Matrix concentration inequalities provide information about the probability
that a random matrix is close to its expectation with respect to the $l_2$
operator norm. This paper uses semigroup methods to derive sharp nonlinear
matrix inequalities. In particular, it is shown that the classic Bakry-\'Emery
curvature criterion implies subgaussian concentration for "matrix Lipschitz"
functions. This argument circumvents the need to develop a matrix version of
the log-Sobolev inequality, a technical obstacle that has blocked previous
attempts to derive matrix concentration inequalities in this setting. The
approach unifies and extends much of the previous work on matrix concentration.
When applied to a product measure, the theory reproduces the matrix Efron-Stein
inequalities due to Paulin et al. It also handles matrix-valued functions on a
Riemannian manifold with uniformly positive Ricci curvature.

使用半群方法推导出非线性的矩阵不等式，并证明了 Bakry-Emery 曲率条件意味着矩阵 Lipschitz 函数的半高斯浓度，从而为推导矩阵浓度不等式提供了一种新的思路。

半群方法下的非线性矩阵集中

Nonlinear Matrix Concentration via Semigroup Methods

Consider any random graph model where potential edges appear independently,
with possibly different probabilities, and assume that the minimum expected
degree is omega(ln n). We prove that the adjacency matrix and the Laplacian of
that random graph are concentrated around the corresponding matrices of the
weighted graph whose edge weights are the probabilities in the random model.
While this may seem surprising, we will see that this matrix concentration
phenomenon is a generalization of known results about the Er\"{o}s-R\'{e}nyi
model. In particular, we will argue that matrix concentration is implicit the
theory of quasi-random graph properties.
We present two main applications of the main result. In bond percolation over
a graph G, we show that the Laplacian of the random subgraph is typically very
close to the Laplacian of G. As a corollary, we improve upon a bound for the
spectral gap due to Chung and Horn that was derived via much more complicated
methods.
In inhomogeneous random graphs, there are points X_1,...,X_n uniformly
distributed on the interval [0,1] and each pair is connected with probability p
kappa(X_i,X_j). We show that if \ln n/n<< p<< 1 and kappa is bounded, then the
adjacency matrix of the random graph is close to an integral operator defined
in terms of kappa.
Our main proof tool is a new concentration inequality for matrix martingales
that generalizes Freedman's inequality for the standard scalar setting.

在随机图中，将边权值视为概率，如果最小期望度数为 ω(ln n)，则随机图的邻接矩阵和拉普拉斯矩阵集中于边权为概率的加权图，应用于债券渗透和不均匀随机图问题中，通过引入矩阵 concenetration 和集中不等式得到新的结论。