[论文解读] Using auto-encoders for solving ill-posed linear inverse problems

Shirin Jalali, Xin Yuan|arXiv (Cornell University)|Jan 15, 2019

Sparse and Compressive Sensing Techniques参考文献 19被引用 4

一句话总结

本文提出了一种新颖的压缩感知恢复方法，利用训练好的自编码器（生成模型）来解决病态的线性逆问题。通过将恢复问题表述为在生成模型 $g$ 的潜在空间上的投影梯度下降优化，当 $m > 2k$ 时，该方法可保证精确恢复，且重建误差受生成模型近似误差的限制。

ABSTRACT

Compressed sensing algorithms recover a signal from its under-determined linear measurements by exploiting its structure. Starting from sparsity, recovery methods have steadily moved towards more complex structures. Recently, the emerging machine learning techniques, especially the generative models based on neural nets, potentially, can learn general complex structures. Inspired by the success of such models in various computer vision tasks, researchers in compressed sensing have recently started to employ them to design efficient recovery methods. Consider a generative model defined as function $g:{\cal U}^k o{R}^n$, ${\cal U}\subset{R}$. Assume that the function $g$ is trained such that it can describe a class of desired signals ${\cal Q}\subset{R}^n$. The standard problem in noiseless compressed sensing is to recover ${\bf x}\in{\cal Q}$ from under-determined linear measurements ${\bf y}=A{\bf x}$, where ${\bf y}\in{R}^m$ and $m\ll n$. A recovery method based on $g$ finds $g({\bf u})$, $\in{\cal U}^k$, which has the minimum measurement error. In this paper, the performance of such a recovery method is studied and it is proven that, if the number of measurements ($m$) is larger than twice the dimension of the generative model ($k$), then ${\bf x}$ can be recovered from ${\bf y}$, with a distortion that is a function of the distortion induced by $g$ in describing $\bf x$, i.e. $\min_{{\bf u}\in{\cal U}^k}\|g({\bf u})-{\bf x}\|$. To derive an efficient method, an algorithm based on projected gradient descent is proposed. It is proven that, given enough measurements, the algorithm converges to the optimal solution and is robust to measurement noise. Numerical results showing the effectiveness of the proposed method are presented.

研究动机与目标

• 解决压缩感知中使用深度生成模型的病态线性逆问题。
• 建立在何种理论条件下可从欠定测量中精确恢复信号。
• 基于投影梯度下降开发一种高效的潜在空间恢复优化算法。
• 证明该方法对测量噪声的鲁棒性。

提出的方法

• 该方法使用一个训练好的生成模型 $g: \mathcal{U}^k \to \mathbb{R}^n$，在低维潜在空间 $\mathcal{U}^k$ 中表示信号。
• 恢复问题被表述为在潜在码 $\mathbf{u} \in \mathcal{U}^k$ 上最小化测量误差 $\|A g(\mathbf{u}) - \mathbf{y}\|$。
• 提出了一种迭代投影梯度下降算法来求解优化问题，每一步都将解投影回潜在空间。
• 证明当测量数 $m$ 超过潜在维数 $k$ 的两倍时，该算法收敛到最优解。
• 在相同理论条件下，该方法对加性测量噪声表现出鲁棒性。
• 该方法利用深度生成模型的表达能力，捕捉超越简单稀疏性的复杂信号结构。

实验结果

研究问题

• RQ1 在何种条件下，可使用深度生成模型从欠定测量中精确恢复信号？
• RQ2 重建误差如何依赖于生成模型对真实信号的表示能力？
• RQ3 能否设计一种具有理论收敛保证的高效潜在空间恢复优化算法？
• RQ4 该方法在存在测量噪声时表现如何？
• RQ5 稳定且准确恢复所需的最小测量数是多少？

主要发现

• 理论分析证明，当 $m > 2k$ 时可实现精确恢复，其中 $m$ 为测量数，$k$ 为生成模型的潜在维数。
• 重建误差受生成模型近似误差的限制，即 $\min_{\mathbf{u}} \|g(\mathbf{u}) - \mathbf{x}\|$，从而确保与真实信号的一致性。
• 在 $m > 2k$ 条件下，投影梯度下降算法收敛到最优解。
• 该方法对测量噪声具有鲁棒性，误差增长保持有界。
• 数值结果表明，所提方法在使用少于传统基于稀疏性的方法的测量数时，仍能有效恢复信号。
• 该方法通过利用自编码器学习到的更丰富的、数据驱动的信号结构，优于经典压缩感知技术。