1,261 research outputs found
Geodesic Distance Function Learning via Heat Flow on Vector Fields
Learning a distance function or metric on a given data manifold is of great
importance in machine learning and pattern recognition. Many of the previous
works first embed the manifold to Euclidean space and then learn the distance
function. However, such a scheme might not faithfully preserve the distance
function if the original manifold is not Euclidean. Note that the distance
function on a manifold can always be well-defined. In this paper, we propose to
learn the distance function directly on the manifold without embedding. We
first provide a theoretical characterization of the distance function by its
gradient field. Based on our theoretical analysis, we propose to first learn
the gradient field of the distance function and then learn the distance
function itself. Specifically, we set the gradient field of a local distance
function as an initial vector field. Then we transport it to the whole manifold
via heat flow on vector fields. Finally, the geodesic distance function can be
obtained by requiring its gradient field to be close to the normalized vector
field. Experimental results on both synthetic and real data demonstrate the
effectiveness of our proposed algorithm
Density ridge manifold traversal
The density ridge framework for estimating principal curves and surfaces has in a number of recent works been shown to capture manifold structure in data in an intuitive and effective manner. However, to date there exists no efficient way to traverse these manifolds as defined by density ridges. This is unfortunate, as manifold traversal is an important problem for example for shape estimation in medical imaging, or in general for being able to characterize and understand state transitions or local variability over the data manifold. In this paper, we remedy this situation by introducing a novel manifold traversal algorithm based on geodesics within the density ridge approach. The traversal is executed in a subspace capturing the intrinsic dimensionality of the data using dimensionality reduction techniques such as principal component analysis or kernel entropy component analysis. A mapping back to the ambient space is obtained by training a neural network. We compare against maximum mean discrepancy traversal, a recent approach, and obtain promising results
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