- The paper presents CPRAND and CPRAND-MIX algorithms that reduce computational complexity and memory usage in CP tensor decomposition.
- It leverages random sampling and mixing techniques, including random sign-flips and FFT, to bypass the full Khatri-Rao product formation.
- Numerical experiments confirm significant speed improvements and stable convergence, making these methods effective for large-scale tensor analysis.
Practical Randomized CP Tensor Decomposition
The paper focuses on improving the efficiency and robustness of the CANDECOMP/PARAFAC (CP) tensor decomposition, a pivotal tool in multiway data analysis across diverse fields such as chemometrics, neuroscience, and signal processing. The researchers address the computational challenges inherent in the standard alternating least squares algorithm (CP-ALS) required for the CP decomposition, which involves solving highly overdetermined least squares problems.
Central to the authors' contribution is the extension of randomized least squares methods to tensors, introducing CPRAND and CPRAND-MIX algorithms. CPRAND leverages random sampling to reduce computational and memory overhead without significantly compromising the quality of the decomposition. This approach does not necessitate forming the entire Khatri-Rao product, which is computationally intensive, thus drastically reducing the workload associated with solving tensor least squares problems.
The CPRAND-MIX algorithm further enhances robustness by incorporating efficient mixing techniques through random sign-flip and orthogonal transformations (e.g., FFT), promoting incoherence before sampling. This preprocessing step ensures that the factor matrices used in the decomposition maintain incoherence, a property beneficial to randomized least squares algorithms.
Numerical experiments conducted on both synthetic and real-world datasets, including large-scale tensors, demonstrate improvements in execution speed, reductions in memory usage, and robustness regarding initialization sensitivity. The randomized algorithms often achieved solutions comparable in quality to CP-ALS but with noticeable speed advantages. These results suggest that randomized CP decomposition methods can be effective in handling large, dense multiway data more efficiently.
Additionally, the authors propose a novel lightweight stopping condition that estimates the model fit error based on sampling entries of the tensor. This approach reduces the computational burden traditionally associated with convergence checks in CP-ALS, thereby further improving the method's practicality in real-world applications.
Overall, the implications of this research are significant for advancing tensor decomposition methodologies. The introduction of randomness and careful sampling suggests potential pathways for adapting other tensor decomposition techniques and possibly extending the framework to sparse data or large-scale distributed computations. Future developments may include exploring the theoretical underpinnings of random sampling schemes within tensor decompositions and refining algorithms to better handle the intricacies of tensor data sparsity and high dimensionality.