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Image Edge detection (ED) is a base task in computer vision. While the performance of the ED algorithm has been improved greatly by introducing CNN-based models, current models still suffer from unsatisfactory precision rates especially when only a low error toleration distance is allowed. Therefore, model architecture for more precise predictions still needs an investigation. On the other hand, the unavoidable noise training data provided by humans would lead to unsatisfactory model predictions even when inputs are edge maps themselves, which also needs a solution. In this paper, more precise ED models are presented with cascaded skipping density blocks (CSDB). Our models obtain state-of-the-art(SOTA) predictions in several datasets, especially in average precision rate (AP), over a high-standard benchmark, which is confirmed by extensive experiments. Also, a novel modification on data augmentation for training is employed, which allows noiseless data to be employed in model training for the first time, and thus further improves the model performance. The relative Python codes can be found on //github.com/Hao-B-Shu/SDPED.

This paper extends various results related to the Gaussian product inequality (GPI) conjecture to the setting of disjoint principal minors of Wishart random matrices. This includes product-type inequalities for matrix-variate analogs of completely monotone functions and Bernstein functions of Wishart disjoint principal minors, respectively. In particular, the product-type inequalities apply to inverse determinant powers. Quantitative versions of the inequalities are also obtained when there is a mix of positive and negative exponents. Furthermore, an extended form of the GPI is shown to hold for the eigenvalues of Wishart random matrices by virtue of their law being multivariate totally positive of order 2 (MTP${}_2$). A new, unexplored avenue of research is presented to study the GPI from the point of view of elliptical distributions.

We develop a numerical method for simulation of incompressible viscous flows by integrating the technology of random vortex method with the core idea of Large Eddy Simulation (LES). Specifically, we utilize the filtering method in LES, interpreted as spatial averaging, along with the integral representation theorem for parabolic equations, to achieve a closure scheme which may be used for calculating solutions of Navier-Stokes equations. This approach circumvents the challenge associated with handling the non-locally integrable 3-dimensional integral kernel in the random vortex method and facilitates the computation of numerical solutions for flow systems via Monte-Carlo method. Numerical simulations are carried out for both laminar and turbulent flows, demonstrating the validity and effectiveness of the method.

We propose a numerical algorithm for the computation of multi-marginal optimal transport (MMOT) problems involving general probability measures that are not necessarily discrete. By developing a relaxation scheme in which marginal constraints are replaced by finitely many linear constraints and by proving a specifically tailored duality result for this setting, we approximate the MMOT problem by a linear semi-infinite optimization problem. Moreover, we are able to recover a feasible and approximately optimal solution of the MMOT problem, and its sub-optimality can be controlled to be arbitrarily close to 0 under mild conditions. The developed relaxation scheme leads to a numerical algorithm which can compute a feasible approximate optimizer of the MMOT problem whose theoretical sub-optimality can be chosen to be arbitrarily small. Besides the approximate optimizer, the algorithm is also able to compute both an upper bound and a lower bound for the optimal value of the MMOT problem. The difference between the computed bounds provides an explicit sub-optimality bound for the computed approximate optimizer. We demonstrate the proposed algorithm in three numerical experiments involving an MMOT problem that stems from fluid dynamics, the Wasserstein barycenter problem, and a large-scale MMOT problem with 100 marginals. We observe that our algorithm is capable of computing high-quality solutions of these MMOT problems and the computed sub-optimality bounds are much less conservative than their theoretical upper bounds in all the experiments.

The field of protein-ligand pose prediction has seen significant advances in recent years, with machine learning-based methods now being commonly used in lieu of classical docking methods or even to predict all-atom protein-ligand complex structures. Most contemporary studies focus on the accuracy and physical plausibility of ligand placement to determine pose quality, often neglecting a direct assessment of the interactions observed with the protein. In this work, we demonstrate that ignoring protein-ligand interaction fingerprints can lead to overestimation of model performance, most notably in recent protein-ligand cofolding models which often fail to recapitulate key interactions.

The digital era has raised many societal challenges, including ICT's rising energy consumption and protecting privacy of personal data processing. This paper considers both aspects in relation to machine learning accuracy in an interdisciplinary exploration. We first present a method to measure the effects of privacy-enhancing techniques on data utility and energy consumption. The environmental-privacy-accuracy trade-offs are discovered through an experimental set-up. We subsequently take a storytelling approach to translate these technical findings to experts in non-ICT fields. We draft two examples for a governmental and auditing setting to contextualise our results. Ultimately, users face the task of optimising their data processing operations in a trade-off between energy, privacy, and accuracy considerations where the impact of their decisions is context-sensitive.

We formulate a uniform tail bound for empirical processes indexed by a class of functions, in terms of the individual deviations of the functions rather than the worst-case deviation in the considered class. The tail bound is established by introducing an initial ``deflation'' step to the standard generic chaining argument. The resulting tail bound is the sum of the complexity of the ``deflated function class'' in terms of a generalization of Talagrand's $\gamma$ functional, and the deviation of the function instance, both of which are formulated based on the natural seminorm induced by the corresponding Cram\'{e}r functions. Leveraging another less demanding natural seminorm, we also show similar bounds, though with implicit dependence on the sample size, in the more general case where finite exponential moments cannot be assumed. We also provide approximations of the tail bounds in terms of the more prevalent Orlicz norms or their ``incomplete'' versions under suitable moment conditions.

Parametric bivariate copula families have been known to flexibly capture various dependence patterns, e.g., either positive or negative dependence in either the lower or upper tails of bivariate distributions. In this paper, our objective is to construct a model that is adaptable enough to capture several of these features simultaneously. We propose a mixture of 4-way rotations of a parametric copula that can achieve this goal. We illustrate the construction using the Clayton family but the concept is general and can be applied to other families. In order to include dynamic dependence regimes, the approach is extended to a time-dependent sequence of mixture copulas in which the mixture probabilities are allowed to evolve in time via a moving average and seasonal types of relationship. The properties of the proposed model and its performance are examined using simulated and real data sets.

Given a finite set of matrices with integer entries, the matrix mortality problem asks if there exists a product of these matrices equal to the zero matrix. We consider a special case of this problem where all entries of the matrices are nonnegative. This case is equivalent to the NFA mortality problem, which, given an NFA, asks for a word $w$ such that the image of every state under $w$ is the empty set. The size of the alphabet of the NFA is then equal to the number of matrices in the set. We study the length of shortest such words depending on the size of the alphabet. We show that for an NFA with $n$ states this length can be at least $2^n - 1$ for an alphabet of size $n$, $2^{(n - 4)/2}$ for an alphabet of size $3$ and $2^{(n - 2)/3}$ for an alphabet of size $2$. We also discuss further open problems related to mortality of NFAs and DFAs.

Fourth-order variational inequalities are encountered in various scientific and engineering disciplines, including elliptic optimal control problems and plate obstacle problems. In this paper, we consider additive Schwarz methods for solving fourth-order variational inequalities. Based on a unified framework of various finite element methods for fourth-order variational inequalities, we develop one- and two-level additive Schwarz methods. We prove that the two-level method is scalable in the sense that the convergence rate of the method depends on $H/h$ and $H/\delta$ only, where $h$ and $H$ are the typical diameters of an element and a subdomain, respectively, and $\delta$ measures the overlap among the subdomains. This proof relies on a new nonlinear positivity-preserving coarse interpolation operator, the construction of which was previously unknown. To the best of our knowledge, this analysis represents the first investigation into the scalability of the two-level additive Schwarz method for fourth-order variational inequalities. Our theoretical results are verified by numerical experiments.