Geodesic distance matrices can reveal shape properties that are largely
invariant to non-rigid deformations, and thus are often used to analyze and
represent 3-D shapes. However, these matrices grow quadratically with the
number of points. Thus for large point sets it is common to use a low-rank
approximation to the distance matrix, which fits in memory and can be
efficiently analyzed using methods such as multidimensional scaling (MDS). In
this paper we present a novel sparse method for efficiently representing
geodesic distance matrices using biharmonic interpolation. This method exploits
knowledge of the data manifold to learn a sparse interpolation operator that
approximates distances using a subset of points. We show that our method is 2x
faster and uses 20x less memory than current leading methods for solving MDS on
large point sets, with similar quality. This enables analyses of large point
sets that were previously infeasible.

本文提出了一种利用双调和插值有效地表示测地距离矩阵的稀疏方法，用于在大数据点集上进行多维缩放分析，相比目前领先的方法，该方法运行速度更快，占用内存更少，可使先前不可行的大数据点集的分析成为可能。

基于经典缩放的流形距离矩阵高效稀疏表示

Efficient, sparse representation of manifold distance matrices for  classical scaling

Inpainting based image compression approaches, especially linear and
non-linear diffusion models, are an active research topic for lossy image
compression. The major challenge in these compression models is to find a small
set of descriptive supporting points, which allow for an accurate
reconstruction of the original image. It turns out in practice that this is a
challenging problem even for the simplest Laplacian interpolation model. In
this paper, we revisit the Laplacian interpolation compression model and
introduce two fast algorithms, namely successive preconditioning primal dual
algorithm and the recently proposed iPiano algorithm, to solve this problem
efficiently. Furthermore, we extend the Laplacian interpolation based
compression model to a more general form, which is based on principles from
bi-level optimization. We investigate two different variants of the Laplacian
model, namely biharmonic interpolation and smoothed Total Variation
regularization. Our numerical results show that significant improvements can be
obtained from the biharmonic interpolation model, and it can recover an image
with very high quality from only 5% pixels.

本文介绍了两种算法来高效地解决 Laplacian interpolation 压缩模型中的支持点选取问题，并将其扩展到了更为一般化的双层优化 Laplacian 模型，其中 biharmonic interpolation 表现出了更好的压缩效果。