Spectral Affine‐Kernel Embeddings

Abstract

In this paper, we propose a controllable embedding method for high‐ and low‐dimensional geometry processing through sparse matrix eigenanalysis. Our approach is equally suitable to perform non‐linear dimensionality reduction on big data, or to offer non‐linear shape editing of 3D meshes and pointsets. At the core of our approach is the construction of a multi‐Laplacian quadratic form that is assembled from local operators whose kernels only contain locally‐affine functions. Minimizing this quadratic form provides an embedding that best preserves all relative coordinates of points within their local neighborhoods. We demonstrate the improvements that our approach brings over existing nonlinear dimensionality reduction methods on a number of datasets, and formulate the first eigen‐based as‐rigid‐as‐possible shape deformation technique by applying our affine‐kernel embedding approach to 3D data augmented with user‐imposed constraints on select vertices.

Parallel Transport Unfolding: A Connection-Based Manifold Learning Approach

“…First, spurious geodesic curvature (i.e., zigzags) in the shortest paths (seen as a piecewise linear curve in D ) between two nodes on the graph introduces inaccuracies in the estimation of geodesic distances. In practice, this issue is exacerbated by the sparse sampling that real applications often have to deal with, even if short paths can be locally rectified through a straightening projection [Budninskiy et al, 2017]. Worse, some computational accelerations of Isomap rely on a subsampling of the initial data [De Silva and Tenenbaum, 2004], making this inaccuracy issue all the more limiting.…”

“…1. In this noiseless case, it turns out that most local methods, such as Modified LLE [Zhang and Wang, 2007], Hessian LLE [Donoho and Grimes, 2003], and SAKE [Budninskiy et al, 2017], do also remarkably well (see Fig. 16), as convexity (or lack thereof) plays basically no role in their embeddings.…”

“…12a &12b verify that PTU (with k = 10) is as resilient to strong noise and outliers as Isomap even without locally adapting neighborhood sizes (K = 25 was used in the Gaussian noise case, and K = 10 in the case of sparse noise). Local methods such as LLE [Roweis and Saul, 2000], Hessian LLE [Donoho and Grimes, 2003], LTSA [Zhang and Zha, 2004], MLLE [Zhang and Wang, 2007] or SAKE [Budninskiy et al, 2017] (with k = 10 for fairness of comparison) fail to unwrap noisy datasets (see Fig. 14) as they do not exploit the geodesic distance estimates between pairs of points that are far apart.…”

“…The manifold assumption, which posits that many high-dimensional datasets actually live on much lower dimensional curved spaces, has been proven surprisingly useful in a variety of contexts. Algorithms working towards extracting reduced dimensional representations of data, also called manifold learning, produce a low count of "intrinsic variables" to describe the high-dimensional input-with which markedly faster computations can be performed in the context of machine learning (to address the "curse of dimensionality"), facial animation [Kim, 2007], skeletal animation [Assa et al, 2005], video editing [Pless, 2003], and nonlinear shape editing [Budninskiy et al, 2017] to mention a few.…”

Automated Chain for Large-Scale 3d Reconstruction of Urban Scenes From Satellite Images

“…From the estimated bare-terrain elevations, we reconstruct the final DTM by applying a least-squares smooth embedding approach to interpolate the surface. We adapted the spectral affine-kernel embedding (SAKE) algorithm proposed in (Budninskiy et al, 2017), and developed its simpler 1 https://www2.jpl.nasa.gov/srtm/index.html expression that we call LAKE, as the Spectral solve in the original approach is replaced by a faster Least-squared solve in our specific application. As it turns out, this approach is particularly appropriate to construct a DTM, as a bare terrain can be thought of as a smooth two-dimensional embedding in 3D of a surface given as a few scattered elevations.…”