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This work focuses on canonical polyadic decomposition (CPD) for large-scale tensors. Many prior works rely on data sparsity to develop scalable CPD algorithms, which are not suitable for handling dense tensor, while dense tensors often arise in applications such as image and video processing. As an alternative, stochastic algorithms utilize data sampling to reduce per-iteration complexity and thus are very scalable, even when handling dense tensors. However, existing stochastic CPD algorithms are facing some challenges. For example, some algorithms are based on randomly sampled tensor entries, and thus each iteration can only updates a small portion of the latent factors. This may result in slow improvement of the estimation accuracy of the latent factors. In addition, the convergence properties of many stochastic CPD algorithms are unclear, perhaps because CPD poses a hard nonconvex problem and is challenging for analysis under stochastic settings. In this work, we propose a stochastic optimization strategy that can effectively circumvent the above challenges. The proposed algorithm updates a whole latent factor at each iteration using sampled fibers of a tensor, which can quickly increase the estimation accuracy. The algorithm is flexible-many commonly used regularizers and constraints can be easily incorporated in the computational framework. The algorithm is also backed by a rigorous convergence theory. Simulations on large-scale dense tensors are employed to showcase the effectiveness of the algorithm. 

Root‐associated fungal communities modify the climatic niches and even the competitive ability of their hosts, yet how the different components of the root microbiome are modified by habitat loss remains a key knowledge gap. Using principles of landscape ecology, we tested how free‐living versus host‐associated microbes differ in their response to landscape heterogeneity. Further, we explore how compartmentalisation of microbes into specialised root structures filters for key fungal symbionts. Our study demonstrates that free‐living fungal community structure correlates with landscape heterogeneity, but that host‐associated fungal communities depart from these patterns. Specifically, biotic filtering in roots, especially via compartmentalisation within specialised root structures, decouples the biogeographic patterns of host‐associated fungal communities from the soil community. In this way, even as habitat loss and fragmentation threaten fungal diversity in the soils, plant hosts exert biotic controls to ensure associations with critical mutualists, helping to preserve the root mycobiome.