We investigate the stability of a Sequential Monte Carlo (SMC) method applied
to the problem of sampling from a target distribution on $\mathbb{R}^d$ for
large $d$. It is well known that using a single importance sampling step one
produces an approximation for the target that deteriorates as the dimension $d$
increases, unless the number of Monte Carlo samples $N$ increases at an
exponential rate in $d$. We show that this degeneracy can be avoided by
introducing a sequence of artificial targets, starting from a `simple' density
and moving to the one of interest, using an SMC method to sample from the
sequence. Using this class of SMC methods with a fixed number of samples, one
can produce an approximation for which the effective sample size (ESS)
converges to a random variable $\varepsilon_N$ as $d\rightarrow\infty$ with
$1<\varepsilon_{N}<N$. The convergence is achieved with a computational cost
proportional to $Nd^2$. If $\varepsilon_N\ll N$, we can raise its value by
introducing a number of resampling steps, say $m$ (where $m$ is independent of
$d$). In this case, ESS converges to a random variable $\varepsilon_{N,m}$ as
$d\rightarrow\infty$ and $\lim_{m\to\infty}\varepsilon_{N,m}=N$. Also, we show
that the Monte Carlo error for estimating a fixed dimensional marginal
expectation is of order $\frac{1}{\sqrt{N}}$ uniformly in $d$. The results
imply that, in high dimensions, SMC algorithms can efficiently control the
variability of the importance sampling weights and estimate fixed dimensional
marginals at a cost which is less than exponential in $d$ and indicate that, in
high dimensions, resampling leads to a reduction in the Monte Carlo error and
increase in the ESS.

研究了在高维问题中，应用序贯蒙特卡洛方法进行采样时普遍出现的效率低下的问题。针对这一问题，提出了使用一系列人工目标密度，通过序贯蒙特卡洛方法进行采样，可用固定数量样本的 SMC 类方法得到精度随维度增加而不降的分布近似，并且引入重采样可以进一步提高重要性权重的可变性，降低蒙特卡洛误差。

高维顺序蒙特卡罗方法的稳定性

On the Stability of Sequential Monte Carlo Methods in High Dimensions

We study generalized density-based clustering in which sharply defined
clusters such as clusters on lower-dimensional manifolds are allowed. We show
that accurate clustering is possible even in high dimensions. We propose two
data-based methods for choosing the bandwidth and we study the stability
properties of density clusters. We show that a simple graph-based algorithm
successfully approximates the high density clusters.

本研究研究了广义密度聚类，提出了两种数据基础方法来选择波宽，并研究了密度聚类的稳定性，表明一种简单的基于图的算法可以成功地近似高密度聚类。