We consider the problem of learning a coefficient vector x_0\in R^N from
noisy linear observation y=Ax_0+w \in R^n. In many contexts (ranging from model
selection to image processing) it is desirable to construct a sparse estimator
x'. In this case, a popular approach consists in solving an L1-penalized least
squares problem known as the LASSO or Basis Pursuit DeNoising (BPDN).
For sequences of matrices A of increasing dimensions, with independent
gaussian entries, we prove that the normalized risk of the LASSO converges to a
limit, and we obtain an explicit expression for this limit. Our result is the
first rigorous derivation of an explicit formula for the asymptotic mean square
error of the LASSO for random instances. The proof technique is based on the
analysis of AMP, a recently developed efficient algorithm, that is inspired
from graphical models ideas.
Simulations on real data matrices suggest that our results can be relevant in
a broad array of practical applications.

本文考虑从具有噪音的线性观测中学习系数向量 x0，通过解决 L1 惩罚的最小二乘问题，即 LASSO 或 BPDN 问题构造一种稀疏估计器 x'，对于随机矩阵序列 A，我们证明了 LASSO 的规范风险趋于极限，并获得了该极限的一个显式表达式，并进行了实际数据矩阵的模拟，表明我们的结果在广泛的实际应用中都是相关的。

高斯矩阵的 LASSO 风险

The LASSO risk for gaussian matrices

The paper proposes a method for constructing a sparse estimator for the
inverse covariance (concentration) matrix in high-dimensional settings. The
estimator uses a penalized normal likelihood approach and forces sparsity by
using a lasso-type penalty. We establish a rate of convergence in the Frobenius
norm as both data dimension $p$ and sample size $n$ are allowed to grow, and
show that the rate depends explicitly on how sparse the true concentration
matrix is. We also show that a correlation-based version of the method exhibits
better rates in the operator norm. We also derive a fast iterative algorithm
for computing the estimator, which relies on the popular Cholesky decomposition
of the inverse but produces a permutation-invariant estimator. The method is
compared to other estimators on simulated data and on a real data example of
tumor tissue classification using gene expression data.

提出一种构建稀疏估计器的方法，用于高维背景下的逆协方差（浓度）矩阵，使用罚函数正态似然方法强制使用 Lasso 型惩罚，并在数据维数 $p$ 和样本量 $n$ 随着增长收敛速率之间建立 Frobenius 范数。同时对真实浓度矩阵进行稀疏度量，提出一种基于相关性的方法，在操作规范下具有更好的收敛速率，推出一种快速迭代算法用于计算估计值，利用常用的 Cholesky 分解反演，得到一种置换不变的估算法，文中将这种方法应用于癌症组织分类的基因表达数据上，并与其他估算方法进行了比较。