671 research outputs found
The Role of Riemannian Manifolds in Computer Vision: From Coding to Deep Metric Learning
A diverse number of tasks in computer vision and machine learning
enjoy from representations of data that are compact yet
discriminative, informative and robust to critical measurements.
Two notable representations are offered by Region Covariance
Descriptors (RCovD) and linear subspaces which are naturally
analyzed through the manifold of Symmetric Positive Definite
(SPD) matrices and the Grassmann manifold, respectively, two
widely used types of Riemannian manifolds in computer vision.
As our first objective, we examine image and video-based
recognition applications where the local descriptors have the
aforementioned Riemannian structures, namely the SPD or linear
subspace structure. Initially, we provide a solution to compute
Riemannian version of the conventional Vector of Locally
aggregated Descriptors (VLAD), using geodesic distance of the
underlying manifold as the nearness measure. Next, by having a
closer look at the resulting codes, we formulate a new concept
which we name Local Difference Vectors (LDV). LDVs enable us to
elegantly expand our Riemannian coding techniques to any
arbitrary metric as well as provide intrinsic solutions to
Riemannian sparse coding and its variants when local structured
descriptors are considered.
We then turn our attention to two special types of covariance
descriptors namely infinite-dimensional RCovDs and rank-deficient
covariance matrices for which the underlying Riemannian
structure, i.e. the manifold of SPD matrices is out of reach to
great extent. %Generally speaking, infinite-dimensional RCovDs
offer better discriminatory power over their low-dimensional
counterparts.
To overcome this difficulty, we propose to approximate the
infinite-dimensional RCovDs by making use of two feature
mappings, namely random Fourier features and the Nystrom method.
As for the rank-deficient covariance matrices, unlike most
existing approaches that employ inference tools by predefined
regularizers, we derive positive definite kernels that can be
decomposed into the kernels on the cone of SPD matrices and
kernels on the Grassmann manifolds and show their effectiveness
for image set classification task.
Furthermore, inspired by attractive properties of Riemannian
optimization techniques, we extend the recently introduced Keep
It Simple and Straightforward MEtric learning (KISSME) method to
the scenarios where input data is non-linearly distributed. To
this end, we make use of the infinite dimensional covariance
matrices and propose techniques towards projecting on the
positive cone in a Reproducing Kernel Hilbert Space (RKHS).
We also address the sensitivity issue of the KISSME to the input
dimensionality. The KISSME algorithm is greatly dependent on
Principal Component Analysis (PCA) as a preprocessing step which
can lead to difficulties, especially when the dimensionality is
not meticulously set.
To address this issue, based on the KISSME algorithm, we develop
a Riemannian framework to jointly learn a mapping performing
dimensionality reduction and a metric in the induced space.
Lastly, in line with the recent trend in metric learning, we
devise end-to-end learning of a generic deep network for metric
learning using our derivation
Statistical and Dynamical Modeling of Riemannian Trajectories with Application to Human Movement Analysis
abstract: The data explosion in the past decade is in part due to the widespread use of rich sensors that measure various physical phenomenon -- gyroscopes that measure orientation in phones and fitness devices, the Microsoft Kinect which measures depth information, etc. A typical application requires inferring the underlying physical phenomenon from data, which is done using machine learning. A fundamental assumption in training models is that the data is Euclidean, i.e. the metric is the standard Euclidean distance governed by the L-2 norm. However in many cases this assumption is violated, when the data lies on non Euclidean spaces such as Riemannian manifolds. While the underlying geometry accounts for the non-linearity, accurate analysis of human activity also requires temporal information to be taken into account. Human movement has a natural interpretation as a trajectory on the underlying feature manifold, as it evolves smoothly in time. A commonly occurring theme in many emerging problems is the need to \emph{represent, compare, and manipulate} such trajectories in a manner that respects the geometric constraints. This dissertation is a comprehensive treatise on modeling Riemannian trajectories to understand and exploit their statistical and dynamical properties. Such properties allow us to formulate novel representations for Riemannian trajectories. For example, the physical constraints on human movement are rarely considered, which results in an unnecessarily large space of features, making search, classification and other applications more complicated. Exploiting statistical properties can help us understand the \emph{true} space of such trajectories. In applications such as stroke rehabilitation where there is a need to differentiate between very similar kinds of movement, dynamical properties can be much more effective. In this regard, we propose a generalization to the Lyapunov exponent to Riemannian manifolds and show its effectiveness for human activity analysis. The theory developed in this thesis naturally leads to several benefits in areas such as data mining, compression, dimensionality reduction, classification, and regression.Dissertation/ThesisDoctoral Dissertation Electrical Engineering 201
Minimal Curvature Trajectories: Riemannian Geometry Concepts for Model Reduction in Chemical Kinetics
In dissipative ordinary differential equation systems different time scales
cause anisotropic phase volume contraction along solution trajectories. Model
reduction methods exploit this for simplifying chemical kinetics via a time
scale separation into fast and slow modes. The aim is to approximate the system
dynamics with a dimension-reduced model after eliminating the fast modes by
enslaving them to the slow ones via computation of a slow attracting manifold.
We present a novel method for computing approximations of such manifolds using
trajectory-based optimization. We discuss Riemannian geometry concepts as a
basis for suitable optimization criteria characterizing trajectories near slow
attracting manifolds and thus provide insight into fundamental geometric
properties of multiple time scale chemical kinetics. The optimization criteria
correspond to a suitable mathematical formulation of "minimal relaxation" of
chemical forces along reaction trajectories under given constraints. We present
various geometrically motivated criteria and the results of their application
to three test case reaction mechanisms serving as examples. We demonstrate that
accurate numerical approximations of slow invariant manifolds can be obtained.Comment: 22 pages, 18 figure
Bayesian Inference on Matrix Manifolds for Linear Dimensionality Reduction
We reframe linear dimensionality reduction as a problem of Bayesian inference
on matrix manifolds. This natural paradigm extends the Bayesian framework to
dimensionality reduction tasks in higher dimensions with simpler models at
greater speeds. Here an orthogonal basis is treated as a single point on a
manifold and is associated with a linear subspace on which observations vary
maximally. Throughout this paper, we employ the Grassmann and Stiefel manifolds
for various dimensionality reduction problems, explore the connection between
the two manifolds, and use Hybrid Monte Carlo for posterior sampling on the
Grassmannian for the first time. We delineate in which situations either
manifold should be considered. Further, matrix manifold models are used to
yield scientific insight in the context of cognitive neuroscience, and we
conclude that our methods are suitable for basic inference as well as accurate
prediction.Comment: All datasets and computer programs are publicly available at
http://www.ics.uci.edu/~babaks/Site/Codes.htm
Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)
The implicit objective of the biennial "international - Traveling Workshop on
Interactions between Sparse models and Technology" (iTWIST) is to foster
collaboration between international scientific teams by disseminating ideas
through both specific oral/poster presentations and free discussions. For its
second edition, the iTWIST workshop took place in the medieval and picturesque
town of Namur in Belgium, from Wednesday August 27th till Friday August 29th,
2014. The workshop was conveniently located in "The Arsenal" building within
walking distance of both hotels and town center. iTWIST'14 has gathered about
70 international participants and has featured 9 invited talks, 10 oral
presentations, and 14 posters on the following themes, all related to the
theory, application and generalization of the "sparsity paradigm":
Sparsity-driven data sensing and processing; Union of low dimensional
subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph
sensing/processing; Blind inverse problems and dictionary learning; Sparsity
and computational neuroscience; Information theory, geometry and randomness;
Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?;
Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website:
http://sites.google.com/site/itwist1
- …