42,881 research outputs found
Tangent space estimation for smooth embeddings of Riemannian manifolds
Numerous dimensionality reduction problems in data analysis involve the
recovery of low-dimensional models or the learning of manifolds underlying sets
of data. Many manifold learning methods require the estimation of the tangent
space of the manifold at a point from locally available data samples. Local
sampling conditions such as (i) the size of the neighborhood (sampling width)
and (ii) the number of samples in the neighborhood (sampling density) affect
the performance of learning algorithms. In this work, we propose a theoretical
analysis of local sampling conditions for the estimation of the tangent space
at a point P lying on a m-dimensional Riemannian manifold S in R^n. Assuming a
smooth embedding of S in R^n, we estimate the tangent space T_P S by performing
a Principal Component Analysis (PCA) on points sampled from the neighborhood of
P on S. Our analysis explicitly takes into account the second order properties
of the manifold at P, namely the principal curvatures as well as the higher
order terms. We consider a random sampling framework and leverage recent
results from random matrix theory to derive conditions on the sampling width
and the local sampling density for an accurate estimation of tangent subspaces.
We measure the estimation accuracy by the angle between the estimated tangent
space and the true tangent space T_P S and we give conditions for this angle to
be bounded with high probability. In particular, we observe that the local
sampling conditions are highly dependent on the correlation between the
components in the second-order local approximation of the manifold. We finally
provide numerical simulations to validate our theoretical findings
Non-Asymptotic Analysis of Tangent Space Perturbation
Constructing an efficient parameterization of a large, noisy data set of
points lying close to a smooth manifold in high dimension remains a fundamental
problem. One approach consists in recovering a local parameterization using the
local tangent plane. Principal component analysis (PCA) is often the tool of
choice, as it returns an optimal basis in the case of noise-free samples from a
linear subspace. To process noisy data samples from a nonlinear manifold, PCA
must be applied locally, at a scale small enough such that the manifold is
approximately linear, but at a scale large enough such that structure may be
discerned from noise. Using eigenspace perturbation theory and non-asymptotic
random matrix theory, we study the stability of the subspace estimated by PCA
as a function of scale, and bound (with high probability) the angle it forms
with the true tangent space. By adaptively selecting the scale that minimizes
this bound, our analysis reveals an appropriate scale for local tangent plane
recovery. We also introduce a geometric uncertainty principle quantifying the
limits of noise-curvature perturbation for stable recovery. With the purpose of
providing perturbation bounds that can be used in practice, we propose plug-in
estimates that make it possible to directly apply the theoretical results to
real data sets.Comment: 53 pages. Revised manuscript with new content addressing application
of results to real data set
The space of essential matrices as a Riemannian quotient manifold
The essential matrix, which encodes the epipolar constraint between points in two projective views,
is a cornerstone of modern computer vision. Previous works have proposed different characterizations
of the space of essential matrices as a Riemannian manifold. However, they either do not consider the
symmetric role played by the two views, or do not fully take into account the geometric peculiarities
of the epipolar constraint. We address these limitations with a characterization as a quotient manifold
which can be easily interpreted in terms of camera poses. While our main focus in on theoretical
aspects, we include applications to optimization problems in computer vision.This work was supported by grants NSF-IIP-0742304, NSF-OIA-1028009, ARL MAST-CTA W911NF-08-2-0004, and ARL RCTA W911NF-10-2-0016, NSF-DGE-0966142, and NSF-IIS-1317788
Trifocal Relative Pose from Lines at Points and its Efficient Solution
We present a new minimal problem for relative pose estimation mixing point
features with lines incident at points observed in three views and its
efficient homotopy continuation solver. We demonstrate the generality of the
approach by analyzing and solving an additional problem with mixed point and
line correspondences in three views. The minimal problems include
correspondences of (i) three points and one line and (ii) three points and two
lines through two of the points which is reported and analyzed here for the
first time. These are difficult to solve, as they have 216 and - as shown here
- 312 solutions, but cover important practical situations when line and point
features appear together, e.g., in urban scenes or when observing curves. We
demonstrate that even such difficult problems can be solved robustly using a
suitable homotopy continuation technique and we provide an implementation
optimized for minimal problems that can be integrated into engineering
applications. Our simulated and real experiments demonstrate our solvers in the
camera geometry computation task in structure from motion. We show that new
solvers allow for reconstructing challenging scenes where the standard two-view
initialization of structure from motion fails.Comment: This material is based upon work supported by the National Science
Foundation under Grant No. DMS-1439786 while most authors were in residence
at Brown University's Institute for Computational and Experimental Research
in Mathematics -- ICERM, in Providence, R
Sufficient Dimension Reduction and Modeling Responses Conditioned on Covariates: An Integrated Approach via Convex Optimization
Given observations of a collection of covariates and responses , sufficient dimension reduction (SDR)
techniques aim to identify a mapping
with such that is independent of . The image
summarizes the relevant information in a potentially large number of covariates
that influence the responses . In many contemporary settings, the number
of responses is also quite large, in addition to a large number of
covariates. This leads to the challenge of fitting a succinctly parameterized
statistical model to , which is a problem that is usually not addressed
in a traditional SDR framework. In this paper, we present a computationally
tractable convex relaxation based estimator for simultaneously (a) identifying
a linear dimension reduction of the covariates that is sufficient with
respect to the responses, and (b) fitting several types of structured
low-dimensional models -- factor models, graphical models, latent-variable
graphical models -- to the conditional distribution of . We analyze the
consistency properties of our estimator in a high-dimensional scaling regime.
We also illustrate the performance of our approach on a newsgroup dataset and
on a dataset consisting of financial asset prices.Comment: 34 pages, 1 figur
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