We introduce ensembles of stochastic neural networks to approximate the Bayesian posterior, combining stochastic methods such as dropout with deep ensembles. The stochastic ensembles are formulated as families of distributions and trained to approximate the Bayesian posterior with variational inference. We implement stochastic ensembles based on Monte Carlo dropout, DropConnect and a novel non-parametric version of dropout and evaluate them on a toy problem and CIFAR image classification. For both tasks, we test the quality of the posteriors directly against Hamiltonian Monte Carlo simulations. Our results show that stochastic ensembles provide more accurate posterior estimates than other popular baselines for Bayesian inference.
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We give an explicit solution formula for the polynomial regression problem in terms of Schur polynomials and Vandermonde determinants. We thereby generalize the work of Chang, Deng, and Floater to the case of model functions of the form $\sum _{i=1}^{n} a_{i} x^{d_{i}}$ for some integer exponents $d_{1} >d_{2} >\dotsc >d_{n} \geq 0$ and phrase the results using Schur polynomials. Even though the solution circumvents the well-known problems with the forward stability of the normal equation, it is only of practical value if $n$ is small because the number of terms in the formula grows rapidly with the number $m$ of data points. The formula can be evaluated essentially without rounding.
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.
Complex interval arithmetic is a powerful tool for the analysis of computational errors. The naturally arising rectangular, polar, and circular (together called primitive) interval types are not closed under simple arithmetic operations, and their use yields overly relaxed bounds. The later introduced polygonal type, on the other hand, allows for arbitrarily precise representation of the above operations for a higher computational cost. We propose the polyarcular interval type as an effective extension of the previous types. The polyarcular interval can represent all primitive intervals and most of their arithmetic combinations precisely and has an approximation capability competing with that of the polygonal interval. In particular, in antenna tolerance analysis it can achieve perfect accuracy for lower computational cost then the polygonal type, which we show in a relevant case study. In this paper, we present a rigorous analysis of the arithmetic properties of all five interval types, involving a new algebro-geometric method of boundary analysis.
Intracranial aneurysms are the leading cause of stroke. One of the established treatment approaches is the embolization induced by coil insertion. However, the prediction of treatment and subsequent changed flow characteristics in the aneurysm, is still an open problem. In this work, we present an approach based on patient specific geometry and parameters including a coil representation as inhomogeneous porous medium. The model consists of the volume-averaged Navier-Stokes equations including the non-Newtonian blood rheology. We solve these equations using a problem-adapted lattice Boltzmann method and present a comparison between fully-resolved and volume-averaged simulations. The results indicate the validity of the model. Overall, this workflow allows for patient specific assessment of the flow due to potential treatment.
The current study investigates the asymptotic spectral properties of a finite difference approximation of nonlocal Helmholtz equations with a Caputo fractional Laplacian and a variable coefficient wave number $\mu$, as it occurs when considering a wave propagation in complex media, characterized by nonlocal interactions and spatially varying wave speeds. More specifically, by using tools from Toeplitz and generalized locally Toeplitz theory, the present research delves into the spectral analysis of nonpreconditioned and preconditioned matrix-sequences. We report numerical evidences supporting the theoretical findings. Finally, open problems and potential extensions in various directions are presented and briefly discussed.
We propose a simple empirical representation of expectations such that: For a number of samples above a certain threshold, drawn from any probability distribution with finite fourth-order statistic, the proposed estimator outperforms the empirical average when tested against the actual population, with respect to the quadratic loss. For datasets smaller than this threshold, the result still holds, but for a class of distributions determined by their first four statistics. Our approach leverages the duality between distributionally robust and risk-averse optimization.
Random features models play a distinguished role in the theory of deep learning, describing the behavior of neural networks close to their infinite-width limit. In this work, we present a thorough analysis of the generalization performance of random features models for generic supervised learning problems with Gaussian data. Our approach, built with tools from the statistical mechanics of disordered systems, maps the random features model to an equivalent polynomial model, and allows us to plot average generalization curves as functions of the two main control parameters of the problem: the number of random features $N$ and the size $P$ of the training set, both assumed to scale as powers in the input dimension $D$. Our results extend the case of proportional scaling between $N$, $P$ and $D$. They are in accordance with rigorous bounds known for certain particular learning tasks and are in quantitative agreement with numerical experiments performed over many order of magnitudes of $N$ and $P$. We find good agreement also far from the asymptotic limits where $D\to \infty$ and at least one between $P/D^K$, $N/D^L$ remains finite.
We consider the classical problems of interpolating a polynomial given a black box for evaluation, and of multiplying two polynomials, in the setting where the bit-lengths of the coefficients may vary widely, so-called unbalanced polynomials. Writing s for the total bit-length and D for the degree, our new algorithms have expected running time $\tilde{O}(s \log D)$, whereas previous methods for (resp.) dense or sparse arithmetic have at least $\tilde{O}(sD)$ or $\tilde{O}(s^2)$ bit complexity.
The morphology and hierarchy of the vascular systems are essential for perfusion in supporting metabolism. In human retina, one of the most energy-demanding organs, retinal circulation nourishes the entire inner retina by an intricate vasculature emerging and remerging at the optic nerve head (ONH). Thus, tracing the vascular branching from ONH through the vascular tree can illustrate vascular hierarchy and allow detailed morphological quantification, and yet remains a challenging task. Here, we presented a novel approach for a robust semi-automatic vessel tracing algorithm on human fundus images by an instance segmentation neural network (InSegNN). Distinct from semantic segmentation, InSegNN separates and labels different vascular trees individually and therefore enable tracing each tree throughout its branching. We have built-in three strategies to improve robustness and accuracy with temporal learning, spatial multi-sampling, and dynamic probability map. We achieved 83% specificity, and 50% improvement in Symmetric Best Dice (SBD) compared to literature, and outperformed baseline U-net. We have demonstrated tracing individual vessel trees from fundus images, and simultaneously retain the vessel hierarchy information. InSegNN paves a way for any subsequent morphological analysis of vascular morphology in relation to retinal diseases.
Retinopathy of prematurity (ROP) is a severe condition affecting premature infants, leading to abnormal retinal blood vessel growth, retinal detachment, and potential blindness. While semi-automated systems have been used in the past to diagnose ROP-related plus disease by quantifying retinal vessel features, traditional machine learning (ML) models face challenges like accuracy and overfitting. Recent advancements in deep learning (DL), especially convolutional neural networks (CNNs), have significantly improved ROP detection and classification. The i-ROP deep learning (i-ROP-DL) system also shows promise in detecting plus disease, offering reliable ROP diagnosis potential. This research comprehensively examines the contemporary progress and challenges associated with using retinal imaging and artificial intelligence (AI) to detect ROP, offering valuable insights that can guide further investigation in this domain. Based on 89 original studies in this field (out of 1487 studies that were comprehensively reviewed), we concluded that traditional methods for ROP diagnosis suffer from subjectivity and manual analysis, leading to inconsistent clinical decisions. AI holds great promise for improving ROP management. This review explores AI's potential in ROP detection, classification, diagnosis, and prognosis.
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