of the main theorems of Selmi et al. in [Multifractal variation for projections of measures, Chaos Solitons Fractals 91 (2016) 414–420] and treat an unsolved case in their work.

Hyper-Reduction Over Nonlinear Manifolds for Large Nonlinear Mechanical Systems

Common trends in model reduction of large nonlinear finite element (FE)-discretized systems involve Galerkin projection of the governing equations onto a low-dimensional linear subspace. Though this reduces the number of unknowns in the system, the computational cost for obtaining the reduced solution could still be high due to the prohibitive computational costs involved in the evaluation of nonlinear terms. Hyper-reduction methods are then used for fast approximation of these nonlinear terms. In the finite element context, the energy conserving sampling and weighing (ECSW) method has emerged as an effective tool for hyper-reduction of Galerkin-projection-based reduced-order models (ROMs). More recent trends in model reduction involve the use of nonlinear manifolds, which involves projection onto the tangent space of the manifold. While there are many methods to identify such nonlinear manifolds, hyper-reduction techniques to accelerate computation in such ROMs are rare. In this work, we propose an extension to ECSW to allow for hyper-reduction using nonlinear mappings, while retaining its desirable stability and structure-preserving properties. As a proof of concept, the proposed hyper-reduction technique is demonstrated over models of a flat plate and a realistic wing structure, whose dynamics have been shown to evolve over a nonlinear (quadratic) manifold. An online speed-up of over one thousand times relative to the full system has been obtained for the wing structure using the proposed method, which is higher than its linear counterpart using the ECSW.

Random affine simplexes

For a fixed k ∈ {1, …, d}, consider arbitrary random vectors X0, …, Xk ∈ ℝd such that the (k + 1)-tuples (UX0, …, UXk) have the same distribution for any rotation U. Let A be any nonsingular d × d matrix. We show that the k-dimensional volume of the convex hull of affinely transformed Xi satisfies \[|{\rm{conv}}(A{X_{\rm{0}}} \ldots ,A{X_k}){\rm{|}}\mathop {\rm{ = }}\limits^{\rm{D}} (|{P_\xi }\varepsilon |/{\kappa _k})|{\rm{conv}}\left( {{X_0}, \ldots ,{X_k}} \right)\] , where ɛ:= {x ∈ ℝd : x┬ (A┬ A)−1x ≤ 1} is an ellipsoid, Pξ denotes the orthogonal projection to a uniformly chosen random k-dimensional linear subspace ξ independent of X0, …, Xk, and κk is the volume of the unit k-dimensional ball. As an application, we derive the following integral geometry formula for ellipsoids: ck,d,p ∫Ad,k |ɛ ∩ E|p+d+1 μd,k(dE) = |ɛ|k+1 ∫Gd,k |PLɛ|pνd,k(dL), where $c_{k,d,p} = \big({\kappa_{d}^{k+1}}/{\kappa_k^{d+1}}\big) ({\kappa_{k(d+p)+k}}/{\kappa_{k(d+p)+d}})$ . Here p > −1 and Ad,k and Gd,k are the affine and the linear Grassmannians equipped with their respective Haar measures. The p = 0 case reduces to an affine version of the integral formula of Furstenberg and Tzkoni (1971).

A likelihood-free estimator of population structure bridging admixture models and principal components analysis

We introduce a simple and computationally efficient method for fitting the admixture model of genetic population structure, called ALStructure. The strategy of ALStructure is to first estimate the low-dimensional linear subspace of the population admixture components and then search for a model within this subspace that is consistent with the admixture model’s natural probabilistic constraints. Central to this strategy is the observation that all models belonging to this constrained space of solutions are risk-minimizing and have equal likelihood, rendering any additional optimization unnecessary. The low-dimensional linear subspace is estimated through a recently introduced principal components analysis method that is appropriate for genotype data, thereby providing a solution that has both principal components and probabilistic admixture interpretations. Our approach differs fundamentally from other existing methods for estimating admixture, which aim to fit the admixture model directly by searching for parameters that maximize the likelihood function or the posterior probability. We observe that ALStructure typically outperforms existing methods both in accuracy and computational speed under a wide array of simulated and real human genotype datasets. Throughout this work we emphasize that the admixture model is a special case of a much broader class of models for which algorithms similar to ALStructure may be successfully employed.

3D Modeling for Environmental Informatics

3D modeling plays an important role in the field of computer vision and image processing. It provides a convenient tool set for many environmental informatics tasks, such as taxonomy and species identification. This chapter discusses a novel way of building the 3D models of objects from their varying 2D views. The appearance of a 3D object depends on both the viewing directions and illumination conditions. What is the set of images of an object under all viewing directions? In this chapter, a novel image representation is proposed, which transforms any n-pixel image of a 3D object to a vector in a 2n-dimensional pose space. In such a pose space, it is proven that the transformed images of a 3D object under all viewing directions form a parametric manifold in a 6-dimensional linear subspace. With in-depth rotations along a single axis in particular, this manifold is an ellipse. Furthermore, it is shown that this parametric pose manifold of a convex object can be estimated from a few images in different poses and used to predict object's appearances under unseen viewing directions. These results immediately suggest a number of approaches to object recognition, scene detection, and 3D modeling, applicable to environmental informatics. Experiments on both synthetic data and real images were reported, which demonstrates the validity of the proposed representation.

3D Modeling for Environmental Informatics

3D modeling plays an important role in the field of computer vision and image processing. It provides a convenient tool set for many environmental informatics tasks, such as taxonomy and species identification. This chapter discusses a novel way of building the 3D models of objects from their varying 2D views. The appearance of a 3D object depends on both the viewing directions and illumination conditions. What is the set of images of an object under all viewing directions? In this chapter, a novel image representation is proposed, which transforms any n-pixel image of a 3D object to a vector in a 2n-dimensional pose space. In such a pose space, it is proven that the transformed images of a 3D object under all viewing directions form a parametric manifold in a 6-dimensional linear subspace. With in-depth rotations along a single axis in particular, this manifold is an ellipse. Furthermore, it is shown that this parametric pose manifold of a convex object can be estimated from a few images in different poses and used to predict object's appearances under unseen viewing directions. These results immediately suggest a number of approaches to object recognition, scene detection, and 3D modeling, applicable to environmental informatics. Experiments on both synthetic data and real images were reported, which demonstrates the validity of the proposed representation.