27 research outputs found
Variational methods and its applications to computer vision
Many computer vision applications such as image segmentation can be formulated in a ''variational'' way as energy minimization problems. Unfortunately, the computational task of minimizing these energies is usually difficult as it generally involves non convex functions in a space with thousands of dimensions and often the associated combinatorial problems are NP-hard to solve. Furthermore, they are ill-posed inverse problems and therefore are extremely sensitive to perturbations (e.g. noise). For this reason in order to compute a physically reliable approximation from given noisy data, it is necessary to incorporate into the mathematical model appropriate regularizations that require complex computations.
The main aim of this work is to describe variational segmentation methods that are particularly effective for curvilinear structures. Due to their complex geometry, classical regularization techniques cannot be adopted because they lead to the loss of most of low contrasted details. In contrast, the proposed method not only better preserves curvilinear structures, but also reconnects some parts that may have been disconnected by noise. Moreover, it can be easily extensible to graphs and successfully applied to different types of data such as medical imagery (i.e. vessels, hearth coronaries etc), material samples (i.e. concrete) and satellite signals (i.e. streets, rivers etc.). In particular, we will show results and performances about an implementation targeting new generation of High Performance Computing (HPC) architectures where different types of coprocessors cooperate. The involved dataset consists of approximately 200 images of cracks, captured in three different tunnels by a robotic machine designed for the European ROBO-SPECT project.Open Acces
Loss Scaling and Step Size in Deep Learning Optimizatio
Deep learning training consumes ever-increasing time and resources, and that isdue to the complexity of the model, the number of updates taken to reach goodresults, and both the amount and dimensionality of the data. In this dissertation,we will focus on making the process of training more efficient by focusing on thestep size to reduce the number of computations for parameters in each update.We achieved our objective in two new ways: we use loss scaling as a proxy forthe learning rate, and we use learnable layer-wise optimizers. Although our workis perhaps not the first to point to the equivalence of loss scaling and learningrate in deep learning optimization, ours is the first to leveraging this relationshiptowards more efficient training. We did not only use it in simple gradient descent,but also we were able to extend it to other adaptive algorithms. Finally, we usemetalearning to shed light on relevant aspects, including learnable lossesand optimizers. In this regard, we developed a novel learnable optimizer andeffectively utilized it to acquire an adaptive rescaling factor and learning rate,resulting in a significant reduction in required memory during training
Communication-constrained distributed quantile regression with optimal statistical guarantees
We address the problem of how to achieve optimal inference in distributed quantile regression without stringent scaling conditions. This is challenging due to the non-smooth nature of the quantile regression (QR) loss function, which invalidates the use of existing methodology. The difficulties are resolved through a double-smoothing approach that is applied to the local (at each data source) and global objective functions. Despite the reliance on a delicate combination of local and global smoothing parameters, the quantile regression model is fully parametric, thereby facilitating interpretation. In the low-dimensional regime, we establish a finite-sample theoretical framework for the sequentially defined distributed QR estimators. This reveals a trade-off between the communication cost and statistical error. We further discuss and compare several alternative confidence set constructions, based on inversion of Wald and score-type tests and resampling techniques, detailing an improvement that is effective for more extreme quantile coefficients. In high dimensions, a sparse framework is adopted, where the proposed doubly-smoothed objective function is complemented with an â1-penalty. We show that the corresponding distributed penalized QR estimator achieves the global convergence rate after a near-constant number of communication rounds. A thorough simulation study further elucidates our findings
Adaptive Algorithms for Coverage Control and Space Partitioning in Mobile Robotic Networks
We consider deployment problems where a mobile robotic network must optimize its configuration in a distributed way in order to minimize a steady-state cost function that depends on the spatial distribution of certain probabilistic events of interest. Three classes of problems are discussed in detail: coverage control problems, spatial partitioning problems, and dynamic vehicle routing problems. Moreover, we assume that the event distribution is a priori unknown, and can only be progressively inferred from the observation of the location of the actual event occurrences. For each problem we present distributed stochastic gradient algorithms that optimize the performance objective. The stochastic gradient view simplifies and generalizes previously proposed solutions, and is applicable to new complex scenarios, for example adaptive coverage involving heterogeneous agents. Finally, our algorithms often take the form of simple distributed rules that could be implemented on resource-limited platforms
Nonlocal Graph-PDEs and Riemannian Gradient Flows for Image Labeling
In this thesis, we focus on the image labeling problem which is the task of performing unique
pixel-wise label decisions to simplify the image while reducing its redundant information. We
build upon a recently introduced geometric approach for data labeling by assignment flows
[
APSS17
] that comprises a smooth dynamical system for data processing on weighted graphs.
Hereby we pursue two lines of research that give new application and theoretically-oriented
insights on the underlying segmentation task.
We demonstrate using the example of Optical Coherence Tomography (OCT), which is the
mostly used non-invasive acquisition method of large volumetric scans of human retinal tis-
sues, how incorporation of constraints on the geometry of statistical manifold results in a novel
purely data driven
geometric
approach for order-constrained segmentation of volumetric data
in any metric space. In particular, making diagnostic analysis for human eye diseases requires
decisive information in form of exact measurement of retinal layer thicknesses that has be done
for each patient separately resulting in an demanding and time consuming task. To ease the
clinical diagnosis we will introduce a fully automated segmentation algorithm that comes up
with a high segmentation accuracy and a high level of built-in-parallelism. As opposed to many
established retinal layer segmentation methods, we use only local information as input without
incorporation of additional global shape priors. Instead, we achieve physiological order of reti-
nal cell layers and membranes including a new formulation of ordered pair of distributions in an
smoothed energy term. This systematically avoids bias pertaining to global shape and is hence
suited for the detection of anatomical changes of retinal tissue structure. To access the perfor-
mance of our approach we compare two different choices of features on a data set of manually
annotated
3
D OCT volumes of healthy human retina and evaluate our method against state of
the art in automatic retinal layer segmentation as well as to manually annotated ground truth
data using different metrics.
We generalize the recent work [
SS21
] on a variational perspective on assignment flows and
introduce a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs.
The G-PDE is derived as nonlocal reparametrization of the assignment flow approach that was
introduced in
J. Math. Imaging & Vision
58(2), 2017. Due to this parameterization, solving the
G-PDE numerically is shown to be equivalent to computing the Riemannian gradient flow with re-
spect to a nonconvex potential. We devise an entropy-regularized difference-of-convex-functions
(DC) decomposition of this potential and show that the basic geometric Euler scheme for inte-
grating the assignment flow is equivalent to solving the G-PDE by an established DC program-
ming scheme. Moreover, the viewpoint of geometric integration reveals a basic way to exploit
higher-order information of the vector field that drives the assignment flow, in order to devise a
novel accelerated DC programming scheme. A detailed convergence analysis of both numerical
schemes is provided and illustrated by numerical experiments
A Framework for Meta-heuristic Parameter Performance Prediction Using Fitness Landscape Analysis and Machine Learning
The behaviour of an optimization algorithm when attempting to solve a problem depends on the values assigned to its control parameters. For an algorithm to obtain desirable performance, its control parameter values must be chosen based on the current problem. Despite being necessary for optimal performance, selecting appropriate control parameter values is time-consuming, computationally expensive, and challenging. As the quantity of control parameters increases, so does the time complexity associated with searching for practical values, which often overshadows addressing the problem at hand, limiting the efficiency of an algorithm. As primarily recognized by the no free lunch theorem, there is no one-size-fits-all to problem-solving; hence from understanding a problem, a tailored approach can substantially help solve it.
To predict the performance of control parameter configurations in unseen environments, this thesis crafts an intelligent generalizable framework leveraging machine learning classification and quantitative characteristics about the problem in question. The proposed parameter performance classifier (PPC) framework is extensively explored by training 84 high-accuracy classifiers comprised of multiple sampling methods, fitness types, and binning strategies. Furthermore, the novel framework is utilized in constructing a new parameter-free particle swarm optimization (PSO) variant called PPC-PSO that effectively eliminates the computational cost of parameter tuning, yields competitive performance amongst other leading methodologies across 99 benchmark functions, and is highly accessible to researchers and practitioners. The success of PPC-PSO shows excellent promise for the applicability of the PPC framework in making many more robust parameter-free meta-heuristic algorithms in the future with incredible generalization capabilities
Probabilistic Ordinary Differential Equation Solvers - Theory and Applications
Ordinary differential equations are ubiquitous in science and engineering, as they provide mathematical models for many physical processes. However, most practical purposes require the temporal evolution of a particular solution. Many relevant ordinary differential equations are known to lack closed-form solutions in terms of simple analytic functions. Thus, users rely on numerical algorithms to compute discrete approximations.
Numerical methods replace the intractable, and thus inaccessible, solution by an approximating model with known computational strategies. This is akin to a process in statistics where an unknown true relationship is modeled with access to instances of said relationship. One branch of statistics, Bayesian modeling, expresses degrees of uncertainty with probability distributions. In recent years, this idea has gained traction for the design and study of numerical algorithms which established probabilistic numerics as a research field in its own right.
The theory part of this thesis is concerned with bridging the gap between classical numerical methods for ordinary differential equations and probabilistic numerics. To this end, an algorithm is presented based on Gaussian processes, a general and versatile model for Bayesian regression. This algorithm is compared to two standard frameworks for the solution of initial value problems. It is shown that the maximum a-posteriori estimator of certain Gaussian process regressors coincide with certain multistep formulae. Furthermore, a particular initialization scheme based on an improper prior model coincides with a Runge-Kutta method for the first discretization step. This analysis provides a higher-order probabilistic numerical algorithm for initial value problems.
Based on the probabilistic description, an estimator of the local integration error is presented, which is used in a step size adaptation scheme. The completed algorithm is evaluated on a benchmark on initial value problems, confirming empirically the theoretically predicted error rates and displaying particularly efficient performance on domains with low accuracy requirements.
To establish the practical benefit of the probabilistic solution, a probabilistic boundary value problem solver is applied to a medical imaging problem. In tractography, diffusion-weighted magnetic resonance imaging data is used to infer connectivity of neural fibers. The first application of the probabilistic solver shows how the quantification of the discretization error can be used in subsequent estimation of fiber density. The second application additionally incorporates the measurement noise of the imaging data into the
tract estimation model. These two extensions of the shortest-path tractography method give more faithful data, modeling and algorithmic uncertainty representations in neural connectivity studies.Gewöhnliche Differentialgleichungen sind allgegenwĂ€rtig in Wissenschaft und Technik, da sie die mathematische Beschreibung vieler physikalischen VorgĂ€nge sind. Jedoch benötigt ein GroĂteil der praktischen Anwendungen die zeitliche Entwicklung einer bestimmten Lösung. Es ist bekannt, dass viele relevante gewöhnliche Differentialgleichungen keine geschlossene Lösung als AusdrĂŒcke einfacher analytischer Funktion besitzen. Daher verlassen sich Anwender auf numerische Algorithmen, um diskrete AnnĂ€herungen zu berechnen.
Numerische Methoden ersetzen die unauswertbare, und daher unzugĂ€ngliche, Lösung durch eine AnnĂ€herung mit bekannten Rechenverfahren. Dies Ă€hnelt einem Vorgang in der Statistik, wobei ein unbekanntes wahres VerhĂ€ltnis mittels Zugang zu Beispielen modeliert wird. Eine Unterdisziplin der Statistik, Bayesâsche Modellierung, stellt graduelle Unsicherheit mittels Wahrscheinlichkeitsverteilungen dar. In den letzten Jahren hat diese Idee an Zugkraft fĂŒr die Konstruktion und Analyse von numerischen Algorithmen gewonnen, was zur Etablierung von probabilistischer Numerik als eigenstĂ€ndiges
Forschungsgebiet fĂŒhrte.
Der Theorieteil dieser Dissertation schlĂ€gt eine BrĂŒcke zwischen herkömmlichen numerischen Verfahren zur Lösung gewöhnlicher Differentialgleichungen und probabilistischer Numerik. Ein auf GauĂâschen Prozessen basierender Algorithmus wird vorgestellt, welche ein generelles und vielseitiges Modell der Bayesschen Regression sind. Dieser Algorithmus wird verglichen mit zwei StandardansĂ€tzen fĂŒr die Lösung von Anfangswertproblemen. Es wird gezeigt, dass der Maximum-a-posteriori-SchĂ€tzer bestimmter GauĂprozess-Regressoren ĂŒbereinstimmt mit bestimmten Mehrschrittverfahren. Weiterhin stimmt ein besonderes Initialisierungsverfahren basierend auf einer uneigentlichen
A-priori-Wahrscheinlichkeit ĂŒberein mit einer Runge-Kutta Methode im ersten Rechenschritt. Diese Analyse fĂŒhrt zu einer probabilistisch-numerischen Methode höherer Ordnung zur Lösung von Anfangswertproblemen.
Basierend auf der probabilistischen Beschreibung wird ein SchÀtzer des lokalen Integrationfehlers prÀsentiert, welcher in einem Schrittweitensteuerungsverfahren verwendet
wird. Der vollstÀndige Algorithmus wird auf einem Satz standardisierter Anfangswertprobleme ausgewertet, um empirisch den von der Theorie vorhergesagten Fehler zu bestÀtigen. Der Test weist dem Verfahren einen besonders effizienten Rechenaufwand im Bereich der niedrigen Genauigkeitsanforderungen aus.
Um den praktischen Nutzen der probabilistischen Lösung nachzuweisen, wird ein probabilistischer Löser fĂŒr Randwertprobleme auf eine Fragestellung der medizinischen Bildgebung angewandt. In der Traktografie werden die Daten der diffusionsgewichteten Magnetresonanzbildgebung verwendet, um die KonnektivitĂ€t neuronaler Fasern zu bestimmen. Die erste Anwendung des probabilistische Lösers demonstriert, wie die Quantifizierung des Diskretisierungsfehlers in einer nachgeschalteten SchĂ€tzung der Faserdichte verwendet werden kann. Die zweite Anwendung integriert zusĂ€tzlich das Messrauschen der Bildgebungsdaten in das StrangschĂ€tzungsmodell. Diese beiden Erweiterungen der KĂŒrzesten-Pfad-Traktografie reprĂ€sentieren die Daten-, Modellierungs- und algorithmische Unsicherheit abbildungstreuer in neuronalen KonnektivitĂ€tsstudien
Investigating hybrids of evolution and learning for real-parameter optimization
In recent years, more and more advanced techniques have been developed in the field
of hybridizing of evolution and learning, this means that more applications with these techniques
can benefit from this progress. One example of these advanced techniques is the
Learnable Evolution Model (LEM), which adopts learning as a guide for the general evolutionary
search. Despite this trend and the progress in LEM, there are still many ideas and
attempts which deserve further investigations and tests. For this purpose, this thesis has
developed a number of new algorithms attempting to combine more learning algorithms
with evolution in different ways. With these developments, we expect to understand the
effects and relations between evolution and learning, and also achieve better performances
in solving complex problems.
The machine learning algorithms combined into the standard Genetic Algorithm (GA)
are the supervised learning method k-nearest-neighbors (KNN), the Entropy-Based Discretization
(ED) method, and the decision tree learning algorithm ID3. We test these algorithms
on various real-parameter function optimization problems, especially the functions
in the special session on CEC 2005 real-parameter function optimization. Additionally, a
medical cancer chemotherapy treatment problem is solved in this thesis by some of our
hybrid algorithms.
The performances of these algorithms are compared with standard genetic algorithms
and other well-known contemporary evolution and learning hybrid algorithms. Some of
them are the CovarianceMatrix Adaptation Evolution Strategies (CMAES), and variants of
the Estimation of Distribution Algorithms (EDA).
Some important results have been derived from our experiments on these developed algorithms.
Among them, we found that even some very simple learning methods hybridized
properly with evolution procedure can provide significant performance improvement; and
when more complex learning algorithms are incorporated with evolution, the resulting algorithms
are very promising and compete very well against the state of the art hybrid algorithms
both in well-defined real-parameter function optimization problems and a practical
evaluation-expensive problem