Additional Regularity Research Articles

Splitting for some classes of homeomorphic and coalescing stochastic flows

The splitting scheme (the Kato-Trotter formula) is applied to stochastic flows with common noise of the type introduced by Th.E. Harris. The case of possibly coalescing flows with continuous infinitesimal covariance is considered and the weak convergence of the corresponding finite-dimensional motions is established. As applications, results for the convergence of the associated pushforward measures and dual flows are given. Similarities between splitting and the Euler-Maruyama scheme yield estimates of the speed of the convergence under additional regularity assumptions.

Neural inverse procedural modeling of knitting yarns from images

We investigate the capabilities of neural inverse procedural modeling to infer high-quality procedural yarn models with fiber-level details from single images of depicted yarn samples. While directly inferring all parameters of the underlying yarn model based on a single neural network may seem an intuitive choice, we show that the complexity of yarn structures in terms of twisting and migration characteristics of the involved fibers can be better encountered in terms of ensembles of networks that focus on individual characteristics. We analyze the effect of different loss functions including a parameter loss to penalize the deviation of inferred parameters to ground truth annotations, a reconstruction loss to enforce similar statistics of the image generated for the estimated parameters in comparison to training images as well as an additional regularization term to explicitly penalize deviations between latent codes of synthetic images and the average latent code of real images in the encoder’s latent space. We demonstrate that the combination of a carefully designed parametric, procedural yarn model with respective network ensembles as well as loss functions even allows robust parameter inference when solely trained on synthetic data. Since our approach relies on the availability of a yarn database with parameter annotations and we are not aware of such a respectively available dataset, we additionally provide, to the best of our knowledge, the first dataset of yarn images with annotations regarding the respective yarn parameters. For this purpose, we use a novel yarn generator that improves the realism of the produced results over previous approaches.

Strichartz estimates for the 2D and 3D massless Dirac-Coulomb equations and applications

In this paper we continue the analysis of the dispersive properties of the 2D and 3D massless Dirac-Coulomb equations that has been started in [7] and [8]. We prove a priori estimates of the solution of the mentioned systems, in particular Strichartz estimates with an additional angular regularity, exploiting the tools developed in the previous works. As an application, we show local well-posedness results for a Dirac-Coulomb equation perturbed with Hartree-type nonlinearities.

DC Neural Networks avoid overfitting in one-dimensional nonlinear regression

In this paper, we analyze Difference of Convex Neural Networks in the context of one-dimensional nonlinear regression. Specifically, we show the surprising ability of the Difference of Convex Multilayer Perceptron (DC-MLP) to avoid overfitting in nonlinear regression. Otherwise said, DC-MLPs self-regularize (do not require additional regularization techniques). Thus, DC-MLPs could result very useful for practical purposes based on one-dimensional nonlinear regression. It turns out that shallow MLPs with a convex activation (ReLU, softplus, etc.) fall in the class of DC-MLPs. On the other hand, we call SQ-MLP the shallow MLP with a Squashing activation (logistic, hyperbolic tangent, etc.). In the numerical experiments, we show that DC-MLPs used for nonlinear regression avoid overfitting, in contrast with SQ-MLPs. We also compare DC-MLPs and SQ-MLPs from a theoretical point of view.

EQUILIBRIUM PROBLEM FOR A TIMOSHENKO PLATE CONTACTING BY THE SIDE AND FACE SURFACES

A new model for a Timoshenko plate contacting by the side surface or the edge of the bottom surface (with respect to the chosen coordinate system) with a rigid obstacle of a given configuration is justified. The non-deformable obstacle is defined by a cylindrical surface, the generators of which are perpendicular to the middle plane of the plate, as well as by a part of the plane that is parallel to the middle plane of the plate. A corresponding variational problem is formulated as a minimization of an energy functional over a non-convex set of admissible displacements. The set of admissible displacements is defined taking into account a condition of fixing and a nonpenetration condition. The nonpenetration condition is given as a system of inequalities describing two cases of possible contacts of the plate and the rigid obstacle. Namely, these two cases correspond to different types of contacts by the plate side edge and by the edge of the plate bottom surface. The solvability of the problem is established. In particular case, when contact zones is previously known, an equivalent differential statement is found under the assumption of additional regularity for the solution to the variational problem.

From Kernel Methods to Neural Networks: A Unifying Variational Formulation

AbstractThe minimization of a data-fidelity term and an additive regularization functional gives rise to a powerful framework for supervised learning. In this paper, we present a unifying regularization functional that depends on an operator $$\textrm{L}$$ L and on a generic Radon-domain norm. We establish the existence of a minimizer and give the parametric form of the solution(s) under very mild assumptions. When the norm is Hilbertian, the proposed formulation yields a solution that involves radial-basis functions and is compatible with the classical methods of machine learning. By contrast, for the total-variation norm, the solution takes the form of a two-layer neural network with an activation function that is determined by the regularization operator. In particular, we retrieve the popular ReLU networks by letting the operator be the Laplacian. We also characterize the solution for the intermediate regularization norms $$\Vert \cdot \Vert =\Vert \cdot \Vert _{L_p}$$ ‖ · ‖ = ‖ · ‖ L p with $$p\in (1,2]$$ p ∈ ( 1 , 2 ] . Our framework offers guarantees of universal approximation for a broad family of regularization operators or, equivalently, for a wide variety of shallow neural networks, including the cases (such as ReLU) where the activation function is increasing polynomially. It also explains the favorable role of bias and skip connections in neural architectures.

A non-regularization self-supervised Retinex approach to low-light image enhancement with parameterized illumination estimation

In current Retinex-based low-light image enhancement (LLIE) methods, fine-tuning regularization parameters for Retinex decomposition and illumination estimation can be cumbersome. To address this, we present a novel non-regularization self-supervised Retinex approach for illumination estimation. Our contributions are twofold: First, we introduce a self-supervised method that incorporates edge-aware smoothness properties in bilateral learning, eliminating the need for regularization terms and simplifying parameter adjustments. Second, to enforce smoothness constraints on the estimated bilateral grid, we propose a bilateral grid parameterization network. This network employs a generative encoder to parameterize the bilateral grid of illumination and a trainable slicing layer guided by a map, reconstructing the grid into an illumination map. Despite the absence of regularization terms, our model excels in generating piece-wise smooth illumination, resulting in enhanced naturalness and improved contrast in images. Our model offers exceptional flexibility by eliminating the need for additional regularization terms and parameter fine-tuning. Moreover, it does not depend on external datasets for training, overcoming dataset collection challenges. Extensive experiments, comparing our model with eight state-of-the-art methods across five public available datasets, unequivocally demonstrate our model’s state-of-the-art performance based on key metrics such as NIQE, NIQMC, and CPCQI. These results reaffirm the effectiveness of our approach in low-light image enhancement. Code will be available at: https://github.com/zhaozunjin/NeurBR.

The Stochastic Klausmeier System and A Stochastic Schauder-Tychonoff Type Theorem

On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.

Stage‐cost design for optimal and model predictive control of linear port‐Hamiltonian systems: Energy efficiency and robustness

AbstractWe consider singular optimal control of port‐Hamiltonian systems with minimal energy supply. We investigate the robustness of different stage‐cost designs w.r.t. time discretization and show that alternative formulations that are equivalent in continuous time, differ strongly in view of discretization. Furthermore, we consider the impact of additional quadratic control regularization and demonstrate that this leads to a considerable increase in energy consumption. Then, we extend our results to the tracking problem within model predictive control and show that the intrinsic but singular choice of the cost functional as the supplied energy leads to a substantial improvement of the closed‐loop performance.

Asymptotic numerical method for cyclic elasto-plasticity and its application to a steam turbine rotor

Flexible operation of the conventional power plants is necessary for the integration of renewables in the energy mix. Steam turbine rotors are one of the most affected components due to the daily thermal load cycling requirements. In order to accurately study the effect of cycling on fatigue, it is necessary to solve cyclic 3D elasto-plasticity problems for a large number of cycles. The established methods are iterative in nature to arrive at the current yield surface for each elasto-plastic step and are computationally extensive. In this paper, we propose the non-iterative Asymptotic Numerical Method (ANM) based solution technique for 3D cyclic elasto-plasticity problems for the first time by a novel regularized Kuhn-Tucker condition. Regularization techniques used in previous works take care of the partial cycle only, i.e., elastic loading, elastic-plastic transition, elasto-plastic flow and elastic unloading for simple linear work hardening models of plasticity. In the present work, the proposed method extends the applicability of ANM by incorporating elastic-plastic transition and elasto-plasticity in reverse loading, second unloading and repeated loading for any number of cycles so that the computationally efficient fatigue analysis can be carried out. In addition, the extension also includes the application of ANM to non-linear kinematic hardening through the Chaboche plasticity model combined with isotropic hardening. For these advancements, additional regularizations are proposed and implemented in a Finite Element Code. The obtained results are then compared to the solutions obtained from the conventional Newton Raphson (NR) solution technique. The proposed ANM is applied to a steam turbine rotor undergoing a typical thermal load cycling and the cyclic elasto-plastic response is compared with that obtained using the conventional Newton Raphson solution technique.

Imaging through a scattering medium via model-driven deep learning.

Imaging through a scattering medium is of great significance in many areas. Especially, speckle correlation imaging has been valued for its noninvasiveness. In this work, we report a deep learning solution that incorporates the physical model and an additional regularization for high-fidelity speckle correlation imaging. Without large-scale data to train, the physical model and regularization prior provide a correct direction for neural network to precisely reconstruct hidden objects from speckle under different scattering scenarios and noise levels. Experimental results demonstrate that the proposed method presents a significant advance in improving generalization and combating the invasion of noise.

Taming Diffusion Models for Music-Driven Conducting Motion Generation

Generating the motion of orchestral conductors from a given piece of symphony music is a challenging task since it requires a model to learn semantic music features and capture the underlying distribution of real conducting motion. Prior works have applied Generative Adversarial Networks (GAN) to this task, but the promising diffusion model, which recently showed its advantages in terms of both training stability and output quality, has not been exploited in this context. This paper presents Diffusion-Conductor, a novel DDIM-based approach for music-driven conducting motion generation, which integrates the diffusion model to a two-stage learning framework. We further propose a random masking strategy to improve the feature robustness, and use a pair of geometric loss functions to impose additional regularizations and increase motion diversity. We also design several novel metrics, including Frechet Gesture Distance (FGD) and Beat Consistency Score (BC) for a more comprehensive evaluation of the generated motion. Experimental results demonstrate the advantages of our model. The code is released at https://github.com/viiika/Diffusion-Conductor.

Adversarial Heterogeneous Graph Neural Network for Robust Recommendation

Recommendation systems play a vital role in identifying the hidden interactions between users and items in online social networks. Recently, graph neural networks (GNNs) have exhibited significant performance gains by modeling the information propagation process in graph-structured data for a recommendation. However, existing GNN-based methods do not have broad applicability to heterogeneous graphs that integrate auxiliary data with diverse types. Moreover, graph structures are susceptible to noise and even unnoticed malicious perturbations, as perturbations from connected nodes can create cumulative effects on a target node in the graph. To enhance the robustness and generalization of GNN-based recommendations, we propose a new optimization model named Adversarial Heterogeneous Graph Neural Network for RECommendation (AHGNNRec). First, AHGNNRec learns user and item embeddings by exploring the distinct contributions of various types of interactions between users and items using a hierarchical heterogeneous graph neural network (HGNN). Second, to produce more robust embeddings for recommendations, we employ the adversarial training (AT) method to optimize the HGNN layers. AT is a min-max optimization training process where the generated adversarial fake nodes from normal nodes with intentional perturbations try to maximally deteriorate the recommendation performance. Following this, we learn about these adversarial user or item nodes by minimizing the impact of an additional regularization term for the recommendation. The experimental outcomes on two real-world benchmark datasets demonstrate the effectiveness of AHGNNRec.

Topic Modeling for Text Structure Assessment: The case of Russian Academic Texts

Background: Automatic assessment of text complexity levels is viewed as an important task, primarily in education. The existing methods of computing text complexity employ simple surface text properties neglecting complexity of text content and structure. The current paradigm of complexity studies can no longer keep up with the challenges of automatic evaluation of text structure.
 Purpose: The aim of the paper is twofold: (1) it introduces a new notion, i.e. complexity of a text topical structure which we define as a quantifiable measure and combination of four parameters, i.e. number of topics, topic coherence, topic distribution, and topic weight. We hypothesize that these parameters are dependent variables of text complexity and aligned with the grade level; (2) the paper is also aimed at justifying applicability of the recently developed methods of topic modeling to measuring complexity of a text topical structure.
 Method: To test this hypothesis, we use Russian Academic Corpus comprising school textbooks, texts of Russian as a foreign language and fiction texts recommended for reading in different grades, and employ it in three versions: (i) Full Texts Corpus, (ii) Corpus of Segments, (iii) Corpus of Paragraphs. The software tools we implement include LDA (Latent Dirichlet Allocation), OnlineLDA and Additive Regularization Of Topic Models with Word2vec-based metric and Normalized Pairwise Mutual Information.
 Results: Our findings include the following: the optimal number of topics in educational texts varies around 20; topic coherence and topic distribution are identified to be functions of grade level complexity; text complexity is suggested to be estimated with structural organization parameters and viewed as a new algorithm complementing the classical approach of text complexity assessment based on linguistic features.
 Conclusion: The results reported and discussed in the article strongly suggest that the theoretical framework and the analytic algorithms used in the study might be fruitfully applied in education and provide a basis for assessing complexity of academic texts.

Analysis of a thermoelastic problem with the Moore–Gibson–Thompson microtemperatures

In this paper, we study, from both an analytical and a numerical point of view, a poro-thermoelastic problem with microtemperatures. The so-called Moore–Gibson–Thompson equation is used to model the contribution for the temperature and microtemperatures. An existence and uniqueness result is proved by using the theory of linear semigroups of contractions and, for the one-dimensional case, the exponential energy decay is found under some conditions on the constitutive coefficients. Then, a fully discrete approximation is introduced by using the finite element method and the implicit Euler scheme. We show that the discrete energy decays and we obtain some a priori error estimates from which, under some adequate additional regularity conditions on the continuous solution, we derive the linear convergence of the approximations. Finally, we perform some numerical simulations to demonstrate the accuracy of the approximations and the behavior of the discrete energy and the solution.

Intruder-free cumulant-truncated driven similarity renormalization group second-order multireference perturbation theory.

Accurate multireference electronic structure calculations are important for constructing potential energy surfaces. Still, even in the case of low-scaling methods, their routine use is limited by the steep growth of the computational and storage costs as the active space grows. This is primarily due to the occurrence of three- and higher-body density matrices or, equivalently, their cumulants. This work examines the effect of various cumulant truncation schemes on the accuracy of the driven similarity renormalization group second-order multireference perturbation theory. We test four different levels of three-body reduced density cumulant truncations that set different classes of cumulant elements to zero. Our test cases include the singlet-triplet gap of CH2, the potential energy curves of the XΣg+1 and AΣu+3 states of N2, and the singlet-triplet splittings of oligoacenes. Our results show that both relative and absolute errors introduced by these cumulant truncations can be as small as 0.5kcal mol-1 or less. At the same time, the amount of memory required is reduced from O(NA6) to O(NA5), where NA is the number of active orbitals. No additional regularization is needed to prevent the intruder state problem in the cumulant-truncated second-order driven similarity renormalization group multireference perturbation theory methods.

A MGT thermoelastic problem with two relaxation parameters

In this paper, we consider, from both analytical and numerical viewpoints, a thermoelastic problem. The so-called MGT model, with two different relaxation parameters, is used for both the displacements and the thermal displacement, leading to a linear coupled system made by two third-order in time partial differential equations. Then, using the theory of linear semi-groups the existence and uniqueness to this problem is proved. If we restrict ourselves to the one-dimensional case, the exponential decay of the energy is obtained assuming some conditions on the constitutive parameters. Then, using the classical finite element method and the implicit Euler scheme, we introduce a fully discrete approximation of a variational formulation of the thermomechanical problem. A main a priori error estimates result is shown, from which we conclude the linear convergence under suitable additional regularity conditions. Finally, we present some one-dimensional numerical simulations to demonstrate the convergence of the fully discrete approximation, the behavior of the discrete energy decay and the dependence on a coupling parameter.

Stability properties of multidimensional symmetric hyperbolic systems with damping, differential constraints and delay

Multidimensional linear hyperbolic systems with constraints and delay are considered. The existence and uniqueness of solutions for rough data are established using Friedrichs method. With additional regularity and compatibility on the initial data and initial history, the stability of such systems are discussed. Under suitable assumptions on the coefficient matrices, we establish standard or regularity-loss type decay estimates. For data that are integrable, better decay rates are provided. The results are applied to the wave, Timoshenko, and linearized Euler–Maxwell systems with delay.

A Hilbertian projection method for constrained level set-based topology optimisation

We present an extension of the projection method proposed by Challis et al. (Int J Solids Struct 45(14–15):4130–4146, 2008) for constrained level set-based topology optimisation that harnesses the Hilbertian velocity extension-regularisation framework. Our Hilbertian projection method chooses a normal velocity for the level set function as a linear combination of (1) an orthogonal projection operator applied to the extended optimisation objective shape sensitivity and (2) a weighted sum of orthogonal basis functions for the extended constraint shape sensitivities. This combination aims for the best possible first-order improvement of the optimisation objective in addition to first-order improvement of the constraints. Our formulation utilising basis orthogonalisation naturally handles linearly dependent constraint shape sensitivities. Furthermore, use of the Hilbertian extension-regularisation framework ensures that the resulting normal velocity is extended away from the boundary and enriched with additional regularity. Our approach is generally applicable to any topology optimisation problem to be solved in the level set framework. We consider several benchmark constrained microstructure optimisation problems and demonstrate that our method is effective with little-to-no parameter tuning. We also find that our method performs well when compared to a Hilbertian sequential linear programming method.

Handling Ill-conditioned Omics Data with Deep Probabilistic Models.

The advent of high-throughput technologies has produced an increase in the dimensionality of omics datasets, which limits the application of machine learning methods due to the great unbalance between the number of observations and features. In this scenario, dimensionality reduction is essential to extract the relevant information within these datasets and project it in a low-dimensional space, and probabilistic latent space models are becoming popular given their capability to capture the underlying structure of the data as well as the uncertainty in the information. This article aims to provide a general classification and dimensionality reduction method based on deep latent space models that tackles two of the main problems that arise in omics datasets: the presence of missing data and the limited number of observations against the number of features. We propose a semi-supervised Bayesian latent space model that infers a low-dimensional embedding driven by the target label: the Deep Bayesian Logistic Regression (DBLR) model. During inference, the model also learns a global vector of weights that allows it to make predictions given the low-dimensional embedding of the observations. Since this kind of dataset is prone to overfitting, we introduce an additional probabilistic regularization method based on the semi-supervised nature of the model. We compared the performance of the DBLR against several state-of-the-art methods for dimensionality reduction, both in synthetic and real datasets with different data types. The proposed model provides more informative low-dimensional representations, outperforms the baseline methods in classification, and can naturally handle missing entries.