Minimizing Quotient Regularization Model

Quotient regularization models (QRMs) are a class of powerful regularization techniques that have gained considerable attention in recent years, due to their ability to handle complex and highly nonlinear data sets. However, the nonconvex nature of QRM poses a significant challenge in finding its optimal solution. We are interested in scenarios where both the numerator and the denominator of QRM are absolutely one-homogeneous functions, which is widely applicable in the fields of signal processing and image processing. In this paper, we utilize a gradient flow to minimize such QRM in combination with a quadratic data fidelity term. Our scheme involves solving a convex problem iteratively.The convergence analysis is conducted on a modified scheme in a continuous formulation, showing the convergence to a stationary point. Numerical experiments demonstrate the effectiveness of the proposed algorithm in terms of accuracy, outperforming the state-of-the-art QRM solvers.Comment: 20 page

Rank-based Decomposable Losses in Machine Learning: A Survey

Recent works have revealed an essential paradigm in designing loss functions that differentiate individual losses vs. aggregate losses. The individual loss measures the quality of the model on a sample, while the aggregate loss combines individual losses/scores over each training sample. Both have a common procedure that aggregates a set of individual values to a single numerical value. The ranking order reflects the most fundamental relation among individual values in designing losses. In addition, decomposability, in which a loss can be decomposed into an ensemble of individual terms, becomes a significant property of organizing losses/scores. This survey provides a systematic and comprehensive review of rank-based decomposable losses in machine learning. Specifically, we provide a new taxonomy of loss functions that follows the perspectives of aggregate loss and individual loss. We identify the aggregator to form such losses, which are examples of set functions. We organize the rank-based decomposable losses into eight categories. Following these categories, we review the literature on rank-based aggregate losses and rank-based individual losses. We describe general formulas for these losses and connect them with existing research topics. We also suggest future research directions spanning unexplored, remaining, and emerging issues in rank-based decomposable losses.Comment: Accepted by IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI

Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1

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

Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference

We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF inference problems. The core of our method is a very efficient bounding procedure, which combines scalable semidefinite programming (SDP) and a cutting-plane method for seeking violated constraints. In order to further speed up the computation, several strategies have been exploited, including model reduction, warm start and removal of inactive constraints. We analyze the performance of the proposed method under different settings, and demonstrate that our method either outperforms or performs on par with state-of-the-art approaches. Especially when the connectivities are dense or when the relative magnitudes of the unary costs are low, we achieve the best reported results. Experiments show that the proposed algorithm achieves better approximation than the state-of-the-art methods within a variety of time budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page

Discrete Tomography by Convex-Concave Regularization using Linear and Quadratic Optimization

Discrete tomography concerns the reconstruction of objects that are made up from a few different materials, each of which comprising a homogeneous density distribution. Under the assumption that these densities are a priori known new algorithms can be developed which typically need less projection data to reveal appealing reconstruction results

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

Collision Avoidance on Unmanned Aerial Vehicles using Deep Neural Networks

Unmanned Aerial Vehicles (UAVs), although hardly a new technology, have recently gained a prominent role in many industries, being widely used not only among enthusiastic consumers but also in high demanding professional situations, and will have a massive societal impact over the coming years. However, the operation of UAVs is full of serious safety risks, such as collisions with dynamic obstacles (birds, other UAVs, or randomly thrown objects). These collision scenarios are complex to analyze in real-time, sometimes being computationally impossible to solve with existing State of the Art (SoA) algorithms, making the use of UAVs an operational hazard and therefore significantly reducing their commercial applicability in urban environments. In this work, a conceptual framework for both stand-alone and swarm (networked) UAVs is introduced, focusing on the architectural requirements of the collision avoidance subsystem to achieve acceptable levels of safety and reliability. First, the SoA principles for collision avoidance against stationary objects are reviewed. Afterward, a novel image processing approach that uses deep learning and optical flow is presented. This approach is capable of detecting and generating escape trajectories against potential collisions with dynamic objects. Finally, novel models and algorithms combinations were tested, providing a new approach for the collision avoidance of UAVs using Deep Neural Networks. The feasibility of the proposed approach was demonstrated through experimental tests using a UAV, created from scratch using the framework developed.Os veículos aéreos não tripulados (VANTs), embora dificilmente considerados uma nova tecnologia, ganharam recentemente um papel de destaque em muitas indústrias, sendo amplamente utilizados não apenas por amadores, mas também em situações profissionais de alta exigência, sendo expectável um impacto social massivo nos próximos anos. No entanto, a operação de VANTs está repleta de sérios riscos de segurança, como colisões com obstáculos dinâmicos (pássaros, outros VANTs ou objetos arremessados). Estes cenários de colisão são complexos para analisar em tempo real, às vezes sendo computacionalmente impossível de resolver com os algoritmos existentes, tornando o uso de VANTs um risco operacional e, portanto, reduzindo significativamente a sua aplicabilidade comercial em ambientes citadinos. Neste trabalho, uma arquitectura conceptual para VANTs autônomos e em rede é apresentada, com foco nos requisitos arquitetônicos do subsistema de prevenção de colisão para atingir níveis aceitáveis de segurança e confiabilidade. Os estudos presentes na literatura para prevenção de colisão contra objectos estacionários são revistos e uma nova abordagem é descrita. Esta tecnica usa técnicas de aprendizagem profunda e processamento de imagem, para realizar a prevenção de colisões em tempo real com objetos móveis. Por fim, novos modelos e combinações de algoritmos são propostos, fornecendo uma nova abordagem para evitar colisões de VANTs usando Redes Neurais Profundas. A viabilidade da abordagem foi demonstrada através de testes experimentais utilizando um VANT, desenvolvido a partir da arquitectura apresentada