Multi-view spectral clustering, which aims at yielding an agreement or
consensus data objects grouping across multi-views with their graph laplacian
matrices, is a fundamental clustering problem. Among the existing methods,
Low-Rank Representation (LRR) based method is quite superior in terms of its
effectiveness, intuitiveness and robustness to noise corruptions. However, it
aggressively tries to learn a common low-dimensional subspace for multi-view
data, while inattentively ignoring the local manifold structure in each view,
which is critically important to the spectral clustering; worse still, the
low-rank minimization is enforced to achieve the data correlation consensus
among all views, failing to flexibly preserve the local manifold structure for
each view. In this paper, 1) we propose a multi-graph laplacian regularized LRR
with each graph laplacian corresponding to one view to characterize its local
manifold structure. 2) Instead of directly enforcing the low-rank minimization
among all views for correlation consensus, we separately impose low-rank
constraint on each view, coupled with a mutual structural consensus constraint,
where it is able to not only well preserve the local manifold structure but
also serve as a constraint for that from other views, which iteratively makes
the views more agreeable. Extensive experiments on real-world multi-view data
sets demonstrate its superiority.

本文提出了一种多图拉普拉斯正则化的低秩表示方法来解决多视图谱聚类问题，该方法不仅在保持本地流形结构方面具有很好的性能，而且还能逐步使不同视图更加一致。在真实世界的多视图数据集上的广泛实验证明了其优越性。

迭代视图协议：基于迭代低秩结构优化方法的多视角谱聚类

Iterative Views Agreement: An Iterative Low-Rank based Structured  Optimization Method to Multi-View Spectral Clustering

We undertake a systematic study of sketching a quadratic form: given an $n
\times n$ matrix $A$, create a succinct sketch $\textbf{sk}(A)$ which can
produce (without further access to $A$) a multiplicative
$(1+\epsilon)$-approximation to $x^T A x$ for any desired query $x \in
\mathbb{R}^n$. While a general matrix does not admit non-trivial sketches,
positive semi-definite (PSD) matrices admit sketches of size
$\Theta(\epsilon^{-2} n)$, via the Johnson-Lindenstrauss lemma, achieving the
"for each" guarantee, namely, for each query $x$, with a constant probability
the sketch succeeds. (For the stronger "for all" guarantee, where the sketch
succeeds for all $x$'s simultaneously, again there are no non-trivial
sketches.)
We design significantly better sketches for the important subclass of graph
Laplacian matrices, which we also extend to symmetric diagonally dominant
matrices. A sequence of work culminating in that of Batson, Spielman, and
Srivastava (SIAM Review, 2014), shows that by choosing and reweighting
$O(\epsilon^{-2} n)$ edges in a graph, one achieves the "for all" guarantee.
Our main results advance this front.
$\bullet$ For the "for all" guarantee, we prove that Batson et al.'s bound is
optimal even when we restrict to "cut queries" $x\in \{0,1\}^n$.
In contrast, previous lower bounds showed the bound only for {\em
spectral-sparsifiers}.
$\bullet$ For the "for each" guarantee, we design a sketch of size $\tilde
O(\epsilon^{-1} n)$ bits for "cut queries" $x\in \{0,1\}^n$. We prove a
nearly-matching lower bound of $\Omega(\epsilon^{-1} n)$ bits. For general
queries $x \in \mathbb{R}^n$, we construct sketches of size
$\tilde{O}(\epsilon^{-1.6} n)$ bits.

本研究对矩阵进行草图，着重探讨了矩阵的不同类型和查询所需的精度。特别地，我们尤其关注了具有特殊特性的正半定矩阵和图拉普拉斯矩阵，为其设计了更优秀的草图，并探讨了草图的实际应用。

关于绘制二次形式的草图

On Sketching Quadratic Forms

In recent years, it has become common practice in neuroscience to use
networks to summarize relational information in a set of measurements,
typically assumed to be reflective of either functional or structural
relationships between regions of interest in the brain. One of the most basic
tasks of interest in the analysis of such data is the testing of hypotheses, in
answer to questions such as "Is there a difference between the networks of
these two groups of subjects?" In the classical setting, where the unit of
interest is a scalar or a vector, such questions are answered through the use
of familiar two-sample testing strategies. Networks, however, are not Euclidean
objects, and hence classical methods do not directly apply. We address this
challenge by drawing on concepts and techniques from geometry, and
high-dimensional statistical inference. Our work is based on a precise
geometric characterization of the space of graph Laplacian matrices and a
nonparametric notion of averaging due to Fr\'echet. We motivate and illustrate
our resulting methodologies for testing in the context of networks derived from
functional neuroimaging data on human subjects from the 1000 Functional
Connectomes Project. In particular, we show that this global test is more
statistical powerful, than a mass-univariate approach. In addition, we have
also provided a method for visualizing the individual contribution of each edge
to the overall test statistic.

本文旨在从几何和高维统计推理的概念和技术入手，通过对图拉普拉斯矩阵空间的精确定义和 Fréchet 的非参数平均概念，为功能神经成像数据的网络构建提供一种新的全局测试方法，并展示了其相较于基于无差别分析的单变量方法更具统计学优势的全局测试方法。此外我们还给出了一种可视化每条边对于总测试统计量的单独贡献的方法。