Projection Pursuit for non-Gaussian Independent Components
Joni Virta, Klaus Nordhausen, Hannu Oja
备注：
47 pages, 11 figures. This work is partially based on the unpublished manuscript "Joint Use of Third and Fourth Cumulants in Independent Component Analysis", Virta et al. (2015), available as an arXiv preprint, https://arxiv.org/abs/1505.02613
In independent component analysis it is assumed that the observed random variables are linear combinations of latent, mutually independent random variables called the independent components. Our model further assumes that only the non-Gaussian independent components are of interest, the Gaussian components being treated as noise. In this paper projection pursuit is used to extract the non-Gaussian components and to separate the corresponding signal and noise subspaces. Our choice for the projection index is a convex combination of squared third and fourth cumulants and we estimate the non-Gaussian components either one-by-one (deflation-based approach) or simultaneously (symmetric approach). The properties of both estimates are considered in detail through the corresponding optimization problems, estimating equations, algorithms and asymptotic properties. Various comparisons of the estimates show that the two approaches separate the signal and noise subspaces equally well but the symmetric one is generally better in extracting the individual non-Gaussian components.
Projection Pursuit through Relative Entropy Minimization
投影寻踪方法允许解决困难的问题，找到一个估计的密度定义在一组非常大的尺寸。在他的开创性文章中，Huber（参见“Projection pursuit”，Annals of Statistics，1985）证明了投影寻踪法的重要性，这要归功于在Kullback-Leibler发散最大化的背景下将密度分解为高斯分量和一些剩余密度。在本文中，我们介绍了一种新的算法，特别是一个测试的因式分解的密度估计从iid样本。
Quantitative asymptotics of graphical projection pursuit
Diaconis和Freedman的一个结果是，在有限的意义上，对于高维数据的大集合，数据的大多数一维投影近似为高斯分布。本文给出了该结果的定量版本。对于$\R^d$中的一组确定性向量$\{x_i\}_{i=1}^n$，其中$n$和$d$固定，令$θ\in\s^{d-1}$为球面的随机点，并令$μ_n^θ$表示将质量$\frac{1}{n}$置于每个点$\inprod{x_1}θ,...,\inprod{x_n}θ$的随机测度。对于固定有界Lipschitz检验函数$f$，$Z$为标准高斯随机变量，$σ^2$为适当的常数，导出了量$\ds¶[|\int f dμ_n^θ-\E f(σZ)|>ε]$的显式界.一个界限也给出了$\ds¶[d_{BL}(μ_n^θ, N(0,σ^2))>ε]$，其中$d_{BL}$表示有界Lipschitz距离，这产生了一个下界的等待时间找到一个非高斯投影的$\{x_i\}$，如果方向是独立的，均匀的$\s^{d-1}$。
Probabilistic Analysis of Least Squares, Orthogonal Projection, and QR Factorization Algorithms Subject to Gaussian Noise
Department of Computer Science, University of Saskatchewan（萨斯喀彻温大学计算机科学系）；Department of Applied Mathematics, University of Colorado Denver（科罗拉多大学丹佛分校应用数学系）；Department of Mathematics, The University of Tulsa（塔尔萨大学数学系）
