论文标题
具有非线性观测和生成先验的广义拉索
The Generalized Lasso with Nonlinear Observations and Generative Priors
论文作者
论文摘要
在本文中，当未知的$ n $维信号在$ l $ -Lipschitz连续生成模型的范围内时，我们研究了噪音非线性测量值的信号估计问题，该范围内有有限的$ k $ j $维输入。我们假设下高斯测量值，这是通过多种测量模型（例如线性，逻辑，1位和其他量化模型）所满足的。此外，我们考虑了对抗性腐败对这些测量结果的影响。我们的分析基于广义的套索方法（Plan and Vershynin，2016年）。我们首先提供非均匀的恢复保证，该保证指出，在I.I.D.〜高斯测量值下，大约$ o \ left（\ frac {k} {k} {ε^2} \ log l \ right）$样品可用于恢复，并具有$ \ ell_2 $ $ ell_2 $ - $ $ $ $ε$，以及该计划是强劲的。然后，我们将此结果应用于神经网络生成模型，并讨论了其他模型和非I.I.D。〜测量的各种扩展。此外，我们表明，在所谓的局部嵌入属性的假设下，我们的结果可以扩展到统一的恢复保证，这是由1位和审查的TOBIT模型所满足的。
In this paper, we study the problem of signal estimation from noisy non-linear measurements when the unknown $n$-dimensional signal is in the range of an $L$-Lipschitz continuous generative model with bounded $k$-dimensional inputs. We make the assumption of sub-Gaussian measurements, which is satisfied by a wide range of measurement models, such as linear, logistic, 1-bit, and other quantized models. In addition, we consider the impact of adversarial corruptions on these measurements. Our analysis is based on a generalized Lasso approach (Plan and Vershynin, 2016). We first provide a non-uniform recovery guarantee, which states that under i.i.d.~Gaussian measurements, roughly $O\left(\frac{k}{ε^2}\log L\right)$ samples suffice for recovery with an $\ell_2$-error of $ε$, and that this scheme is robust to adversarial noise. Then, we apply this result to neural network generative models, and discuss various extensions to other models and non-i.i.d.~measurements. Moreover, we show that our result can be extended to the uniform recovery guarantee under the assumption of a so-called local embedding property, which is satisfied by the 1-bit and censored Tobit models.
