This paper addresses the problem of nonlinear filtering, i.e., computing the
conditional distribution of the state of a stochastic dynamical system given a
history of noisy partial observations. The primary focus is on scenarios
involving degenerate likelihoods or high-dimensional states, where traditional
sequential importance resampling (SIR) particle filters face the weight
degeneracy issue. Our proposed method builds on an optimal transport
interpretation of nonlinear filtering, leading to a simulation-based and
likelihood-free algorithm that estimates the Brenier optimal transport map from
the current distribution of the state to the distribution at the next time
step. Our formulation allows us to harness the approximation power of neural
networks to model complex and multi-modal distributions and employ stochastic
optimization algorithms to enhance scalability. Extensive numerical experiments
are presented that compare our method to the SIR particle filter and the
ensemble Kalman filter, demonstrating the superior performance of our method in
terms of sample efficiency, high-dimensional scalability, and the ability to
capture complex and multi-modal distributions.

基于最优输运解释的非线性滤波方法通过模拟和无似然算法估计从当前状态分布到下一个时间步的 Brenier 最优输运映射，利用神经网络建模复杂和多模态分布以及采用随机优化算法提高可伸缩性，通过广泛的数值实验与 SIR 粒子滤波器和集合卡尔曼滤波器进行比较，展示了我们方法在样本效率、高维可伸缩性和捕捉复杂和多模态分布等方面的优越性。

基于最优传输的高维非线性滤波

Optimal Transport-based Nonlinear Filtering in High-dimensional Settings

Computational units in artificial neural networks compute a linear
combination of their inputs, and then apply a nonlinear filter, often a ReLU
shifted by some bias, and if the inputs come themselves from other units, they
were already filtered with their own biases. In a layer, multiple units share
the same inputs, and each input was filtered with a unique bias, resulting in
output values being based on shared input biases rather than individual optimal
ones. To mitigate this issue, we introduce DAC, a new computational unit based
on preactivation and multiple biases, where input signals undergo independent
nonlinear filtering before the linear combination. We provide a Keras
implementation and report its computational efficiency. We test DAC
convolutions in ResNet architectures on CIFAR-10, CIFAR-100, Imagenette, and
Imagewoof, and achieve performance improvements of up to 1.73%. We exhibit
examples where DAC is more efficient than its standard counterpart as a
function approximator, and we prove a universal representation theorem.

本研究介绍了一种基于预激活和多个偏差的新型计算单元 DAC，用于减轻神经网络中多个单元共享输入偏差的问题，并在 ResNet 架构中测试 DAC 卷积，取得了最高 1.73% 的性能提升。

通过激活树突连接改善神经网络性能

Improving Performance in Neural Networks by Dendrites-Activated  Connections

We present a novel approximate inference method for diffusion processes,
based on the Wasserstein gradient flow formulation of the diffusion. In this
formulation, the time-dependent density of the diffusion is derived as the
limit of implicit Euler steps that follow the gradients of a particular free
energy functional. Existing methods for computing Wasserstein gradient flows
rely on discretization of the domain of the diffusion, prohibiting their
application to domains in more than several dimensions. We propose instead a
discretization-free inference method that computes the Wasserstein gradient
flow directly in a space of continuous functions. We characterize approximation
properties of the proposed method and evaluate it on a nonlinear filtering
task, finding performance comparable to the state-of-the-art for filtering
diffusions.

该研究论文介绍了一种基于 Wasserstein 梯度流的扩散过程的新近似推理方法，该方法直接在连续函数空间中计算 Wasserstein 梯度流，并具有可比拟的过滤能力。

Wasserstein 梯度流的近似推断

Approximate inference with Wasserstein gradient flows

In this chapter, we analyze nonlinear filtering problems in distributed
environments, e.g., sensor networks or peer-to-peer protocols. In these
scenarios, the agents in the environment receive measurements in a streaming
fashion, and they are required to estimate a common (nonlinear) model by
alternating local computations and communications with their neighbors. We
focus on the important distinction between single-task problems, where the
underlying model is common to all agents, and multitask problems, where each
agent might converge to a different model due to, e.g., spatial dependencies or
other factors. Currently, most of the literature on distributed learning in the
nonlinear case has focused on the single-task case, which may be a strong
limitation in real-world scenarios. After introducing the problem and reviewing
the existing approaches, we describe a simple kernel-based algorithm tailored
for the multitask case. We evaluate the proposal on a simulated benchmark task,
and we conclude by detailing currently open problems and lines of research.

本文用于分析分布式情况下的非线性滤波问题，提出了一种针对多任务场景的基于核的算法，并对其进行了模拟基准测试，最后详细说明了目前尚未解决的问题和未来的研究方向。