Smoothness Assumptions Research Articles

Hausdorff approximations and volume of tubes of singular algebraic sets

We prove bounds for the volume of neighborhoods of algebraic sets, in the euclidean space or the sphere, in terms of the degree of the defining polynomials, the number of variables and the dimension of the algebraic set, without any smoothness assumption. This generalizes previous work of Lotz (Proc Am Math Soc 143(5):1875–1889, 2015) on smooth complete intersections in the euclidean space and of Bürgisser et al. (Math Comp 77(263):1559–1583, 2008) on hypersurfaces in the sphere, and gives a complete solution to Bürgisser and Cucker (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 349, Springer, Heidelberg, 2013, Problem 17).

On solutions arising from radial spatial dynamics of some semilinear elliptic equations

We consider the semilinear elliptic equation $$\Delta u+f(x,u)=0, $$ where \(x\in\mathbb{R}^N\setminus\{0\}\), \(N\geq 2,\) and \(f \) satisfies certain smoothness and structural assumptions. We construct solutions of the form \(u(r,\phi)=r^{(2-N)/2} \tilde{u}(\log r,\phi)\), where \(r=|x|>0|0\), \(\phi\in\mathbb{S}^{N-1}\), and \(\tilde{u}\) is quasiperiodic in its first argument with two nonresonant frequencies. These solutions are found using some recent developments in the theory of spatial dynamics, in which the radial variable r takes the role of time, combined with classical results from dynamical systems and the KAM theory. For more information see https://ejde.math.txstate.edu/conf-proc/26/v1/abstr.html

Specialization for the pro-étale fundamental group

For a formal scheme$\mathfrak {X}$of finite type over a complete rank-one valuation ring, we construct a specialization morphism\[ \pi^{\mathrm{dJ}}_1(\mathfrak{X}_\eta) \to \pi^{{\textrm{pro}}\unicode{x00E9}{\textrm{t}}}_1(\mathfrak{X}_k) \]from the de Jong fundamental group of the rigid generic fiber to the Bhatt–Scholze pro-étale fundamental group of the special fiber. The construction relies on an interplay between admissible blowups of$\mathfrak {X}$and normalizations of the irreducible components of$\mathfrak {X}_k$, and employs the Berthelot tubes of these irreducible components in an essential way. Using related techniques, we show that under certain smoothness and semistability assumptions, covering spaces in the sense of de Jong of a smooth rigid space which are tame satisfy étale descent.

Improved bounds for discretization of Langevin diffusions: Near-optimal rates without convexity

Discretizations of the Langevin diffusion have been proven very useful for developing and analyzing algorithms for sampling and stochastic optimization. We present an improved non-asymptotic analysis of the Euler-Maruyama discretization of the Langevin diffusion. Our analysis does not require global contractivity, and yields polynomial dependence on the time horizon. Compared to existing approaches, we make an additional smoothness assumption, and improve the existing rate in discretization step size from O(η) to O(η2) in terms of the KL divergence. This result matches the correct order for numerical SDEs, without suffering from exponential time dependence. When applied to MCMC, this result simultaneously improves on the analyses of a range of sampling algorithms that are based on Dalalyan’s approach.

A joint learning framework for Gaussian processes regression and graph learning

In the traditional Gaussian process regression (GPR), covariance matrix of outputs is dominated by a given kernel function, that generally depends on pairwise distance or correlation between sample inputs. Nevertheless, this kind of models hardly utilize high-order statistical properties or globally topological information among sample inputs, undermining their prediction capability. To remedy this defect, we propose in this paper a novel GPR framework combining the MLE of Gaussian processes with graph learning. In our model, sample inputs are modeled by a weighted graph, whose topology is directly inferred from sample inputs based on either the smoothness assumption or the self-representative property. Such global information can be viewed as a kind of knowledge a prior, guiding the process of learning hyper-parameters of the chosen kernel function and the construction of covariance matrix of GPR model outputs. In practice, hyper-parameters of the GPR model and adjacency matrix of the graph can be trained by the alternating optimization. Theoretical analyses regarding solutions to graph learning are also presented to reduce computational complexity. Experimental results demonstrate that the proposed framework can achieve competitive performance in terms of prediction accuracies and computational efficiency, compared to state-of-the-art GPR algorithms.

Heavy-Tailed Density Estimation

A novel statistical method is proposed and investigated for estimating a heavy tailed density under mild smoothness assumptions. Statistical analyses of heavy-tailed distributions are susceptible to the problem of sparse information in the tail of the distribution getting washed away by unrelated features of a hefty bulk. The proposed Bayesian method avoids this problem by incorporating smoothness and tail regularization through a carefully specified semiparametric prior distribution, and is able to consistently estimate both the density function and its tail index at near minimax optimal rates of contraction. A joint, likelihood driven estimation of the bulk and the tail is shown to help improve uncertainty assessment in estimating the tail index parameter and offer more accurate and reliable estimates of the high tail quantiles compared to thresholding methods. Supplementary materials for this article are available online.

Optimal error bound for immersed weak Galerkin finite element method for elliptic interface problems

In Mu and Zhang (2019), an immersed weak Galerkin finite element method (IWG-FEM) is developed for solving elliptic interface problems and it is proved that this method has optimal a-priori error estimate in an energy norm under artificial smoothness assumption on the solution. In this study, we prove that IWG-FEM converges optimally in energy norm under natural smoothness assumption on solution. Furthermore, we show that IWG-FEM converges optimally in the L2 norm which did not present in Mu and Zhang (2019) because of the artificial H3 smoothness requirement. A series of numerical experiments are conducted and reported to verify the theoretical finding.

Bayesian Optimization for Expensive Smooth-Varying Functions

Bayesian optimization (BO) is a powerful class of data-driven techniques for the maximization of expensive-to-evaluate objective functions. These techniques construct a Gaussian process (GP) regression for representing the objective function according to the latest available function evaluations and sequentially select samples and evaluate the function by maximizing an acquisition function. The primary assumption in most BO policies is that the objective function has a uniform level of smoothness over the input space, modeled by a kernel function. However, the uniform smoothness assumption is likely to be violated in a wide range of practical problems, primary domains in which the objective function is evaluated differently at various regions of input space (e.g., through different experiments, software, or approximators). This article develops a BO framework capable of optimizing expensive smooth-varying functions. Unlike the existing techniques that rely on a single GP model, the proposed framework constructs a set of local and global GP models to represent the objective function. The predictive mean and variance at any given sample in the input space are computed according to the posterior probabilities of the local and global GP models. Local and global models are adaptively controlled through a single parameter, which can be optimized along with other GP models’ parameters during the optimization process. Using the predicted local and global values, the expected improvement acquisition function is employed as one of the possible acquisition functions for the selection process. The performance of the proposed framework is assessed extensively through two optimization benchmark problems.

Conditional Local Convolution for Spatio-Temporal Meteorological Forecasting

Spatio-temporal forecasting is challenging attributing to the high nonlinearity in temporal dynamics as well as complex location-characterized patterns in spatial domains, especially in fields like weather forecasting. Graph convolutions are usually used for modeling the spatial dependency in meteorology to handle the irregular distribution of sensors' spatial location. In this work, a novel graph-based convolution for imitating the meteorological flows is proposed to capture the local spatial patterns. Based on the assumption of smoothness of location-characterized patterns, we propose conditional local convolution whose shared kernel on nodes' local space is approximated by feedforward networks, with local representations of coordinate obtained by horizon maps into cylindrical-tangent space as its input. The established united standard of local coordinate system preserves the orientation on geography. We further propose the distance and orientation scaling terms to reduce the impacts of irregular spatial distribution. The convolution is embedded in a Recurrent Neural Network architecture to model the temporal dynamics, leading to the Conditional Local Convolution Recurrent Network (CLCRN). Our model is evaluated on real-world weather benchmark datasets, achieving state-of-the-art performance with obvious improvements. We conduct further analysis on local pattern visualization, model's framework choice, advantages of horizon maps and etc. The source code is available at https://github.com/BIRD-TAO/CLCRN.

Data smoothing with applications to edge detection

Abstract The aim of this paper is to present a new stable method for smoothing and differentiating noisy data defined on a bounded domain Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} with N ≥ 1 N\ge 1 . The proposed method stems from the smoothing properties of the classical diffusion equation; the smoothed data are obtained by solving a diffusion equation with the noisy data imposed as the initial condition. We analyze the stability and convergence of the proposed method and we give optimal convergence rates. One of the main advantages of this method lies in multivariable problems, where some of the other approaches are not easily generalized. Moreover, this approach does not require strong smoothness assumptions on the underlying data, which makes it appealing for detecting data corners or edges. Numerical examples demonstrate the feasibility and robustness of the method even with the presence of a large amounts of noise.

Cyclic coordinate descent in the Hölder smooth setting

We provide a convergence analysis of cyclic block coordinate descent under Hölder smoothness assumptions. We consider both convex and non-convex objectives, obtaining a convergence rate of O(p/kγ) to the optimal value for convex objectives and a rate of O(p11+γ/kγ1+γ) to a stationary point for non-convex objectives, where γ is the Hölder exponent. These rates recover those obtained in existing literature for full gradient descent and/or when the objective is simply Lipschitz smooth.

Semi-uniform input-to-state stability of infinite-dimensional systems

We introduce the notions of semi-uniform input-to-state stability and its subclass, polynomial input-to-state stability, for infinite-dimensional systems. We establish a characterization of semi-uniform input-to-state stability based on attractivity properties as in the uniform case. Sufficient conditions for linear systems to be polynomially input-to-state stable are provided, which restrict the range of the input operator depending on the rate of polynomial decay of the product of the semigroup and the resolvent of its generator. We also show that a class of bilinear systems are polynomially integral input-to-state stable under a certain smoothness assumption on nonlinear operators.

Neural network interpolation operators optimized by Lagrange polynomial

In this paper, we introduce a new type of interpolation operators by using Lagrange polynomials of degree r, which can be regarded as feedforward neural networks with four layers. The approximation rate of the new operators can be estimated by the (r+1)-th modulus of smoothness of the objective functions. By adding some smooth assumptions on the activation function, we establish two important inequalities of the derivatives of the operators. With these two inequalities, by using the K-functional and Berens–Lorentz lemma in approximation theory, we establish the converse theorem of approximation. We also give the Voronovskaja-type asymptotic estimation of the operators for smooth functions. Furthermore, we extend our operators to the multivariate case, and investigate their approximation properties for multivariate functions. Finally, some numerical examples are given to demonstrate the validity of the theoretical results obtained and the superiority of the operators.

On the monoidal invariance of the cohomological dimension of Hopf algebras

We discuss the question of whether the global dimension is a monoidal invariant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidal categories of comodules, then their global dimensions should be equal. We provide several positive new answers to this question, under various assumptions of smoothness, cosemisimplicity or finite dimension. We also discuss the comparison between the global dimension and the Gerstenhaber-Schack cohomological dimension in the cosemisimple case, obtaining equality in the case the latter is finite. One of our main tools is the new concept of twisted separable functor.

GRAPH-THEORETICAL APPROACH TO ROBUST 3D NORMAL EXTRACTION OF LIDAR DATA

Abstract. Low dimensional primitive feature extraction from LiDAR point clouds (such as planes) forms the basis of majority of LiDAR data processing tasks. A major challenge in LiDAR data analysis arises from the irregular nature of LiDAR data that forces practitioners to either regularize the data using some form of gridding or utilize a triangular mesh such as triangulated irregular network (TIN). While there have been a handful applications using LiDAR data as a connected graph, a principled treatment of utilizing graph-theoretical approach for LiDAR data modelling is still lacking. In this paper, we try to bridge this gap by utilizing graphical approach for normal estimation from LiDAR point clouds. We formulate the normal estimation problem in an optimization framework, where we find the corresponding normal vector for each LiDAR point by utilizing its nearest neighbors and simultaneously enforcing a graph smoothness assumption based on point samples. This is a non-linear constrained convex optimization problem which can then be solved using projected conjugate gradient descent to yield an unique solution. As an enhancement to our optimization problem, we also provide different weighted solutions based on the dot product of the normals and Euclidean distance between the points. In order to assess the performance of our proposed normal extraction method and weighting strategies, we first provide a detailed analysis on repeated randomly generated datasets with four different noise levels and four different tuning parameters. Finally, we benchmark our proposed method against existing state-of-the-art approaches on a large scale synthetic plane extraction dataset. The code for the proposed approach along with the simulations and benchmarking is available at https://github.com/arpan-kusari/graph-plane-extraction-simulation.

Boundedness of the nodal domains of additive Gaussian fields

We study the connectivity of the excursion sets of additive Gaussian fields, i.e. stationary centred Gaussian fields whose covariance function decomposes into a sum of terms that depend separately on the coordinates. Our main result is that, under mild smoothness and correlation decay assumptions, the excursion sets { f ≤ ℓ } \{f \le \ell \} of additive planar Gaussian fields are bounded almost surely at the critical level ℓ c = 0 \ell _c = 0 . Since we do not assume positive correlations, this provides the first examples of continuous non-positively-correlated stationary planar Gaussian fields for which the boundedness of the nodal domains has been confirmed. By contrast, in dimension d ≥ 3 d \ge 3 the excursion sets have unbounded components at all levels.

ScSGL: kernelized signed graph learning for single-cell gene regulatory network inference.

Elucidating the topology of gene regulatory networks (GRNs) from large single-cell RNA sequencing datasets, while effectively capturing its inherent cell-cycle heterogeneity and dropouts, is currently one of the most pressing problems in computational systems biology. Recently, graph learning (GL) approaches based on graph signal processing have been developed to infer graph topology from signals defined on graphs. However, existing GL methods are not suitable for learning signed graphs, a characteristic feature of GRNs, which are capable of accounting for both activating and inhibitory relationships in the gene network. They are also incapable of handling high proportion of zero values present in the single cell datasets. To this end, we propose a novel signed GL approach, scSGL, that learns GRNs based on the assumption of smoothness and non-smoothness of gene expressions over activating and inhibitory edges, respectively. scSGL is then extended with kernels to account for non-linearity of co-expression and for effective handling of highly occurring zero values. The proposed approach is formulated as a non-convex optimization problem and solved using an efficient ADMM framework. Performance assessment using simulated datasets demonstrates the superior performance of kernelized scSGL over existing state of the art methods in GRN recovery. The performance of scSGL is further investigated using human and mouse embryonic datasets. The scSGL code and analysis scripts are available on https://github.com/Single-Cell-Graph-Learning/scSGL. Supplementary data are available at Bioinformatics online.

Moreau Envelope Augmented Lagrangian Method for Nonconvex Optimization with Linear Constraints

The augmented Lagrangian method (ALM) is one of the most useful methods for constrained optimization. Its convergence has been well established under convexity assumptions or smoothness assumptions, or under both assumptions. ALM may experience oscillations and divergence when the underlying problem is simultaneously nonconvex and nonsmooth. In this paper, we consider the linearly constrained problem with a nonconvex (in particular, weakly convex) and nonsmooth objective. We modify ALM to use a Moreau envelope of the augmented Lagrangian and establish its convergence under conditions that are weaker than those in the literature. We call it the Moreau envelope augmented Lagrangian (MEAL) method. We also show that the iteration complexity of MEAL is $$o(\varepsilon ^{-2})$$ to yield an $$\varepsilon $$ -accurate first-order stationary point. We establish its whole sequence convergence (regardless of the initial guess) and a rate when a Kurdyka–Łojasiewicz property is assumed. Moreover, when the subproblem of MEAL has no closed-form solution and is difficult to solve, we propose two practical variants of MEAL, an inexact version called iMEAL with an approximate proximal update, and a linearized version called LiMEAL for the constrained problem with a composite objective. Their convergence is also established.

Edgeworth expansions for network moments

Network method of moments (Ann. Statist. 39 (2011) 2280–2301) is an important tool for nonparametric network inference. However, there has been little investigation on accurate descriptions of the sampling distributions of network moment statistics. In this paper, we present the first higher-order accurate approximation to the sampling CDF of a studentized network moment by Edgeworth expansion. In sharp contrast to classical literature on noiseless U-statistics, we show that the Edgeworth expansion of a network moment statistic as a noisy U-statistic can achieve higher-order accuracy without nonlattice or smoothness assumptions but just requiring weak regularity conditions. Behind this result is our surprising discovery that the two typically-hated factors in network analysis, namely, sparsity and edgewise observational errors, jointly play a blessing role, contributing a crucial self-smoothing effect in the network moment statistic and making it analytically tractable. Our assumptions match the minimum requirements in related literature. For sparse networks, our theory shows that our empirical Edgeworth expansion and a simple normal approximation both achieve the same gradually depreciating Berry–Esseen-type bound as the network becomes sparser. This result also significantly refines the best previous theoretical result. For practitioners, our empirical Edgeworth expansion is highly accurate and computationally efficient. It is also easy to implement and convenient for parallel computing. We demonstrate the clear advantage of our method by several comprehensive simulation studies. As a byproduct, we also provide a finite-sample analysis of the network jackknife. We showcase three applications of our results in network inference. We prove, to our knowledge, the first theoretical guarantee of higher-order accuracy for some network bootstrap schemes, and moreover, the first theoretical guidance for selecting the subsample size for network subsampling. We also derive a one-sample test and the Cornish–Fisher confidence interval for a given moment with higher-order accurate controls of confidence level and type I error, respectively.

ELUCIDATING THE FRACTAL NATURE OF POWDER BED IN SELECTIVE LASER MELTING OF METALLIC COMPONENTS

In this work, the fractal nature of Selective Laser Melting (SLM) additive manufacturing process (AM) is elucidated. Fractal dimension and lacunarity of metallic powders are calculated from Scanning Electron Microscopy (SEM) images adapted from literature. The complexity and homogeneity of the textures of the powder beds are also studied through the comparison of fractal dimension and lacunarity. It is found that better densification results are obtained when the powder bed’s fractal dimension is closer to the golden mean number of 1.618. Furthermore, this finding is extended to expressions for predicting the component’s bulk density produced via SLM by setting the [Formula: see text] exponent equal to the golden mean value and finding the proportionality constant, [Formula: see text], using a nonlinear least squares method. The proposed approach works well since theoretical prediction and experimental data compare well with root-mean-square-error (RMSE) values that do not exceed [Formula: see text]. This work sheds new light on enhancing additive manufacturing technologies considering the fractal nature of SLM since its process mathematical models are constructed around Euclidian space-time with continuous smooth assumptions that should be adapted to include the fractal nature of the manufacturing process aiming to improve their precision. The underlying interweaving of SLM, as a fractal process, and the golden mean number is revealed.