Summary of Computational and Statistical Guarantees For Tensor-on-tensor Regression with Tensor Train Decomposition, by Zhen Qin and Zhihui Zhu
Computational and Statistical Guarantees for Tensor-on-Tensor Regression with Tensor Train Decomposition
by Zhen Qin, Zhihui Zhu
First submitted to arxiv on: 10 Jun 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: Signal Processing (eess.SP); Optimization and Control (math.OC)
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary The proposed tensor-on-tensor regression model has been shown to be effective in scenarios like scalar-on-tensor regression and tensor-on-vector regression. However, the exponential growth in tensor complexity poses challenges for storage and computation in ToT regression. The introduction of tensor decompositions, specifically the tensor train (TT)-based ToT model, has addressed these issues by reducing memory requirements, improving computational efficiency, and decreasing sampling complexity. Despite the practical benefits, a disparity exists between theoretical analysis and real-world performance. This paper delves into the theoretical and algorithmic aspects of the TT-based ToT regression model, including an error analysis for the solution to a constrained least-squares optimization problem. The proposed optimization algorithms, iterative hard thresholding (IHT) and factorization approach using Riemannian gradient descent (RGD), are shown to have linear convergence rates when the restricted isometry property (RIP) is satisfied. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary Tensor-on-tensor regression models can be used for various tasks, like predicting scalar values based on tensors or vice versa. However, as tensor complexity grows, these models become less efficient due to increased storage and computation needs. To overcome this challenge, researchers have introduced tensor decompositions, which reduce memory requirements, speed up computations, and simplify sampling processes. This paper explores the theoretical and algorithmic aspects of a specific type of tensor decomposition called the tensor train (TT). The authors analyze the error bounds for solutions to optimization problems using TT and propose two efficient algorithms, IHT and RGD, which can be used to find these solutions. |
Keywords
» Artificial intelligence » Gradient descent » Optimization » Regression
