On Graduated Optimization for Stochastic Non-Convex Problems

The graduated optimization approach, also known as the continuation method, is a popular heuristic to solving non-convex problems that has received renewed interest over the last decade. Despite its popularity, very little is known in terms of theoretical convergence analysis. In this paper we describe a new first-order algorithm based on graduated optimiza- tion and analyze its performance. We characterize a parameterized family of non- convex functions for which this algorithm provably converges to a global optimum. In particular, we prove that the algorithm converges to an {\epsilon}-approximate solution within O(1/\epsilon^2) gradient-based steps. We extend our algorithm and analysis to the setting of stochastic non-convex optimization with noisy gradient feedback, attaining the same convergence rate. Additionally, we discuss the setting of zero-order optimization, and devise a a variant of our algorithm which converges at rate of O(d^2/\epsilon^4).Comment: 17 page

On the Link between Gaussian Homotopy Continuation and Convex Envelopes

Abstract. The continuation method is a popular heuristic in computer vision for nonconvex optimization. The idea is to start from a simpli-fied problem and gradually deform it to the actual task while tracking the solution. It was first used in computer vision under the name of graduated nonconvexity. Since then, it has been utilized explicitly or im-plicitly in various applications. In fact, state-of-the-art optical flow and shape estimation rely on a form of continuation. Despite its empirical success, there is little theoretical understanding of this method. This work provides some novel insights into this technique. Specifically, there are many ways to choose the initial problem and many ways to progres-sively deform it to the original task. However, here we show that when this process is constructed by Gaussian smoothing, it is optimal in a specific sense. In fact, we prove that Gaussian smoothing emerges from the best affine approximation to Vese’s nonlinear PDE. The latter PDE evolves any function to its convex envelope, hence providing the optimal convexification

Structural Brain Changes in Early‐Onset Alzheimer's Disease Subjects Using the LONI Pipeline Environment

BACKGROUND AND PURPOSEThis study investigates 36 subjects aged 55‐65 from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database to expand our knowledge of early‐onset (EO) Alzheimer's Disease (EO‐AD) using neuroimaging biomarkers.METHODSNine of the subjects had EO‐AD, and 27 had EO mild cognitive impairment (EO‐MCI). The structural ADNI data were parcellated using BrainParser, and the 15 most discriminating neuroimaging markers between the two cohorts were extracted using the Global Shape Analysis (GSA) Pipeline workflow. Then the Local Shape Analysis (LSA) Pipeline workflow was used to conduct local (per‐vertex) post‐hoc statistical analyses of the shape differences based on the participants’ diagnoses (EO‐MCI+EO‐AD). Tensor‐based Morphometry (TBM) and multivariate regression models were used to identify the significance of the structural brain differences based on the participants’ diagnoses.RESULTSThe significant between‐group regional differences using GSA were found in 15 neuroimaging markers. The results of the LSA analysis workflow were based on the subject diagnosis, age, years of education, apolipoprotein E (ε4), Mini‐Mental State Examination, visiting times, and logical memory as regressors. All the variables had significant effects on the regional shape measures. Some of these effects survived the false discovery rate (FDR) correction. Similarly, the TBM analysis showed significant effects on the Jacobian displacement vector fields, but these effects were reduced after FDR correction.CONCLUSIONSThese results may explain some of the differences between EO‐AD and EO‐MCI, and some of the characteristics of the EO cognitive impairment subjects.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/113121/1/jon12252.pd

Variational approximation of a second order free discontinuity problem in computer vision

We consider a functional, proposed by Blake and Zisserman for computer vision problems, which depends on free discontinuities, free gradient discontinuities, and second order derivatives. We show how this functional can be approximated by elliptic functionals defined on Sobolev spaces. The approximation takes place in a variational sense, the De Giorgi Γ-convergence, and extends to this second order model an approximation of the Mumford–Shah functional obtained by Ambrosio and Tortorelli. For the purpose of illustration an algorithm based on the Γ-convergent approximation is applied to the problem of computing depth from stereo images and some numerical examples are presented

Optimal multi-scale matching

Abstract The coarse-to-fine search strategy is extensively used in current reported research

Recent Ostracod Fauna of the Western Ross Sea (Antarctica): A Poorly Known Ingredient of Polar Carbonate Factories

Ostracoda are a minor but recurrent component of Southern Ocean marine carbonate factories, and their low-Mg calcitic skeletal mineralogy helps in ensuring a noteworthy post-mortem resilience. Our study, based upon surface sediment occurrences, contributes to the better definition of their distribution vs. potential controlling factors in Antarctic waters. The ostracod fauna from the Western Ross Sea Shelf appears dominated by Australicythere polylyca, Australicythere devexa, Xestoleberis rigusa, Loxoreticulatum fallax, Cativella bensoni, Austrotrachyleberis antarctica and Patagonacythere longiducta, colonizing a variety of shelf environments along a wide bathymetric range. The abundance and richness values correlate well to nutrient distribution and sediment supply, primarily related to the circulation of different oceanographic regimes affecting the floor of the Ross Sea Shelf. Circumpolar Deep Water could represent the main factor controlling the distribution of ostracods. Similar results (high abundance and richness in ostracod values) were also recorded in the Terra Nova Bay and in a nearby area characterized by warm water rich in nutrients and composed of water of circumpolar origin flowing from the open ocean southwards onto the continental shelf. Particulate Fe (pFe), in suspended particulate matter (SPM), and other particulate trace metals in TNB could support the hypothesis that biogenic iron may significantly contribute to the bioavailable iron pool, sustaining both primary production and ostracod fauna richness in this area

Automatic Segmentation of the Retinal Nerve Fiber Layer by Means of Mathematical Morphology and Deformable Models in 2D Optical Coherence Tomography Imaging

[EN] Glaucoma is a neurodegenerative disease process that leads to progressive damage of the optic nerve to produce visual impairment and blindness. Spectral-domain OCT technology enables peripapillary circular scans of the retina and the measurement of the thickness of the retinal nerve fiber layer (RNFL) for the assessment of the disease status or progression in glaucoma patients. This paper describes a new approach to segment and measure the retinal nerve fiber layer in peripapillary OCT images. The proposed method consists of two stages. In the first one, morphological operators robustly detect the coarse location of the layer boundaries, despite the speckle noise and diverse artifacts in the OCT image. In the second stage, deformable models are initialized with the results of the previous stage to perform a fine segmentation of the boundaries, providing an accurate measurement of the entire RNFL. The results of the RNFL segmentation were qualitatively assessed by ophthalmologists, and the measurements of the thickness of the RNFL were quantitatively compared with those provided by the OCT inbuilt software as well as the state-of-the-art methods.This work was partially funded by Spanish National projects AES2017-PI17/00771 and AES2017-PI17/00821 (Instituto de Salud Carlos III), PID2019-105142RB-C21 (AI4SKIN) (Spanish Ministry of Economy and Competitiveness), PTA2017-14610-I (State Research Spanish Agency), regional project 20901/PI/18 (Fundacion Seneca) and Polytechnic University of Valencia (PAID-01-20).Berenguer-Vidal, R.; Verdú-Monedero, R.; Morales-Sánchez, J.; Sellés-Navarro, I.; Del Amor, R.; García-Pardo, JG.; Naranjo Ornedo, V. (2021). Automatic Segmentation of the Retinal Nerve Fiber Layer by Means of Mathematical Morphology and Deformable Models in 2D Optical Coherence Tomography Imaging. Sensors. 21(23):1-30. https://doi.org/10.3390/s21238027S130212

Filling the signed distance field by fitting local quadrics

Abstract We propose a method of filling unmeasured regions of shape models integrated from multiple measurements of surface shapes. We use the signed distance field (SDF

Scaling priors for intrinsic Gaussian Markov random fields applied to blood pressure data

An Intrinsic Gaussian Markov Random Field (IGMRF) can be used to induce conditional dependence in Bayesian hierarchical models. IGMRFs have both a precision matrix, which defines the neighborhood structure of the model, and a precision, or scaling, parameter. Previous studies have shown the importance of selecting the prior for this scaling parameter appropriately for different types of IGMRF, as it can have a substantial impact on posterior estimates. Here, we focus on cases in one and two dimensions, where tuning of the prior is achieved by mapping it to the marginal SD of an IGMRF of corresponding dimensionality. We compare the effects of scaling various IGMRFs, including an application to real two‐dimensional blood pressure data using MCMC methods