Data assimilation (DA) methods use priors arising from differential equations
to robustly interpolate and extrapolate data. Popular techniques such as
ensemble methods that handle high-dimensional, nonlinear PDE priors focus
mostly on state estimation, however can have difficulty learning the parameters
accurately. On the other hand, machine learning based approaches can naturally
learn the state and parameters, but their applicability can be limited, or
produce uncertainties that are hard to interpret. Inspired by the Integrated
Nested Laplace Approximation (INLA) method in spatial statistics, we propose an
alternative approach to DA based on iteratively linearising the dynamical
model. This produces a Gaussian Markov random field at each iteration, enabling
one to use INLA to infer the state and parameters. Our approach can be used for
arbitrary nonlinear systems, while retaining interpretability, and is
furthermore demonstrated to outperform existing methods on the DA task. By
providing a more nuanced approach to handling nonlinear PDE priors, our
methodology offers improved accuracy and robustness in predictions, especially
where data sparsity is prevalent.

基于迭代线性化动力学模型的集成嵌套拉普拉斯近似方法，通过产生每次迭代的高斯马尔科夫随机场，利用 INLA 推断状态和参数，从而提供了处理非线性偏微分方程先验的更细致的方法，改进了在数据稀疏性普遍存在的情况下的预测准确性和鲁棒性。

非线性动力系统中的迭代 INLA 用于状态和参数估计

Iterated INLA for State and Parameter Estimation in Nonlinear Dynamical  Systems

In this paper, we model the dependencies among the items that are recommended
to a user in a collaborative-filtering problem via a Gaussian Markov Random
Field (MRF). We build upon Besag's auto-normal parameterization and
pseudo-likelihood, which not only enables computationally efficient learning,
but also connects the areas of MRFs and sparse inverse covariance estimation
with autoencoders and neighborhood models, two successful approaches in
collaborative filtering. We propose a novel approximation for learning sparse
MRFs, where the trade-off between recommendation-accuracy and training-time can
be controlled. At only a small fraction of the training-time compared to
various baselines, including deep nonlinear models, the proposed approach
achieved competitive ranking-accuracy on all three well-known data-sets used in
our experiments, and notably a 20% gain in accuracy on the data-set with the
largest number of items.

本文研究了协同过滤问题中推荐给用户的物品之间的依赖关系模型，提出了一种基于高斯马尔科夫随机场的自动归一参数化和伪似然函数的方法，可以实现计算效率和推荐准确性之间的平衡。通过几个数据集的实验证明了该方法的有效性。

随机马尔可夫场在协同过滤中的应用

Markov Random Fields for Collaborative Filtering

The L1-regularized Gaussian maximum likelihood estimator (MLE) has been shown
to have strong statistical guarantees in recovering a sparse inverse covariance
matrix, or alternatively the underlying graph structure of a Gaussian Markov
Random Field, from very limited samples. We propose a novel algorithm for
solving the resulting optimization problem which is a regularized
log-determinant program. In contrast to recent state-of-the-art methods that
largely use first order gradient information, our algorithm is based on
Newton's method and employs a quadratic approximation, but with some
modifications that leverage the structure of the sparse Gaussian MLE problem.
We show that our method is superlinearly convergent, and present experimental
results using synthetic and real-world application data that demonstrate the
considerable improvements in performance of our method when compared to other
state-of-the-art methods.

本研究提出了一种基于牛顿法的新型算法，用于解决优化问题，该问题是一个正则化的对数行列式程序，能够从非常有限的样本中恢复稀疏逆协方差矩阵，或者是高斯马尔科夫随机场的基础图结构，并通过合成和真实的应用数据实验结果表明，与其他最先进的方法相比，我们的方法在性能上有了显着的改进。