By removing irrelevant and redundant features, feature selection aims to find
a good representation of the original features. With the prevalence of
unlabeled data, unsupervised feature selection has been proven effective in
alleviating the so-called curse of dimensionality. Most existing matrix
factorization-based unsupervised feature selection methods are built upon
subspace learning, but they have limitations in capturing nonlinear structural
information among features. It is well-known that kernel techniques can capture
nonlinear structural information. In this paper, we construct a model by
integrating kernel functions and kernel alignment, which can be equivalently
characterized as a matrix factorization problem. However, such an extension
raises another issue: the algorithm performance heavily depends on the choice
of kernel, which is often unknown a priori. Therefore, we further propose a
multiple kernel-based learning method. By doing so, our model can learn both
linear and nonlinear similarity information and automatically generate the most
appropriate kernel. Experimental analysis on real-world data demonstrates that
the two proposed methods outperform other classic and state-of-the-art
unsupervised feature selection methods in terms of clustering results and
redundancy reduction in almost all datasets tested.

通过特征选择方法，将无关和冗余特征删除，以期获得原始特征的良好表示。本文构建了一个模型，通过集成核函数和核对齐，来解决非线性结构信息的捕捉问题。此外，还提出了一种多核学习方法，通过学习线性和非线性相似性信息，并自动生成最合适的核函数。在实验中，这两种方法在聚类结果和冗余减少方面表现优于其他经典和最先进的无监督特征选择方法。

基于矩阵分解的无监督特征选择的核对齐

Kernel Alignment for Unsupervised Feature Selection via Matrix  Factorization

Kernel techniques are among the most influential approaches in data science
and statistics. Under mild conditions, the reproducing kernel Hilbert space
associated to a kernel is capable of encoding the independence of $M\ge 2$
random variables. Probably the most widespread independence measure relying on
kernels is the so-called Hilbert-Schmidt independence criterion (HSIC; also
referred to as distance covariance in the statistics literature). Despite
various existing HSIC estimators designed since its introduction close to two
decades ago, the fundamental question of the rate at which HSIC can be
estimated is still open. In this work, we prove that the minimax optimal rate
of HSIC estimation on $\mathbb R^d$ for Borel measures containing the Gaussians
with continuous bounded translation-invariant characteristic kernels is
$\mathcal O\!\left(n^{-1/2}\right)$. Specifically, our result implies the
optimality in the minimax sense of many of the most-frequently used estimators
(including the U-statistic, the V-statistic, and the Nystr\"om-based one) on
$\mathbb R^d$.

在本研究中，我们证明了具有连续有界平移不变特征核的 Borel 测度中，Hilbert-Schmidt 独立性准则（HSIC）在 R^d 上的最优最小化估计率为 O (n^(-1/2))，从而证实了许多经常使用的估计器（包括 U 统计量、V 统计量和 Nystrom-based 统计量）在 R^d 上的最小化最优性。