arXiv 1506.021812015-06-06math.STcs.ITmath.ITstat.TH

具有非线性测量的LASSO等效于具有线性测量的PDKEEPAAA

The LASSO with Non-linear Measurements is Equivalent to One With Linear Measurements

Chrtistos Thrampoulidis, Ehsan Abbasi, Babak Hassibi

摘要：

考虑从$m$测量值$y_i=g_i(a_i^Tx_0)$估计未知但结构化的信号$x_0\in R^n$，其中$a_i$是已知测量矩阵$A$的行，并且$g$是（潜在未知的）非线性和随机链接函数。这样的测量功能可能出现在测量设备具有非线性和不确定性的应用中。它也可以通过设计产生，例如，$g_i(x)=\text{sign}(x+z_i)$对应于有噪声的1比特量化测量。受Brillinger的经典工作以及Plan和Vershynin最近的工作的启发，我们通过求解某些正则化参数$λ>0$和某些（通常为非光滑）凸结构诱导正则化函数的广义LASSO来估计$x_0$。虽然这种方法似乎天真地忽略了非线性函数$g$，但Brillinger（在非约束情况下）和Plan和Vershynin都表明，当$A$的条目是iid标准正态时，这是$x_0$的良好估计，直到比例常数$μ$，它只取决于$g$。在这项工作中，我们大大加强了这些结果得到明确的表达式的平方误差，为\emph{regularized} LASSO，这是渐进的\emph{precise}当$m$和$n$增长大。主要结果是，非线性观测广义LASSO的估计性能为\emph{asymptotically the same}，其观测值为线性$y_i=μa_i^Tx_0 + σz_i$，具有$μ= Eγg(γ)$和$σ^2 = E(g(γ)-μγ)^2$，$γ$为标准正态.据我们所知，推导出的估计性能的表达式是第一个已知的精确结果在这种情况下。我们的结果的一个有趣的后果是，最佳量化器的测量，最大限度地减少估计误差的LASSO是著名的最大量化器。

英文摘要：

Consider estimating an unknown, but structured, signal $x_0\in R^n$ from $m$ measurement $y_i=g_i(a_i^Tx_0)$, where the $a_i$'s are the rows of a known measurement matrix $A$, and, $g$ is a (potentially unknown) nonlinear and random link-function. Such measurement functions could arise in applications where the measurement device has nonlinearities and uncertainties. It could also arise by design, e.g., $g_i(x)=\text{sign}(x+z_i)$, corresponds to noisy 1-bit quantized measurements. Motivated by the classical work of Brillinger, and more recent work of Plan and Vershynin, we estimate $x_0$ via solving the Generalized-LASSO for some regularization parameter $λ>0$ and some (typically non-smooth) convex structure-inducing regularizer function. While this approach seems to naively ignore the nonlinear function $g$, both Brillinger (in the non-constrained case) and Plan and Vershynin have shown that, when the entries of $A$ are iid standard normal, this is a good estimator of $x_0$ up to a constant of proportionality $μ$, which only depends on $g$. In this work, we considerably strengthen these results by obtaining explicit expressions for the squared error, for the \emph{regularized} LASSO, that are asymptotically \emph{precise} when $m$ and $n$ grow large. A main result is that the estimation performance of the Generalized LASSO with non-linear measurements is \emph{asymptotically the same} as one whose measurements are linear $y_i=μa_i^Tx_0 + σz_i$, with $μ= Eγg(γ)$ and $σ^2 = E(g(γ)-μγ)^2$, and, $γ$ standard normal. To the best of our knowledge, the derived expressions on the estimation performance are the first-known precise results in this context. One interesting consequence of our result is that the optimal quantizer of the measurements that minimizes the estimation error of the LASSO is the celebrated Lloyd-Max quantizer.

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英文摘要：

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