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1. Practical Consequences of the Bias in the Laplace Approximation to Marginal Likelihood for Hierarchical Models

   https://doi.org/10.3390/e27030289  

   Lele, Subhash R; Glen, C George; Ponciano, José Miguel (March 2025, Entropy)

   Due to the high dimensional integration over latent variables, computing marginal likelihood and posterior distributions for the parameters of a general hierarchical model is a difficult task. The Markov Chain Monte Carlo (MCMC) algorithms are commonly used to approximate the posterior distributions. These algorithms, though effective, are computationally intensive and can be slow for large, complex models. As an alternative to the MCMC approach, the Laplace approximation (LA) has been successfully used to obtain fast and accurate approximations to the posterior mean and other derived quantities related to the posterior distribution. In the last couple of decades, LA has also been used to approximate the marginal likelihood function and the posterior distribution. In this paper, we show that the bias in the Laplace approximation to the marginal likelihood has substantial practical consequences. 

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2. Discrepancy in modular arithmetic progressions

   https://doi.org/10.1112/s0010437x22007758  

   Fox, Jacob; Xu, Max Wenqiang; Zhou, Yunkun (November 2022, Compositio Mathematica)

   Celebrated theorems of Roth and of Matoušek and Spencer together show that the discrepancy of arithmetic progressions in the first $$n$$ positive integers is $$\Theta (n^{1/4})$$ . We study the analogous problem in the $$\mathbb {Z}_n$$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $$\mathbb {Z}_n$$ for all positive integer $$n$$ . We further determine up to a constant factor the discrepancy of arithmetic progressions in $$\mathbb {Z}_n$$ for many $$n$$ . For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $$\mathbb {Z}_n$$ is $$\Theta (n^{1/3+r_k/(6k)})$$ , where $$r_k \in \{0,1,2\}$$ is the remainder when $$k$$ is divided by $$3$$ . This solves a problem of Hebbinghaus and Srivastav. 

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3. Optimal estimation of bacterial growth rates based on a permuted monotone matrix

   https://doi.org/10.1093/biomet/asaa082  

   Ma, Rong; Cai, T Tony; Li, Hongzhe (October 2020, Biometrika)

   Summary Motivated by the problem of estimating bacterial growth rates for genome assemblies from shotgun metagenomic data, we consider the permuted monotone matrix model $$Y=\Theta\Pi+Z$$ where $$Y\in \mathbb{R}^{n\times p}$$ is observed, $$\Theta\in \mathbb{R}^{n\times p}$$ is an unknown approximately rank-one signal matrix with monotone rows, $$\Pi \in \mathbb{R}^{p\times p}$$ is an unknown permutation matrix, and $$Z\in \mathbb{R}^{n\times p}$$ is the noise matrix. In this article we study estimation of the extreme values associated with the signal matrix $$\Theta$$, including its first and last columns and their difference. Treating these estimation problems as compound decision problems, minimax rate-optimal estimators are constructed using the spectral column-sorting method. Numerical experiments on simulated and synthetic microbiome metagenomic data are conducted, demonstrating the superiority of the proposed methods over existing alternatives. The methods are illustrated by comparing the growth rates of gut bacteria in inflammatory bowel disease patients and control subjects. 

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4. Large-time behavior of solutions of parabolic equations on the real line with convergent initial data III: unstable limit at infinity

   Pauthier, Antoine; Poláčik, Peter (August 2022, Partial Differential Equations and Applications)

   Abstract This is a continuation, and conclusion, of our study of bounded solutions u of the semilinear parabolic equation $$u_t=u_{xx}+f(u)$$ u t = u xx + f ( u ) on the real line whose initial data $$u_0=u(\cdot ,0)$$ u 0 = u ( · , 0 ) have finite limits $$\theta ^\pm $$ θ ± as $$x\rightarrow \pm \infty $$ x → ± ∞ . We assume that f is a locally Lipschitz function on $$\mathbb {R}$$ R satisfying minor nondegeneracy conditions. Our goal is to describe the asymptotic behavior of u ( x ,  t ) as $$t\rightarrow \infty $$ t → ∞ . In the first two parts of this series we mainly considered the cases where either $$\theta ^-\ne \theta ^+$$ θ - ≠ θ + ; or $$\theta ^\pm =\theta _0$$ θ ± = θ 0 and $$f(\theta _0)\ne 0$$ f ( θ 0 ) ≠ 0 ; or else $$\theta ^\pm =\theta _0$$ θ ± = θ 0 , $$f(\theta _0)=0$$ f ( θ 0 ) = 0 , and $$\theta _0$$ θ 0 is a stable equilibrium of the equation $${{\dot{\xi }}}=f(\xi )$$ ξ ˙ = f ( ξ ) . In all these cases we proved that the corresponding solution u is quasiconvergent—if bounded—which is to say that all limit profiles of $$u(\cdot ,t)$$ u ( · , t ) as $$t\rightarrow \infty $$ t → ∞ are steady states. The limit profiles, or accumulation points, are taken in $$L^\infty _{loc}(\mathbb {R})$$ L loc ∞ ( R ) . In the present paper, we take on the case that $$\theta ^\pm =\theta _0$$ θ ± = θ 0 , $$f(\theta _0)=0$$ f ( θ 0 ) = 0 , and $$\theta _0$$ θ 0 is an unstable equilibrium of the equation $${{\dot{\xi }}}=f(\xi )$$ ξ ˙ = f ( ξ ) . Our earlier quasiconvergence theorem in this case involved some restrictive technical conditions on the solution, which we now remove. Our sole condition on $$u(\cdot ,t)$$ u ( · , t ) is that it is nonoscillatory (has only finitely many critical points) at some $$t\ge 0$$ t ≥ 0 . Since it is known that oscillatory bounded solutions are not always quasiconvergent, our result is nearly optimal. 

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5. AE-OT-GAN: Training GANs from data specific latent distribution

   Dongsheng An, Yang Guo (January 2020, European Conference on Computer Vision (ECCV2020))

   Though generative adversarial networks (GANs) are prominent models to generate realistic and crisp images, they are unstable to train and suffer from the mode collapse problem. The problems of GANs come from approximating the intrinsic discontinuous distribution transform map with continuous DNNs. The recently proposed AE-OT model addresses the discontinuity problem by explicitly computing the discontinuous optimal transform map in the latent space of the autoencoder. Though have no mode collapse, the generated images by AE-OT are blurry. In this paper, we propose the AE-OT-GAN model to utilize the advantages of the both models: generate high quality images and at the same time overcome the mode collapse problems. Specifically, we firstly embed the low dimensional image manifold into the latent space by autoencoder (AE). Then the extended semi-discrete optimal transport (SDOT) map is used to generate new latent codes. Finally, our GAN model is trained to generate high quality images from the latent distribution induced by the extended SDOT map. The distribution transform map from this dataset related latent distribution to the data distribution will be continuous, and thus can be well approximated by the continuous DNNs. Additionally, the paired data between the latent codes and the real images gives us further restriction about the generator and stabilizes the training process. Experiments on simple MNIST dataset and complex datasets like CIFAR10 and CelebA show the advantages of the proposed method. 

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