3,843 research outputs found
A robust spectral method for finding lumpings and meta stable states of non-reversible Markov chains
A spectral method for identifying lumping in large Markov chains is
presented. Identification of meta stable states is treated as a special case.
The method is based on spectral analysis of a self-adjoint matrix that is a
function of the original transition matrix. It is demonstrated that the
technique is more robust than existing methods when applied to noisy
non-reversible Markov chains.Comment: 10 pages, 7 figure
Geometry of the ergodic quotient reveals coherent structures in flows
Dynamical systems that exhibit diverse behaviors can rarely be completely
understood using a single approach. However, by identifying coherent structures
in their state spaces, i.e., regions of uniform and simpler behavior, we could
hope to study each of the structures separately and then form the understanding
of the system as a whole. The method we present in this paper uses trajectory
averages of scalar functions on the state space to: (a) identify invariant sets
in the state space, (b) form coherent structures by aggregating invariant sets
that are similar across multiple spatial scales. First, we construct the
ergodic quotient, the object obtained by mapping trajectories to the space of
trajectory averages of a function basis on the state space. Second, we endow
the ergodic quotient with a metric structure that successfully captures how
similar the invariant sets are in the state space. Finally, we parametrize the
ergodic quotient using intrinsic diffusion modes on it. By segmenting the
ergodic quotient based on the diffusion modes, we extract coherent features in
the state space of the dynamical system. The algorithm is validated by
analyzing the Arnold-Beltrami-Childress flow, which was the test-bed for
alternative approaches: the Ulam's approximation of the transfer operator and
the computation of Lagrangian Coherent Structures. Furthermore, we explain how
the method extends the Poincar\'e map analysis for periodic flows. As a
demonstration, we apply the method to a periodically-driven three-dimensional
Hill's vortex flow, discovering unknown coherent structures in its state space.
In the end, we discuss differences between the ergodic quotient and
alternatives, propose a generalization to analysis of (quasi-)periodic
structures, and lay out future research directions.Comment: Submitted to Elsevier Physica D: Nonlinear Phenomen
Temporal Feature Selection with Symbolic Regression
Building and discovering useful features when constructing machine learning models is the central task for the machine learning practitioner. Good features are useful not only in increasing the predictive power of a model but also in illuminating the underlying drivers of a target variable. In this research we propose a novel feature learning technique in which Symbolic regression is endowed with a ``Range Terminal\u27\u27 that allows it to explore functions of the aggregate of variables over time. We test the Range Terminal on a synthetic data set and a real world data in which we predict seasonal greenness using satellite derived temperature and snow data over a portion of the Arctic. On the synthetic data set we find Symbolic regression with the Range Terminal outperforms standard Symbolic regression and Lasso regression. On the Arctic data set we find it outperforms standard Symbolic regression, fails to beat the Lasso regression, but finds useful features describing the interaction between Land Surface Temperature, Snow, and seasonal vegetative growth in the Arctic
Unsupervised Object Discovery and Localization in the Wild: Part-based Matching with Bottom-up Region Proposals
This paper addresses unsupervised discovery and localization of dominant
objects from a noisy image collection with multiple object classes. The setting
of this problem is fully unsupervised, without even image-level annotations or
any assumption of a single dominant class. This is far more general than
typical colocalization, cosegmentation, or weakly-supervised localization
tasks. We tackle the discovery and localization problem using a part-based
region matching approach: We use off-the-shelf region proposals to form a set
of candidate bounding boxes for objects and object parts. These regions are
efficiently matched across images using a probabilistic Hough transform that
evaluates the confidence for each candidate correspondence considering both
appearance and spatial consistency. Dominant objects are discovered and
localized by comparing the scores of candidate regions and selecting those that
stand out over other regions containing them. Extensive experimental
evaluations on standard benchmarks demonstrate that the proposed approach
significantly outperforms the current state of the art in colocalization, and
achieves robust object discovery in challenging mixed-class datasets.Comment: CVPR 201
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