Learned reconstruction methods with convergence guarantees

In recent years, deep learning has achieved remarkable empirical success for image reconstruction. This has catalyzed an ongoing quest for precise characterization of correctness and reliability of data-driven methods in critical use-cases, for instance in medical imaging. Notwithstanding the excellent performance and efficacy of deep learning-based methods, concerns have been raised regarding their stability, or lack thereof, with serious practical implications. Significant advances have been made in recent years to unravel the inner workings of data-driven image recovery methods, challenging their widely perceived black-box nature. In this article, we will specify relevant notions of convergence for data-driven image reconstruction, which will form the basis of a survey of learned methods with mathematically rigorous reconstruction guarantees. An example that is highlighted is the role of ICNN, offering the possibility to combine the power of deep learning with classical convex regularization theory for devising methods that are provably convergent. This survey article is aimed at both methodological researchers seeking to advance the frontiers of our understanding of data-driven image reconstruction methods as well as practitioners, by providing an accessible description of useful convergence concepts and by placing some of the existing empirical practices on a solid mathematical foundation

A Linear-Time Kernel Goodness-of-Fit Test

We propose a novel adaptive test of goodness-of-fit, with computational cost linear in the number of samples. We learn the test features that best indicate the differences between observed samples and a reference model, by minimizing the false negative rate. These features are constructed via Stein's method, meaning that it is not necessary to compute the normalising constant of the model. We analyse the asymptotic Bahadur efficiency of the new test, and prove that under a mean-shift alternative, our test always has greater relative efficiency than a previous linear-time kernel test, regardless of the choice of parameters for that test. In experiments, the performance of our method exceeds that of the earlier linear-time test, and matches or exceeds the power of a quadratic-time kernel test. In high dimensions and where model structure may be exploited, our goodness of fit test performs far better than a quadratic-time two-sample test based on the Maximum Mean Discrepancy, with samples drawn from the model.Comment: Accepted to NIPS 201

Visual Analytics Methods for Exploring Geographically Networked Phenomena

abstract: The connections between different entities define different kinds of networks, and many such networked phenomena are influenced by their underlying geographical relationships. By integrating network and geospatial analysis, the goal is to extract information about interaction topologies and the relationships to related geographical constructs. In the recent decades, much work has been done analyzing the dynamics of spatial networks; however, many challenges still remain in this field. First, the development of social media and transportation technologies has greatly reshaped the typologies of communications between different geographical regions. Second, the distance metrics used in spatial analysis should also be enriched with the underlying network information to develop accurate models. Visual analytics provides methods for data exploration, pattern recognition, and knowledge discovery. However, despite the long history of geovisualizations and network visual analytics, little work has been done to develop visual analytics tools that focus specifically on geographically networked phenomena. This thesis develops a variety of visualization methods to present data values and geospatial network relationships, which enables users to interactively explore the data. Users can investigate the connections in both virtual networks and geospatial networks and the underlying geographical context can be used to improve knowledge discovery. The focus of this thesis is on social media analysis and geographical hotspots optimization. A framework is proposed for social network analysis to unveil the links between social media interactions and their underlying networked geospatial phenomena. This will be combined with a novel hotspot approach to improve hotspot identification and boundary detection with the networks extracted from urban infrastructure. Several real world problems have been analyzed using the proposed visual analytics frameworks. The primary studies and experiments show that visual analytics methods can help analysts explore such data from multiple perspectives and help the knowledge discovery process.Dissertation/ThesisDoctoral Dissertation Computer Science 201

NPCirc: An R Package for Nonparametric Circular Methods

Nonparametric density and regression estimation methods for circular data are included in the R package NPCirc. Specifically, a circular kernel density estimation procedure is provided, jointly with different alternatives for choosing the smoothing parameter. In the regression setting, nonparametric estimation for circular-linear, circular-circular and linear-circular data is also possible via the adaptation of the classical Nadaraya-Watson and local linear estimators. In order to assess the significance of the features observed in the smooth curves, both for density and regression with a circular covariate and a linear response, a SiZer technique is developed for circular data, namely CircSiZer. Some data examples are also included in the package, jointly with a routine that allows generating mixtures of different circular distributions

Weighted-Average Least Squares (WALS): Confidence and Prediction Intervals

We consider inference for linear regression models estimated by weighted-average least squares (WALS), a frequentist model averaging approach with a Bayesian flavor. We propose a new simulation method that yields re-centered confidence and prediction intervals by exploiting the bias-corrected posterior mean as a frequentist estimator of a normal location parameter. We investigate the performance of WALS and several alternative estimators in an extensive set of Monte Carlo experiments that allow for increasing complexity of the model space and heteroskedastic, skewed, and thick-tailed regression errors. In addition to WALS, we include unrestricted and fully restricted least squares, two post-selection estimators based on classical information criteria, a penalization estimator, and Mallows and jackknife model averaging estimators. We show that, compared to the other approaches, WALS performs well in terms of the mean squared error of point estimates, and also in terms of coverage errors and lengths of confidence and prediction intervals