4,473 research outputs found
Fast Exact Inference for Recursive Cardinality Models
Cardinality potentials are a generally useful class of high order potential that affect probabilities based on how many of D binary variables are active. Maximum a posteriori (MAP) inference for cardinality potential models is well-understood, with efficient computations taking O(D log D) time. Yet efficient marginalization and sampling have not been addressed as thoroughly in the machine learning community. We show that there exists a simple algorithm for computing marginal probabilities and drawing exact joint samples that runs in O(D log2 D) time, and we show how to frame the algorithm as efficient belief propagation in a low order tree-structured model that includes additional auxiliary variables. We then develop a new, more general class of models, termed Recursive Cardinality models, which take advantage of this efficiency. Finally, we show how to do efficient exact inference in models composed of a tree structure and a cardinality potential. We explore the expressive power of Recursive Cardinality models and empirically demonstrate their utility.Engineering and Applied Science
WARP: Wavelets with adaptive recursive partitioning for multi-dimensional data
Effective identification of asymmetric and local features in images and other
data observed on multi-dimensional grids plays a critical role in a wide range
of applications including biomedical and natural image processing. Moreover,
the ever increasing amount of image data, in terms of both the resolution per
image and the number of images processed per application, requires algorithms
and methods for such applications to be computationally efficient. We develop a
new probabilistic framework for multi-dimensional data to overcome these
challenges through incorporating data adaptivity into discrete wavelet
transforms, thereby allowing them to adapt to the geometric structure of the
data while maintaining the linear computational scalability. By exploiting a
connection between the local directionality of wavelet transforms and recursive
dyadic partitioning on the grid points of the observation, we obtain the
desired adaptivity through adding to the traditional Bayesian wavelet
regression framework an additional layer of Bayesian modeling on the space of
recursive partitions over the grid points. We derive the corresponding
inference recipe in the form of a recursive representation of the exact
posterior, and develop a class of efficient recursive message passing
algorithms for achieving exact Bayesian inference with a computational
complexity linear in the resolution and sample size of the images. While our
framework is applicable to a range of problems including multi-dimensional
signal processing, compression, and structural learning, we illustrate its work
and evaluate its performance in the context of 2D and 3D image reconstruction
using real images from the ImageNet database. We also apply the framework to
analyze a data set from retinal optical coherence tomography
On the number of ranked species trees producing anomalous ranked gene trees
Analysis of probability distributions conditional on species trees has
demonstrated the existence of anomalous ranked gene trees (ARGTs), ranked gene
trees that are more probable than the ranked gene tree that accords with the
ranked species tree. Here, to improve the characterization of ARGTs, we study
enumerative and probabilistic properties of two classes of ranked labeled
species trees, focusing on the presence or avoidance of certain subtree
patterns associated with the production of ARGTs. We provide exact enumerations
and asymptotic estimates for cardinalities of these sets of trees, showing that
as the number of species increases without bound, the fraction of all ranked
labeled species trees that are ARGT-producing approaches 1. This result extends
beyond earlier existence results to provide a probabilistic claim about the
frequency of ARGTs
A Novel Convex Relaxation for Non-Binary Discrete Tomography
We present a novel convex relaxation and a corresponding inference algorithm
for the non-binary discrete tomography problem, that is, reconstructing
discrete-valued images from few linear measurements. In contrast to state of
the art approaches that split the problem into a continuous reconstruction
problem for the linear measurement constraints and a discrete labeling problem
to enforce discrete-valued reconstructions, we propose a joint formulation that
addresses both problems simultaneously, resulting in a tighter convex
relaxation. For this purpose a constrained graphical model is set up and
evaluated using a novel relaxation optimized by dual decomposition. We evaluate
our approach experimentally and show superior solutions both mathematically
(tighter relaxation) and experimentally in comparison to previously proposed
relaxations
Correlation Decay in Random Decision Networks
We consider a decision network on an undirected graph in which each node
corresponds to a decision variable, and each node and edge of the graph is
associated with a reward function whose value depends only on the variables of
the corresponding nodes. The goal is to construct a decision vector which
maximizes the total reward. This decision problem encompasses a variety of
models, including maximum-likelihood inference in graphical models (Markov
Random Fields), combinatorial optimization on graphs, economic team theory and
statistical physics. The network is endowed with a probabilistic structure in
which costs are sampled from a distribution. Our aim is to identify sufficient
conditions to guarantee average-case polynomiality of the underlying
optimization problem. We construct a new decentralized algorithm called Cavity
Expansion and establish its theoretical performance for a variety of models.
Specifically, for certain classes of models we prove that our algorithm is able
to find near optimal solutions with high probability in a decentralized way.
The success of the algorithm is based on the network exhibiting a correlation
decay (long-range independence) property. Our results have the following
surprising implications in the area of average case complexity of algorithms.
Finding the largest independent (stable) set of a graph is a well known NP-hard
optimization problem for which no polynomial time approximation scheme is
possible even for graphs with largest connectivity equal to three, unless P=NP.
We show that the closely related maximum weighted independent set problem for
the same class of graphs admits a PTAS when the weights are i.i.d. with the
exponential distribution. Namely, randomization of the reward function turns an
NP-hard problem into a tractable one
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