300 research outputs found
Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces II: non-compact symmetric spaces
Gaussian processes are arguably the most important class of spatiotemporal
models within machine learning. They encode prior information about the modeled
function and can be used for exact or approximate Bayesian learning. In many
applications, particularly in physical sciences and engineering, but also in
areas such as geostatistics and neuroscience, invariance to symmetries is one
of the most fundamental forms of prior information one can consider. The
invariance of a Gaussian process' covariance to such symmetries gives rise to
the most natural generalization of the concept of stationarity to such spaces.
In this work, we develop constructive and practical techniques for building
stationary Gaussian processes on a very large class of non-Euclidean spaces
arising in the context of symmetries. Our techniques make it possible to (i)
calculate covariance kernels and (ii) sample from prior and posterior Gaussian
processes defined on such spaces, both in a practical manner. This work is
split into two parts, each involving different technical considerations: part I
studies compact spaces, while part II studies non-compact spaces possessing
certain structure. Our contributions make the non-Euclidean Gaussian process
models we study compatible with well-understood computational techniques
available in standard Gaussian process software packages, thereby making them
accessible to practitioners
Positive Semidefinite Metric Learning with Boosting
The learning of appropriate distance metrics is a critical problem in image
classification and retrieval. In this work, we propose a boosting-based
technique, termed \BoostMetric, for learning a Mahalanobis distance metric. One
of the primary difficulties in learning such a metric is to ensure that the
Mahalanobis matrix remains positive semidefinite. Semidefinite programming is
sometimes used to enforce this constraint, but does not scale well.
\BoostMetric is instead based on a key observation that any positive
semidefinite matrix can be decomposed into a linear positive combination of
trace-one rank-one matrices. \BoostMetric thus uses rank-one positive
semidefinite matrices as weak learners within an efficient and scalable
boosting-based learning process. The resulting method is easy to implement,
does not require tuning, and can accommodate various types of constraints.
Experiments on various datasets show that the proposed algorithm compares
favorably to those state-of-the-art methods in terms of classification accuracy
and running time.Comment: 11 pages, Twenty-Third Annual Conference on Neural Information
Processing Systems (NIPS 2009), Vancouver, Canad
Positive Semidefinite Metric Learning Using Boosting-like Algorithms
The success of many machine learning and pattern recognition methods relies
heavily upon the identification of an appropriate distance metric on the input
data. It is often beneficial to learn such a metric from the input training
data, instead of using a default one such as the Euclidean distance. In this
work, we propose a boosting-based technique, termed BoostMetric, for learning a
quadratic Mahalanobis distance metric. Learning a valid Mahalanobis distance
metric requires enforcing the constraint that the matrix parameter to the
metric remains positive definite. Semidefinite programming is often used to
enforce this constraint, but does not scale well and easy to implement.
BoostMetric is instead based on the observation that any positive semidefinite
matrix can be decomposed into a linear combination of trace-one rank-one
matrices. BoostMetric thus uses rank-one positive semidefinite matrices as weak
learners within an efficient and scalable boosting-based learning process. The
resulting methods are easy to implement, efficient, and can accommodate various
types of constraints. We extend traditional boosting algorithms in that its
weak learner is a positive semidefinite matrix with trace and rank being one
rather than a classifier or regressor. Experiments on various datasets
demonstrate that the proposed algorithms compare favorably to those
state-of-the-art methods in terms of classification accuracy and running time.Comment: 30 pages, appearing in Journal of Machine Learning Researc
State Space Approaches for Modeling Activities in Video Streams
The objective is to discern events and behavior in activities using video sequences, which conform to common human experience. It has several applications such as recognition, temporal segmentation, video indexing and anomaly detection. Activity modeling offers compelling challenges to computational vision systems at several levels ranging from low-level vision tasks for detection and segmentation to high-level models for extracting perceptually salient information. With a focus on the latter, the following approaches are presented: event detection in discrete state space, epitomic representation in continuous state space, temporal segmentation using mixed state models, key frame detection using antieigenvalues and spatio-temporal activity volumes.
Significant changes in motion properties are said to be events. We present an event probability sequence representation in which the probability of event occurrence is computed using stable changes at the state level of the discrete state hidden Markov model that generates the observed trajectories. Reliance on a trained model however, can be a limitation. A data-driven antieigenvalue-based approach is proposed for detecting changes. Antieigenvalues are sensitive to turnings whereas eigenvalues capture directions of maximum variance in the data. In both these approaches, events are assumed to be instantaneous quantities. This is relaxed using an epitomic representation in continuous state space.
Video sequences are segmented using a sliding window within which the dynamics of each object is assumed to be linear. The system matrix, initial state value and the input signal statistics are said to form an epitome. The system matrices are decomposed using the Iwasawa matrix decomposition to isolate the effect of rotation, scaling and projection of the state vector. It is used to compute physically meaningful distances between epitomes. Epitomes reveal dominant primitives of activities that have an abstracted interpretation. A mixed state approach for activities is presented in which higher-level primitives of behavior is encoded in the discrete state component and observed dynamics in the continuous state component. The effectiveness of mixed state models is demonstrated using temporal segmentation. In addition to motion trajectories, the volume carved out in an xyt cube by a moving object is characterized using Morse functions
Modelling, Measuring and Compensating Color Weak Vision
We use methods from Riemann geometry to investigate transformations between
the color spaces of color-normal and color weak observers. The two main
applications are the simulation of the perception of a color weak observer for
a color normal observer and the compensation of color images in a way that a
color weak observer has approximately the same perception as a color normal
observer. The metrics in the color spaces of interest are characterized with
the help of ellipsoids defined by the just-noticable-differences between color
which are measured with the help of color-matching experiments. The constructed
mappings are isometries of Riemann spaces that preserve the perceived
color-differences for both observers. Among the two approaches to build such an
isometry, we introduce normal coordinates in Riemann spaces as a tool to
construct a global color-weak compensation map. Compared to previously used
methods this method is free from approximation errors due to local
linearizations and it avoids the problem of shifting locations of the origin of
the local coordinate system. We analyse the variations of the Riemann metrics
for different observers obtained from new color matching experiments and
describe three variations of the basic method. The performance of the methods
is evaluated with the help of semantic differential (SD) tests.Comment: Full resolution color pictures are available from the author
The microscopic dynamics of quantum space as a group field theory
We provide a rather extended introduction to the group field theory approach
to quantum gravity, and the main ideas behind it. We present in some detail the
GFT quantization of 3d Riemannian gravity, and discuss briefly the current
status of the 4-dimensional extensions of this construction. We also briefly
report on recent results obtained in this approach and related open issues,
concerning both the mathematical definition of GFT models, and possible avenues
towards extracting interesting physics from them.Comment: 60 pages. Extensively revised version of the contribution to
"Foundations of Space and Time: Reflections on Quantum Gravity", edited by G.
Ellis, J. Murugan, A. Weltman, published by Cambridge University Pres
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