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：我们认为,这种非标准空间最近被证明是执行信号和图像处理应用中像素适应常规化的适当功能框架。然而,与其在Hilbert环境中的使用情况相比,在可变表象空间的Banach空间框架中用于解决线性反问题(例如CT图像重建)的SGD算法并非直截了当,特别是因为缺少封闭式表达方式和基本规范的不可分离性属性。在这个手稿中,我们表明SGD的迭代可有效地使用相关的模块功能。模拟和真实CT数据的数值验证表明,与SGD解决方案相比,在Hilbert和其他Banach环境中的SGD(p_n)}(\mathbb{R})的应用都有很大改进,特别是在数据中观测到非伽西或混合噪音时。</s>