Distance Shrinkage and Euclidean Embedding via Regularized Kernel Estimation

Abstract

Although recovering an Euclidean distance matrix from noisy observations is a common problem in practice, how well this could be done remains largely unknown. To fill in this void, we study a simple distance matrix estimate based upon the so-called regularized kernel estimate. We show that such an estimate can be characterized as simply applying a constant amount of shrinkage to all observed pairwise distances. This fact allows us to establish risk bounds for the estimate implying that the true distances can be estimated consistently in an average sense as the number of objects increases. In addition, such a characterization suggests an efficient algorithm to compute the distance matrix estimator, as an alternative to the usual second order cone programming known not to scale well for large problems. Numerical experiments and an application in visualizing the diversity of Vpu protein sequences from a recent HIV-1 study further demonstrate the practical merits of the proposed method.

A Fast Matrix Majorization-Projection Method for Penalized Stress Minimization With Box Constraints

“…For example, it could be applied to image data for dimensionality reduction as done in [7] and problems studied in [45], [46]. It also remains to be seen whether the developed techniques can be used for the variants of the stress function considered in [3] and for outlier removal in the robust MDS [47]- [49].…”

Robust Euclidean embedding via EDM optimization

“…Extensions of Newton's method for the model (24) with more constraints including general weights W i j , the rank constraint rank(J D J ) ≤ r or the box constraints (7) can be found in [3,11,40]. A recent application of the model (24) with a regularization term to Statistics is [55], where the problem is solved by an SDP, similar to that proposed by Toh [48].…”

Penalized Nonparametric Scalar-on-Function Regression via Principal Coordinates

“…But as in most of the kernel literature, Corrada Bravo et al (2009) require a positive semidefinite matrix K . The “pseudo-attributes” entered into their smoothing spline ANOVA model have the same form as the PCo’s (7), but arise from a positive semidefinite K , which they obtain by regularized kernel estimation (Lu et al, 2005; Zhang et al, 2016). A simpler way to derive a positive semidefinite K from a non-Euclidean distance matrix D is to add a suitable positive constant to the non-diagonal entries of D , and then apply (5).…”