896 research outputs found
DIMAL: Deep Isometric Manifold Learning Using Sparse Geodesic Sampling
This paper explores a fully unsupervised deep learning approach for computing
distance-preserving maps that generate low-dimensional embeddings for a certain
class of manifolds. We use the Siamese configuration to train a neural network
to solve the problem of least squares multidimensional scaling for generating
maps that approximately preserve geodesic distances. By training with only a
few landmarks, we show a significantly improved local and nonlocal
generalization of the isometric mapping as compared to analogous non-parametric
counterparts. Importantly, the combination of a deep-learning framework with a
multidimensional scaling objective enables a numerical analysis of network
architectures to aid in understanding their representation power. This provides
a geometric perspective to the generalizability of deep learning.Comment: 10 pages, 11 Figure
LOCA: LOcal Conformal Autoencoder for standardized data coordinates
We propose a deep-learning based method for obtaining standardized data
coordinates from scientific measurements.Data observations are modeled as
samples from an unknown, non-linear deformation of an underlying Riemannian
manifold, which is parametrized by a few normalized latent variables. By
leveraging a repeated measurement sampling strategy, we present a method for
learning an embedding in that is isometric to the latent
variables of the manifold. These data coordinates, being invariant under smooth
changes of variables, enable matching between different instrumental
observations of the same phenomenon. Our embedding is obtained using a LOcal
Conformal Autoencoder (LOCA), an algorithm that constructs an embedding to
rectify deformations by using a local z-scoring procedure while preserving
relevant geometric information. We demonstrate the isometric embedding
properties of LOCA on various model settings and observe that it exhibits
promising interpolation and extrapolation capabilities. Finally, we apply LOCA
to single-site Wi-Fi localization data, and to -dimensional curved surface
estimation based on a -dimensional projection
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