Tensors play a crucial role in numerous scientific and engineering fields. This paper addresses the low-multilinear-rank tensor completion problem, a fundamental task in tensor-related applications. By exploiting the manifold structure inherent to the fixed-multilinear-rank tensor set, we introduce a simple yet highly effective preconditioned Riemannian metric and propose the Preconditioned Riemannian Gradient Descent (PRGD) algorithm. Compared to the standard Riemannian Gradient Descent (RGD), PRGD achieves faster convergence while maintaining the same order of per-iteration computational complexity. Theoretically, we provide the recovery guarantee for PRGD under near-optimal sampling complexity. Numerical results highlight the efficiency of PRGD, outperforming state-of-the-art methods on both synthetic data and real-world video inpainting tasks.

Finding low-rank matrix weights is a key technique for addressing the high memory usage and computational demands of large models. Most existing algorithms rely on the factorization of the low-rank matrix weights, which is non-unique and redundant. Their convergence is slow especially when the target low-rank matrices are ill-conditioned, because the convergence rate depends on the condition number of the Jacobian operator for the factorization and the Hessian of the loss function with respect to the weight matrix. To address this challenge, we adopt the Riemannian gradient descent (RGD) algorithm on the Riemannian manifold of fixed-rank matrices to update the entire low-rank weight matrix. This algorithm completely avoids the factorization, thereby eliminating the negative impact of the Jacobian condition number. Furthermore, by leveraging the geometric structure of the Riemannian manifold and selecting an appropriate metric, it mitigates the negative impact of the Hessian condition number. Ultimately, this results in our two plug-and-play optimizers: RAdaGrad and RAdamW, which are RGD with metrics adapted from AdaGrad and AdamW and restricted to the manifold. Our algorithms can be seamlessly integrated with various deep neural network architectures without any modifications. We evaluate the effectiveness of our algorithms through fine-tuning experiments on large language models and diffusion models. Experimental results consistently demonstrate that our algorithms provide superior performance compared to state-of-the-art methods. Additionally, our algorithm is not only effective for fine-tuning large models but is also applicable to deep neural network (DNN) compression.