We consider the problem of estimating an unknown signal $x_0$ from noisy
linear observations $y = Ax_0 + z\in R^m$. In many practical instances, $x_0$
has a certain structure that can be captured by a structure inducing convex
function $f(\cdot)$. For example, $\ell_1$ norm can be used to encourage a
sparse solution. To estimate $x_0$ with the aid of $f(\cdot)$, we consider the
well-known LASSO method and provide sharp characterization of its performance.
We assume the entries of the measurement matrix $A$ and the noise vector $z$
have zero-mean normal distributions with variances $1$ and $\sigma^2$
respectively. For the LASSO estimator $x^*$, we attempt to calculate the
Normalized Square Error (NSE) defined as $\frac{\|x^*-x_0\|_2^2}{\sigma^2}$ as
a function of the noise level $\sigma$, the number of observations $m$ and the
structure of the signal. We show that, the structure of the signal $x_0$ and
choice of the function $f(\cdot)$ enter the error formulae through the summary
parameters $D(cone)$ and $D(\lambda)$, which are defined as the Gaussian
squared-distances to the subdifferential cone and to the $\lambda$-scaled
subdifferential, respectively. The first LASSO estimator assumes a-priori
knowledge of $f(x_0)$ and is given by $\arg\min_{x}\{{\|y-Ax\|_2}~\text{subject
to}~f(x)\leq f(x_0)\}$. We prove that its worst case NSE is achieved when
$\sigma\rightarrow 0$ and concentrates around $\frac{D(cone)}{m-D(cone)}$.
Secondly, we consider $\arg\min_{x}\{\|y-Ax\|_2+\lambda f(x)\}$, for some
$\lambda\geq 0$. This time the NSE formula depends on the choice of $\lambda$
and is given by $\frac{D(\lambda)}{m-D(\lambda)}$. We then establish a mapping
between this and the third estimator $\arg\min_{x}\{\frac{1}{2}\|y-Ax\|_2^2+
\lambda f(x)\}$. Finally, for a number of important structured signal classes,
we translate our abstract formulae to closed-form upper bounds on the NSE.

本文研究了使用结构感知凸函数去估计稀疏信号，并提供了 LASSO 方法性能的尖锐特征化，具体表现为通过集合距离的形式来描述估算误差，该误差与信号结构和函数选择有关。

广义 LASSO 的平方误差：精确分析

The Squared-Error of Generalized LASSO: A Precise Analysis

A well-known analysis of Tropp and Gilbert shows that orthogonal matching
pursuit (OMP) can recover a k-sparse n-dimensional real vector from 4 k log(n)
noise-free linear measurements obtained through a random Gaussian measurement
matrix with a probability that approaches one as n approaches infinity. This
work strengthens this result by showing that a lower number of measurements, 2
k log(n - k), is in fact sufficient for asymptotic recovery. More generally,
when the sparsity level satisfies kmin <= k <= kmax but is unknown, 2 kmax
log(n - kmin) measurements is sufficient. Furthermore, this number of
measurements is also sufficient for detection of the sparsity pattern (support)
of the vector with measurement errors provided the signal-to-noise ratio (SNR)
scales to infinity. The scaling 2 k log(n - k) exactly matches the number of
measurements required by the more complex lasso method for signal recovery with
a similar SNR scaling.

本文使用正交匹配追踪算法研究了稀疏信号恢复的问题并发现，使用高斯测量矩阵的低噪声情况下，仅需较少的测量次数和无限趋近于 1 的概率即可实现稀疏向量的恢复，该结果还与 lasso 方法的恢复次数相匹配。