Stochastic optimisation in Riemannian manifolds, especially the Riemannian stochastic gradient method, has attracted much recent attention. The present work applies stochastic optimisation to the task of recursive estimation of a statistical parameter which belongs to a Riemannian manifold. Roughly, this task amounts to stochastic minimisation of a statistical divergence function. The following problem is considered: how to obtain fast, asymptotically efficient, recursive estimates, using a Riemannian stochastic optimisation algorithm with decreasing step sizes. In solving this problem, several original results are introduced. First, without any convexity assumptions on the divergence function, we proved that, with an adequate choice of step sizes, the algorithm computes recursive estimates which achieve a fast non-asymptotic rate of convergence. Second, the asymptotic normality of these recursive estimates is proved by employing a novel linearisation technique. Third, it is proved that, when the Fisher information metric is used to guide the algorithm, these recursive estimates achieve an optimal asymptotic rate of convergence, in the sense that they become asymptotically efficient. These results, while relatively familiar in the Euclidean context, are here formulated and proved for the first time in the Riemannian context. In addition, they are illustrated with a numerical application to the recursive estimation of elliptically contoured distributions.

Color transfer, an essential technique in image editing, has recently received significant attention. However, achieving a balance between holistic color style transfer and local detail refinement remains a challenging task. This paper proposes an innovative color transfer method, named BHL, which stands for Balanced consideration of both Holistic transformation and Local refinement. The BHL method employs a statistical framework to address the challenge of achieving a balance between holistic color transfer and the preservation of fine details during the color transfer process. Holistic color transformation is achieved using optimal transport theory within the generalized Gaussian modeling framework. The local refinement module adjusts color and texture details on a per-pixel basis using a Gaussian Mixture Model (GMM). To address the high computational complexity inherent in complex statistical modeling, a parameter estimation method called the unit-wise Riemannian information gradient (uRIG) method is introduced. The uRIG method significantly reduces the computational burden through the second-order acceleration effect of the Fisher information metric. Comprehensive experiments demonstrate that the BHL method outperforms state-of-the-art techniques in both visual quality and objective evaluation criteria, even under stringent time constraints. Remarkably, the BHL method processes high-resolution images in an average of 4.874 s, achieving the fastest processing time compared to the baselines. The BHL method represents a significant advancement in the field of color transfer, offering a balanced approach that combines holistic transformation and local refinement while maintaining efficiency and high visual quality.

Accurate estimation of Vegetation Water Content (VWC) is crucial for understanding terrestrial ecosystem dynamics. However, conventional remote sensing retrieval methods struggle to capture the complex, non-linear relationships between VWC and variables like Vegetation Optical Depth (VOD), Leaf Area Index (LAI), and Vegetation Height (VH) across diverse land cover types. While machine learning provides potential solutions, models with fixed architectures often exhibit poor generalization. This study introduced a novel framework for VWC estimation using a Conditional Adaptive Neural Network (CANN), which dynamically tailored its network parameters to input data characteristics. The model predicted VWC using VOD, LAI, VH, and the International Geosphere-Biosphere Programme (IGBP) land cover type as inputs. Results show that the proposed VWC estimates in this study were highly consistent with the reference VWC, by achieving a correlation coefficient (R) of 0.94 (RMSE=1.13 kg/m$^{2}$) against LFMC-VWC (VWC derived by Globe-Live Fuel Moisture Content) and an R of 0.95 (RMSE=2.48kg/m$^{2}$) against NDVI-VWC(VWC derived by NDVI). In addition, the temporal dynamics of CANN-VWC closely followed the SMAPVEX in situ VWC measurements, exhibiting more competitive performance than the NDVI-VWC estimates.