813 research outputs found

    Non-convex Optimization for Machine Learning

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    A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low rank are frequently imposed or else the objective itself is designed to be a non-convex function. This is especially true of algorithms that operate in high-dimensional spaces or that train non-linear models such as tensor models and deep networks. The freedom to express the learning problem as a non-convex optimization problem gives immense modeling power to the algorithm designer, but often such problems are NP-hard to solve. A popular workaround to this has been to relax non-convex problems to convex ones and use traditional methods to solve the (convex) relaxed optimization problems. However this approach may be lossy and nevertheless presents significant challenges for large scale optimization. On the other hand, direct approaches to non-convex optimization have met with resounding success in several domains and remain the methods of choice for the practitioner, as they frequently outperform relaxation-based techniques - popular heuristics include projected gradient descent and alternating minimization. However, these are often poorly understood in terms of their convergence and other properties. This monograph presents a selection of recent advances that bridge a long-standing gap in our understanding of these heuristics. The monograph will lead the reader through several widely used non-convex optimization techniques, as well as applications thereof. The goal of this monograph is to both, introduce the rich literature in this area, as well as equip the reader with the tools and techniques needed to analyze these simple procedures for non-convex problems.Comment: The official publication is available from now publishers via http://dx.doi.org/10.1561/220000005

    Weirdest Martensite: Smectic Liquid Crystal Microstructure And Weyl-poincaré Invariance

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    Smectic liquid crystals are remarkable, beautiful examples of materials microstructure, with ordered patterns of geometrically perfect ellipses and hyperbolas. The solution of the complex problem of filling three-dimensional space with domains of focal conics under constraining boundary conditions yields a set of strict rules, which are similar to the compatibility conditions in a martensitic crystal. Here we present the rules giving compatible conditions for the concentric circle domains found at two-dimensional smectic interfaces with planar boundary conditions. Using configurations generated by numerical simulations, we develop a clustering algorithm to decompose the planar boundaries into domains. The interfaces between different domains agree well with the smectic compatibility conditions. We also discuss generalizations of our approach to describe the full three-dimensional smectic domains, where the variant symmetry group is the Weyl-Poincaré group of Lorentz boosts, translations, rotations, and dilatations. © 2016 American Physical Society.11

    Variational methods and its applications to computer vision

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    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

    Influence of chemistry and structure on interfacial segregation in NbMoTaW with high-throughput atomistic simulations

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    Refractory multi-principal element alloys exhibiting promising mechanical properties such as excellent strength retention at elevated temperatures have been attracting increasing attention. Although their inherent chemical complexity is considered a defining feature, a challenge arises in predicting local chemical ordering, particularly in grain boundary regions with enhanced structural disorder. In this study, we use atomistic simulations of a large group of bicrystal models to sample a wide variety of interfacial sites (grain boundary) in NbMoTaW and explore emergent trends in interfacial segregation and the underlying structural and chemical driving factors. Sampling hundreds of bicrystals along the [001] symmetric tilt axis and analyzing more than one hundred and thirty thousand grain boundary sites with a variety of local atomic environments, we uncover segregation trends in NbMoTaW. While Nb is the dominant segregant, more notable are the segregation patterns that deviate from expected behavior and mark situations where local structural and chemical driving forces lead to interesting segregation events. For example, incomplete depletion of Ta in low-angle boundaries results from chemical pinning due to favorable local compositional environments associated with chemical short-range ordering. Finally, machine learning models capturing and comparing the structural and chemical features of interfacial sites are developed to weigh their relative importance and contributions to segregation tendency, revealing a significant increase in predictive capability when including local chemical information. Overall, this work, highlighting the complex interplay between local grain boundary structure and chemical short-range ordering, suggest tunable segregation and chemical ordering by tailoring grain boundary structure in multi-principal element alloys

    29th International Symposium on Algorithms and Computation: ISAAC 2018, December 16-19, 2018, Jiaoxi, Yilan, Taiwan

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    Cox-McFadden Partial and Marginal Likelihoods for the Proportional Hazard Model with Random Effects

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    In survival analysis, Cox\u27s name is associated with the partial likelihood technique that allows consistent estimation of proportional hazard scale parameters without specifying a duration dependence baseline. In discrete choice analysis, McFadden\u27s name is associated with the generalized extreme-value (GEV) class of logistic choice models that relax the independence of irrelevant alternatives assumption. This paper shows that the mixed class of proportional hazard specifications allowing consistent estimation of scale and mixing parameters using partial likelihood is isomorphic to the GEV class. Independent censoring is allowed and I discuss approximations to the partial likelihood in the presence of ties. Finally, the partial likelihood score vector can be used to construct log-rank tests that do not require the independence of observations involved
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