Local reconstruction analysis of inverting the Radon transform in the plane from noisy discrete data
In this paper, we investigate the reconstruction error, $N_\e^{\text{rec}}(x)$, when a linear, filtered back-projection (FBP) algorithm is applied to noisy, discrete Radon transform data with sampling step size $\epsilon$ in two-dimensions. Specifically, we analyze $N_\e^{\text{rec}}(x)$ for $x$ in small, $O(\e)$-sized neighborhoods around a generic fixed point, $x_0$, in the plane, where the measurement noise values, $\eta_{k,j}$ (i.e., the errors in the sinogram space), are random variables. The latter are independent, but not necessarily identically distributed. We show, under suitable assumptions on the first three moments of the $\eta_{k,j}$, that the following limit exists: $N^{\text{rec}}(\chx;x_0) = \lim_{\e\to0}N_\e^{\text{rec}}(x_0+\e\chx)$, for $\check x$ in a bounded domain. Here, $N_\e^{\text{rec}}$ and $ N^{\text{rec}}$ are viewed as continuous random variables, and the limit is understood in the sense of distributions. Once the limit is established, we prove that $N^{\text{rec}}$ is a zero mean Gaussian random field and compute explicitly its covariance. In addition, we validate our theory using numerical simulations and pseudo random noise.
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In this work, we introduce a novel approach to regularization in multivariable regression problems. Our regularizer, called DLoss, penalises differences between the model's derivatives and derivatives of the data generating function as estimated from the training data. We call these estimated derivatives data derivatives. The goal of our method is to align the model to the data, not only in terms of target values but also in terms of the derivatives involved. To estimate data derivatives, we select (from the training data) 2-tuples of input-value pairs, using either nearest neighbour or random, selection. On synthetic and real datasets, we evaluate the effectiveness of adding DLoss, with different weights, to the standard mean squared error loss. The experimental results show that with DLoss (using nearest neighbour selection) we obtain, on average, the best rank with respect to MSE on validation data sets, compared to no regularization, L2 regularization, and Dropout.
Exploring the semantic context in scene images is essential for indoor scene recognition. However, due to the diverse intra-class spatial layouts and the coexisting inter-class objects, modeling contextual relationships to adapt various image characteristics is a great challenge. Existing contextual modeling methods for scene recognition exhibit two limitations: 1) They typically model only one kind of spatial relationship among objects within scenes in an artificially predefined manner, with limited exploration of diverse spatial layouts. 2) They often overlook the differences in coexisting objects across different scenes, suppressing scene recognition performance. To overcome these limitations, we propose SpaCoNet, which simultaneously models Spatial relation and Co-occurrence of objects guided by semantic segmentation. Firstly, the Semantic Spatial Relation Module (SSRM) is constructed to model scene spatial features. With the help of semantic segmentation, this module decouples the spatial information from the scene image and thoroughly explores all spatial relationships among objects in an end-to-end manner. Secondly, both spatial features from the SSRM and deep features from the Image Feature Extraction Module are allocated to each object, so as to distinguish the coexisting object across different scenes. Finally, utilizing the discriminative features above, we design a Global-Local Dependency Module to explore the long-range co-occurrence among objects, and further generate a semantic-guided feature representation for indoor scene recognition. Experimental results on three widely used scene datasets demonstrate the effectiveness and generality of the proposed method.
We develop an inferential toolkit for analyzing object-valued responses, which correspond to data situated in general metric spaces, paired with Euclidean predictors within the conformal framework. To this end we introduce conditional profile average transport costs, where we compare distance profiles that correspond to one-dimensional distributions of probability mass falling into balls of increasing radius through the optimal transport cost when moving from one distance profile to another. The average transport cost to transport a given distance profile to all others is crucial for statistical inference in metric spaces and underpins the proposed conditional profile scores. A key feature of the proposed approach is to utilize the distribution of conditional profile average transport costs as conformity score for general metric space-valued responses, which facilitates the construction of prediction sets by the split conformal algorithm. We derive the uniform convergence rate of the proposed conformity score estimators and establish asymptotic conditional validity for the prediction sets. The finite sample performance for synthetic data in various metric spaces demonstrates that the proposed conditional profile score outperforms existing methods in terms of both coverage level and size of the resulting prediction sets, even in the special case of scalar and thus Euclidean responses. We also demonstrate the practical utility of conditional profile scores for network data from New York taxi trips and for compositional data reflecting energy sourcing of U.S. states.
The paper explores the Biased Random-Key Genetic Algorithm (BRKGA) in the domain of logistics and vehicle routing. Specifically, the application of the algorithm is contextualized within the framework of the Vehicle Routing Problem with Occasional Drivers and Time Window (VRPODTW) that represents a critical challenge in contemporary delivery systems. Within this context, BRKGA emerges as an innovative solution approach to optimize routing plans, balancing cost-efficiency with operational constraints. This research introduces a new BRKGA, characterized by a variable mutant population which can vary from generation to generation, named BRKGA-VM. This novel variant was tested to solve a VRPODTW. For this purpose, an innovative specific decoder procedure was proposed and implemented. Furthermore, a hybridization of the algorithm with a Variable Neighborhood Descent (VND) algorithm has also been considered, showing an improvement of problem-solving capabilities. Computational results show a better performances in term of effectiveness over a previous version of BRKGA, denoted as MP. The improved performance of BRKGA-VM is evident from its ability to optimize solutions across a wide range of scenarios, with significant improvements observed for each type of instance considered. The analysis also reveals that VM achieves preset goals more quickly compared to MP, thanks to the increased variability induced in the mutant population which facilitates the exploration of new regions of the solution space. Furthermore, the integration of VND has shown an additional positive impact on the quality of the solutions found.
In this article, we develop a new class of multivariate distributions adapted for count data, called Tree P{\'o}lya Splitting. This class results from the combination of a univariate distribution and singular multivariate distributions along a fixed partition tree. As we will demonstrate, these distributions are flexible, allowing for the modeling of complex dependencies (positive, negative, or null) at the observation level. Specifically, we present the theoretical properties of Tree P{\'o}lya Splitting distributions by focusing primarily on marginal distributions, factorial moments, and dependency structures (covariance and correlations). The abundance of 17 species of Trichoptera recorded at 49 sites is used, on one hand, to illustrate the theoretical properties developed in this article on a concrete case, and on the other hand, to demonstrate the interest of this type of models, notably by comparing them to classical approaches in ecology or microbiome.
In this paper, the construction of $C^{1}$ cubic quasi-interpolants on a three-direction mesh of $\RR^{2}$ is addressed. The quasi-interpolating splines are defined by directly setting their Bernstein-B\'{e}zier coefficients relative to each triangle from point and gradient values in order to reproduce the polynomials of the highest possible degree. Moreover, additional global properties are required. Finally, we provide some numerical tests confirming the approximation properties.
In this paper we compare two numerical methods to integrate Riemannian cubic polynomials on the Stiefel manifold $\textbf{St}_{n,k}$. The first one is the adjusted de Casteljau algorithm, and the second one is a symplectic integrator constructed through discretization maps. In particular, we choose the cases of $n=3$ together with $k=1$ and $k=2$. The first case is diffeomorphic to the sphere and the quasi-geodesics appearing in the adjusted de Casteljau algorithm are actually geodesics. The second case is an example where we have a pure quasi-geodesic different from a geodesic. We provide a numerical comparison of both methods and discuss the obtained results to highlight the benefits of each method.
In this paper, to address the optimization problem on a compact matrix manifold, we introduce a novel algorithmic framework called the Transformed Gradient Projection (TGP) algorithm, using the projection onto this compact matrix manifold. Compared with the existing algorithms, the key innovation in our approach lies in the utilization of a new class of search directions and various stepsizes, including the Armijo, nonmonotone Armijo, and fixed stepsizes, to guide the selection of the next iterate. Our framework offers flexibility by encompassing the classical gradient projection algorithms as special cases, and intersecting the retraction-based line-search algorithms. Notably, our focus is on the Stiefel or Grassmann manifold, revealing that many existing algorithms in the literature can be seen as specific instances within our proposed framework, and this algorithmic framework also induces several new special cases. Then, we conduct a thorough exploration of the convergence properties of these algorithms, considering various search directions and stepsizes. To achieve this, we extensively analyze the geometric properties of the projection onto compact matrix manifolds, allowing us to extend classical inequalities related to retractions from the literature. Building upon these insights, we establish the weak convergence, convergence rate, and global convergence of TGP algorithms under three distinct stepsizes. In cases where the compact matrix manifold is the Stiefel or Grassmann manifold, our convergence results either encompass or surpass those found in the literature. Finally, through a series of numerical experiments, we observe that the TGP algorithms, owing to their increased flexibility in choosing search directions, outperform classical gradient projection and retraction-based line-search algorithms in several scenarios.
In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue $\lambda$ and eigenfunction $u$ separately, we treat the eigenpair $(\lambda, u)$ as one element in a product space $\mathbb R \times H_0^1(\Omega)$. Then in the presented multigrid method, only one discrete linear boundary value problem needs to be solved for each level of the multigrid sequence. Because we avoid solving large-scale nonlinear eigenvalue problems directly, the overall efficiency is significantly improved. The optimal error estimate and linear computational complexity can be derived simultaneously. In addition, we also provide an improved multigrid method coupled with a mixing scheme to further guarantee the convergence and stability of the iteration scheme. More importantly, we prove convergence for the residuals after each iteration step. For nonlinear eigenvalue problems, such theoretical analysis is missing from the existing literatures on the mixing iteration scheme.
The consistency of the maximum likelihood estimator for mixtures of elliptically-symmetric distributions for estimating its population version is shown, where the underlying distribution $P$ is nonparametric and does not necessarily belong to the class of mixtures on which the estimator is based. In a situation where $P$ is a mixture of well enough separated but nonparametric distributions it is shown that the components of the population version of the estimator correspond to the well separated components of $P$. This provides some theoretical justification for the use of such estimators for cluster analysis in case that $P$ has well separated subpopulations even if these subpopulations differ from what the mixture model assumes.
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