Constrained Best Euclidean Distance Embedding on a Sphere: A Matrix Optimization Approach

Abstract

Abstract. The problem of data representation on a sphere of unknown radius arises from various disciplines such as Statistics (spatial data representation), Psychology (constrained multidimensional scaling), and Computer Science (machine learning and pattern recognition). The best representation often needs to minimize a distance function of the data on a sphere as well as to satisfy some Euclidean distance constraints. It is those spherical and Euclidean distance constraints that present an enormous challenge to the existing algorithms. In this paper, we reformulate the problem as an Euclidean distance matrix optimization problem with a low rank constraint. We then propose an iterative algorithm that uses a quadratically convergent Newton-CG method at its each step. We study fundamental issues including constraint nondegeneracy and the nonsingularity of generalized Jacobian that ensure the quadratic convergence of the Newton method. We use some classic examples from the spherical multidimensional scaling to demonstrate the flexibility of the algorithm in incorporating various constraints. We also present an interesting application to the circle fitting problem.Key words. Euclidean distance matrix, Matrix optimization, Lagrangian duality, Spherical multidimensional scaling, Semismooth Newton-CG method.AMS subject classifications. 49M45, 90C25, 90C331. Introduction. In this paper, we are mainly concerned with placing n points {x 1 , . . . , x n } in a best way on a sphere in IR r . The primary information that we use is an incomplete/complete set of pairwise Euclidean distances (often with noises) among the n points. In such a setting, IR r is often a low-dimensional space (e.g., r takes 2 or 3 for data visualization) and is known as the embedding space. The center of the sphere is unknown. For some applications, the center can be put at origin in IR r . Furthermore, the radius of the sphere is also unknown. In our matrix optimization formulation of the problem, we treat both the center and the radius as unknown variables. We develop a fast numerical method for this problem and present a few of interesting applications taken from existing literature.The problem described above has long appeared in the constrained Multi-Dimensional Scaling (MDS) when r ≤ 3, which is mainly for the purpose of data visualization, see [9, Sect. 4.6] and [4, Sect. 10.3] for more details. In particular, it is known as the spherical MDS when r = 3 and the circular MDS when r = 2. Most numerical methods in this part took advantages of r being 2 or 3. For example, two of the earliest circular MDS were by Borg and Lingoes [5] and Lee and Bentler [28], where they introduced a new point x 0 ∈ IR r as the center of the sphere (i.e., circles in their case) and further forced the following constraints to hold:

Conditional Quadratic Semidefinite Programming: Examples and Methods

“…Before we move on to give a new application, we would like to point out the difference between this work and our previous work [33,2], which made most of the subspace V = e ⊥ . In contrast, our analysis here is for general subspace V and the assumptions are more general.…”

“…Problem (4) was studied by Hayden and Wells [21]. Like the positive semidefiniteness case (2), it also has a closed form solution. Let Q ∈ IR n×n be the product of r Householder transformations such that…”

“…Because J defined in (17) is the orthogonal projection matrix onto e ⊥ , both the researches by Torgerson and Gower lead to Schoenberg condition (15). That is, the matrix −D (2) is conditionally positive semidefinite on V = e ⊥ . Another important piece of contribution is by Micchelli [29] on connections between EDMs and reproducing kernels, which play an essential role in machine learning research (e.g., support vector machines).…”

“…4.5 and by following the steps in [2,Thm. 3.9], which deals with the nonsingularity issue of a semismooth Newton method for a Euclidean distance matrix problem with spherical constraints.…”

“…For example, in the nearest Euclidean distance matrix problem [33], both the strong Slater condition and constraint nondegeneracy hold. For the spherical EDM problem in [2], the strong Slater condition holds while constraint nondegeneracy is satisfied under some (weak) condition.…”