Several statistical approaches based on reproducing kernels have been
proposed to detect abrupt changes arising in the full distribution of the
observations and not only in the mean or variance. Some of these approaches
enjoy good statistical properties (oracle inequality, \ldots). Nonetheless,
they have a high computational cost both in terms of time and memory. This
makes their application difficult even for small and medium sample sizes ($n<
10^4$). This computational issue is addressed by first describing a new
efficient and exact algorithm for kernel multiple change-point detection with
an improved worst-case complexity that is quadratic in time and linear in
space. It allows dealing with medium size signals (up to $n \approx 10^5$).
Second, a faster but approximation algorithm is described. It is based on a
low-rank approximation to the Gram matrix. It is linear in time and space. This
approximation algorithm can be applied to large-scale signals ($n \geq 10^6$).
These exact and approximation algorithms have been implemented in \texttt{R}
and \texttt{C} for various kernels. The computational and statistical
performances of these new algorithms have been assessed through empirical
experiments. The runtime of the new algorithms is observed to be faster than
that of other considered procedures. Finally, simulations confirmed the higher
statistical accuracy of kernel-based approaches to detect changes that are not
only in the mean. These simulations also illustrate the flexibility of
kernel-based approaches to analyze complex biological profiles made of DNA copy
number and allele B frequencies. An R package implementing the approach will be
made available on github.

本文提出了两种既适用于小规模信号也适用于大规模信号的，基于再生核方法的突变检测算法。新算法在计算时间和内存使用方面具有较高的效率和精度，并且被证明是一种高度灵活的方法，可以用于复杂的生物数据分析。

带核方法的多变点检测的新高效算法

New efficient algorithms for multiple change-point detection with  kernels

Recently, a general method for analyzing the statistical accuracy of the EM
algorithm has been developed and applied to some simple latent variable models
[Balakrishnan et al. 2016]. In that method, the basin of attraction for valid
initialization is required to be a ball around the truth. Using Stein's Lemma,
we extend these results in the case of estimating the centers of a
two-component Gaussian mixture in $d$ dimensions. In particular, we
significantly expand the basin of attraction to be the intersection of a half
space and a ball around the origin. If the signal-to-noise ratio is at least a
constant multiple of $ \sqrt{d\log d} $, we show that a random initialization
strategy is feasible.

本研究拓展了 EM 算法在高斯混合模型中的应用，并通过随机初始化策略提高了统计精确度。