We consider a stochastic gradient descent (SGD) algorithm for solving linear inverse problems (e.g., CT image reconstruction) in the Banach space framework of variable exponent Lebesgue spaces $\ell^{(p_n)}(\mathbb{R})$. Such non-standard spaces have been recently proved to be the appropriate functional framework to enforce pixel-adaptive regularisation in signal and image processing applications. Compared to its use in Hilbert settings, however, the application of SGD in the Banach setting of $\ell^{(p_n)}(\mathbb{R})$ is not straightforward, due, in particular to the lack of a closed-form expression and the non-separability property of the underlying norm. In this manuscript, we show that SGD iterations can effectively be performed using the associated modular function. Numerical validation on both simulated and real CT data show significant improvements in comparison to SGD solutions both in Hilbert and other Banach settings, in particular when non-Gaussian or mixed noise is observed in the data.

翻译：本文考虑在变指数Lebesgue空间$\ell^{(p_n)}(\mathbb{R})$的巴拿赫空间框架下，求解线性反问题（如CT图像重建）的随机梯度下降（SGD）算法。此类非标准空间最近被证明是信号与图像处理应用中实现像素自适应正则化的适当函数框架。然而，与希尔伯特空间的SGD算法相比，在$\ell^{(p_n)}(\mathbb{R})$的巴拿赫空间中应用SGD并非易事，这尤其源于其底层范数缺乏闭式表达式且不可分离的特性。本文证明，利用关联模函数可有效执行SGD迭代。模拟数据和真实CT数据的数值验证表明，与希尔伯特及其他巴拿赫空间中的SGD解相比，本方法在数据中存在非高斯噪声或混合噪声时具有显著优势。