1,155 research outputs found
Latent tree models
Latent tree models are graphical models defined on trees, in which only a
subset of variables is observed. They were first discussed by Judea Pearl as
tree-decomposable distributions to generalise star-decomposable distributions
such as the latent class model. Latent tree models, or their submodels, are
widely used in: phylogenetic analysis, network tomography, computer vision,
causal modeling, and data clustering. They also contain other well-known
classes of models like hidden Markov models, Brownian motion tree model, the
Ising model on a tree, and many popular models used in phylogenetics. This
article offers a concise introduction to the theory of latent tree models. We
emphasise the role of tree metrics in the structural description of this model
class, in designing learning algorithms, and in understanding fundamental
limits of what and when can be learned
Calibrating Generative Models: The Probabilistic Chomsky-Schützenberger Hierarchy
A probabilistic Chomsky–Schützenberger hierarchy of grammars is introduced and studied, with the aim of understanding the expressive power of generative models. We offer characterizations of the distributions definable at each level of the hierarchy, including probabilistic regular, context-free, (linear) indexed, context-sensitive, and unrestricted grammars, each corresponding to familiar probabilistic machine classes. Special attention is given to distributions on (unary notations for) positive integers. Unlike in the classical case where the "semi-linear" languages all collapse into the regular languages, using analytic tools adapted from the classical setting we show there is no collapse in the probabilistic hierarchy: more distributions become definable at each level. We also address related issues such as closure under probabilistic conditioning
TVL<sub>1</sub> Planarity Regularization for 3D Shape Approximation
The modern emergence of automation in many industries has given impetus to extensive research into mobile robotics. Novel perception technologies now enable cars to drive autonomously, tractors to till a field automatically and underwater robots to construct pipelines. An essential requirement to facilitate both perception and autonomous navigation is the analysis of the 3D environment using sensors like laser scanners or stereo cameras. 3D sensors generate a very large number of 3D data points when sampling object shapes within an environment, but crucially do not provide any intrinsic information about the environment which the robots operate within.
This work focuses on the fundamental task of 3D shape reconstruction and modelling from 3D point clouds. The novelty lies in the representation of surfaces by algebraic functions having limited support, which enables the extraction of smooth consistent implicit shapes from noisy samples with a heterogeneous density. The minimization of total variation of second differential degree makes it possible to enforce planar surfaces which often occur in man-made environments. Applying the new technique means that less accurate, low-cost 3D sensors can be employed without sacrificing the 3D shape reconstruction accuracy
Tensor network and (-adic) AdS/CFT
We use the tensor network living on the Bruhat-Tits tree to give a concrete
realization of the recently proposed -adic AdS/CFT correspondence (a
holographic duality based on the -adic number field ). Instead
of assuming the -adic AdS/CFT correspondence, we show how important features
of AdS/CFT such as the bulk operator reconstruction and the holographic
computation of boundary correlators are automatically implemented in this
tensor network.Comment: 59 pages, 18 figures; v3: improved presentation, added figures and
reference
Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations
It is well-known that any sum of squares (SOS) program can be cast as a
semidefinite program (SDP) of a particular structure and that therein lies the
computational bottleneck for SOS programs, as the SDPs generated by this
procedure are large and costly to solve when the polynomials involved in the
SOS programs have a large number of variables and degree. In this paper, we
review SOS optimization techniques and present two new methods for improving
their computational efficiency. The first method leverages the sparsity of the
underlying SDP to obtain computational speed-ups. Further improvements can be
obtained if the coefficients of the polynomials that describe the problem have
a particular sparsity pattern, called chordal sparsity. The second method
bypasses semidefinite programming altogether and relies instead on solving a
sequence of more tractable convex programs, namely linear and second order cone
programs. This opens up the question as to how well one can approximate the
cone of SOS polynomials by second order representable cones. In the last part
of the paper, we present some recent negative results related to this question.Comment: Tutorial for CDC 201
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