60 research outputs found
Subspace Evolution and Transfer (SET) for Low-Rank Matrix Completion
We describe a new algorithm, termed subspace evolution and transfer (SET),
for solving low-rank matrix completion problems. The algorithm takes as its
input a subset of entries of a low-rank matrix, and outputs one low-rank matrix
consistent with the given observations. The completion task is accomplished by
searching for a column space on the Grassmann manifold that matches the
incomplete observations. The SET algorithm consists of two parts -- subspace
evolution and subspace transfer. In the evolution part, we use a gradient
descent method on the Grassmann manifold to refine our estimate of the column
space. Since the gradient descent algorithm is not guaranteed to converge, due
to the existence of barriers along the search path, we design a new mechanism
for detecting barriers and transferring the estimated column space across the
barriers. This mechanism constitutes the core of the transfer step of the
algorithm. The SET algorithm exhibits excellent empirical performance for both
high and low sampling rate regimes
Deformable and articulated 3D reconstruction from monocular video sequences
PhDThis thesis addresses the problem of deformable and articulated structure from motion from
monocular uncalibrated video sequences. Structure from motion is defined as the problem of
recovering information about the 3D structure of scenes imaged by a camera in a video sequence.
Our study aims at the challenging problem of non-rigid shapes (e.g. a beating heart or a smiling
face). Non-rigid structures appear constantly in our everyday life, think of a bicep curling, a
torso twisting or a smiling face. Our research seeks a general method to perform 3D shape
recovery purely from data, without having to rely on a pre-computed model or training data.
Open problems in the field are the difficulty of the non-linear estimation, the lack of a real-time
system, large amounts of missing data in real-world video sequences, measurement noise and
strong deformations. Solving these problems would take us far beyond the current state of the
art in non-rigid structure from motion. This dissertation presents our contributions in the field
of non-rigid structure from motion, detailing a novel algorithm that enforces the exact metric
structure of the problem at each step of the minimisation by projecting the motion matrices
onto the correct deformable or articulated metric motion manifolds respectively. An important
advantage of this new algorithm is its ability to handle missing data which becomes crucial
when dealing with real video sequences. We present a generic bilinear estimation framework,
which improves convergence and makes use of the manifold constraints. Finally, we demonstrate
a sequential, frame-by-frame estimation algorithm, which provides a 3D model and camera
parameters for each video frame, while simultaneously building a model of object deformation
Structure from sound with incomplete data
In this paper, we consider the problem of jointly localizing a microphone array and identifying the direction of arrival of acoustic events. Under the assumption that the sources are in the far field, this problem can be formulated as a constrained low-rank matrix factorization with an unknown column offset. Our focus is on handling missing entries, particularly when the measurement matrix does not contain a single complete column. This case has not received attention in the literature and is not handled by existing algorithms, however it is prevalent in practice. We propose an iterative algorithm that works with pairwise differences between the measurements eliminating the dependence on the unknown offset. We demonstrate state-of-the-art performance both in terms of accuracy and versatility
Guarantees of Riemannian Optimization for Low Rank Matrix Completion
We study the Riemannian optimization methods on the embedded manifold of low
rank matrices for the problem of matrix completion, which is about recovering a
low rank matrix from its partial entries. Assume entries of an
rank matrix are sampled independently and uniformly with replacement. We
first prove that with high probability the Riemannian gradient descent and
conjugate gradient descent algorithms initialized by one step hard thresholding
are guaranteed to converge linearly to the measured matrix provided
\begin{align*} m\geq C_\kappa n^{1.5}r\log^{1.5}(n), \end{align*} where
is a numerical constant depending on the condition number of the
underlying matrix. The sampling complexity has been further improved to
\begin{align*} m\geq C_\kappa nr^2\log^{2}(n) \end{align*} via the resampled
Riemannian gradient descent initialization. The analysis of the new
initialization procedure relies on an asymmetric restricted isometry property
of the sampling operator and the curvature of the low rank matrix manifold.
Numerical simulation shows that the algorithms are able to recover a low rank
matrix from nearly the minimum number of measurements
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