122 research outputs found
Inference and Model Parameter Learning for Image Labeling by Geometric Assignment
Image labeling is a fundamental problem in the area of low-level image analysis. In this work, we present novel approaches to maximum a posteriori (MAP) inference and model
parameter learning for image labeling, respectively. Both approaches are formulated in a smooth geometric setting, whose respective solution space is a simple Riemannian manifold. Optimization
consists of multiplicative updates that geometrically integrate the resulting Riemannian gradient flow.
Our novel approach to MAP inference is based on discrete graphical models. By utilizing local Wasserstein distances for coupling assignment measures across edges of the
underlying graph, we smoothly approximate a given discrete objective function and restrict it to the
assignment manifold. A corresponding update scheme combines geometric integration of the resulting gradient flow, and rounding to integral solutions that represent
valid labelings. This formulation constitutes an inner relaxation of the discrete labeling problem, i.e. throughout this process local marginalization constraints known from the established linear programming relaxation are satisfied.
Furthermore, we study the inverse problem of model parameter learning using the linear assignment flow and training data with ground truth. This is accomplished by a Riemannian gradient flow on the manifold of parameters that determine the regularization properties of the assignment flow. This smooth formulation enables us to tackle the model parameter learning problem from the perspective of parameter estimation of dynamical systems. By using symplectic partitioned Runge--Kutta methods for numerical integration, we show that deriving the sensitivity conditions of the parameter learning problem and its discretization commute. A favorable property of our approach is that learning is based on exact inference
Variational Approaches for Image Labeling on the Assignment Manifold
The image labeling problem refers to the task of assigning to each pixel a single element from a finite predefined set of labels. In classical approaches the labeling task is formulated as a minimization problem of specifically structured objective functions.
Assignment flows for contextual image labeling are a recently proposed alternative formulation via spatially coupled replicator equations.
In this work, the classical and dynamical viewpoint of image labeling are combined into a variational formulation. This is accomplished by following the induced Riemannian gradient descent flow on an elementary statistical manifold with respect to the underlying information geometry.
Convergence and stability behavior of this approach are investigated using the log-barrier method. A novel parameterization of the assignment flow by its dominant component is derived, revealing a Riemannian gradient flow structure that clearly identifies the two governing processes of the flow: spatial regularization of assignments and gradual enforcement of unambiguous label decisions. Also, a continuous-domain formulation of the corresponding potential is presented and well-posedness of the related optimization problem is established. Furthermore, an alternative smooth variational approach to maximum a-posteriori inference based on discrete graphical models is derived by utilizing local Wasserstein distances. Following the resulting Riemannian gradient flow leads to an inference process which always satisfies the local marginalization constraints and incorporates a smooth rounding mechanism towards unambiguous assignments
New Convex Relaxations and Global Optimality in Variational Imaging
Variational methods constitute the basic building blocks for solving many image analysis tasks, be it segmentation, depth estimation, optical flow, object detection etc. Many of these problems can be expressed in the framework of Markov Random Fields (MRF) or as continuous labelling problems. Finding the Maximum A-Posteriori (MAP) solutions of suitably constructed MRFs or the optimizers of the labelling problems give solutions to the aforementioned tasks. In either case, the associated optimization problem amounts to solving structured energy minimization problems.
In this thesis we study novel extensions applicable to Markov Random Fields and continuous labelling problems through which we are able to incorporate statistical global constraints. To this end, we devise tractable relaxations of the resulting energy minimization problem and efficient algorithms to tackle them. Second, we propose a general mechanism to find partial optimal solutions to the problem of finding a MAP-solution of an MRF, utilizing only standard relxations
Geometric Numerical Integration of the Assignment Flow
The assignment flow is a smooth dynamical system that evolves on an
elementary statistical manifold and performs contextual data labeling on a
graph. We derive and introduce the linear assignment flow that evolves
nonlinearly on the manifold, but is governed by a linear ODE on the tangent
space. Various numerical schemes adapted to the mathematical structure of these
two models are designed and studied, for the geometric numerical integration of
both flows: embedded Runge-Kutta-Munthe-Kaas schemes for the nonlinear flow,
adaptive Runge-Kutta schemes and exponential integrators for the linear flow.
All algorithms are parameter free, except for setting a tolerance value that
specifies adaptive step size selection by monitoring the local integration
error, or fixing the dimension of the Krylov subspace approximation. These
algorithms provide a basis for applying the assignment flow to machine learning
scenarios beyond supervised labeling, including unsupervised labeling and
learning from controlled assignment flows
Algebraic and Geometric Models for Space Networking
In this paper we introduce some new algebraic and geometric perspectives on
networked space communications. Our main contribution is a novel definition of
a time-varying graph (TVG), defined in terms of a matrix with values in subsets
of the real line P(R). We leverage semi-ring properties of P(R) to model
multi-hop communication in a TVG using matrix multiplication and a truncated
Kleene star. This leads to novel statistics on the communication capacity of
TVGs called lifetime curves, which we generate for large samples of randomly
chosen STARLINK satellites, whose connectivity is modeled over day-long
simulations. Determining when a large subsample of STARLINK is temporally
strongly connected is further analyzed using novel metrics introduced here that
are inspired by topological data analysis (TDA). To better model networking
scenarios between the Earth and Mars, we introduce various semi-rings capable
of modeling propagation delay as well as protocols common to Delay Tolerant
Networking (DTN), such as store-and-forward. Finally, we illustrate the
applicability of zigzag persistence for featurizing different space networks
and demonstrate the efficacy of K-Nearest Neighbors (KNN) classification for
distinguishing Earth-Mars and Earth-Moon satellite systems using time-varying
topology alone.Comment: 43 pages, 18 figures, comments welcom
A Survey on Deep Semi-supervised Learning
Deep semi-supervised learning is a fast-growing field with a range of
practical applications. This paper provides a comprehensive survey on both
fundamentals and recent advances in deep semi-supervised learning methods from
model design perspectives and unsupervised loss functions. We first present a
taxonomy for deep semi-supervised learning that categorizes existing methods,
including deep generative methods, consistency regularization methods,
graph-based methods, pseudo-labeling methods, and hybrid methods. Then we offer
a detailed comparison of these methods in terms of the type of losses,
contributions, and architecture differences. In addition to the past few years'
progress, we further discuss some shortcomings of existing methods and provide
some tentative heuristic solutions for solving these open problems.Comment: 24 pages, 6 figure
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