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. Themore »
Fiber-Sampled Stochastic Mirror Descent for Tensor Decomposition with β-Divergence
Canonical polyadic decomposition (CPD) has been a workhorse for multimodal data analytics. This work puts forth a stochastic algorithmic framework for CPD under β-divergence, which is well-motivated in statistical learning—where the Euclidean distance is typically not preferred. Despite the existence of a series of prior works addressing this topic, pressing computational and theoretical challenges, e.g., scalability and convergence issues, still remain. In this paper, a unified stochastic mirror descent framework is developed for large-scale β-divergence CPD. Our key contribution is the integrated design of a tensor fiber sampling strategy and a flexible stochastic Bregman divergence-based mirror descent iterative procedure, which significantly reduces the computation and memory cost per iteration for various β. Leveraging the fiber sampling scheme and the multilinear algebraic structure of low-rank tensors, the proposed lightweight algorithm also ensures global convergence to a stationary point under mild conditions. Numerical results on synthetic and real data show that our framework attains significant computational saving compared with state-of-the-art methods.
- Award ID(s):
- Publication Date:
- NSF-PAR ID:
- Journal Name:
- IEEE ICASSP 2021
- Page Range or eLocation-ID:
- 2925 to 2929
- Sponsoring Org:
- National Science Foundation
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