# Towards Efficient and Accurate Approximation: Tensor Decomposition Based on Randomized Block Krylov Iteration

arXiv: https://arxiv.org/abs/2211.14828 · HTML: https://mlsanity.com/paper/2211.14828/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 52 references that are in the map.
- First posted on arXiv 27 Nov 2022.

## Builds on (most cited first, 12 of 12)

- [Sketching as a Tool for Numerical Linear Algebra](https://mlsanity.com/paper/1411.4357/) (2014)
- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [Randomized Block Krylov Methods for Stronger and Faster Approximate Singular Value Decomposition](https://mlsanity.com/paper/1504.05477/) (2015)
- [Practical sketching algorithms for low-rank matrix approximation](https://mlsanity.com/paper/1609.00048/) (2016)
- [Fast and Guaranteed Tensor Decomposition via Sketching](https://mlsanity.com/paper/1506.04448/) (2015)
- [Tucker decomposition-based Temporal Knowledge Graph Completion](https://mlsanity.com/paper/2011.07751/) (2020)
- [Randomized Tensor Ring Decomposition and Its Application to Large-scale Data Reconstruction](https://mlsanity.com/paper/1901.01652/) (2019)
- [Fast and Accurate Randomized Algorithms for Low-rank Tensor Decompositions](https://mlsanity.com/paper/2104.01101/) (2021)
- [More Efficient Sampling for Tensor Decomposition With Worst-Case Guarantees](https://mlsanity.com/paper/2110.07631/) (2021)
- [Low-Rank Approximation with 1/ε1/3 Matrix-Vector Products](https://mlsanity.com/paper/2202.05120/) (2022)
- [A Practical Guide to Randomized Matrix Computations with MATLAB Implementations](https://mlsanity.com/paper/1505.07570/) (2015)
- [An Improved Frequent Directions Algorithm for Low-Rank Approximation via Block Krylov Iteration](https://mlsanity.com/paper/2109.11703/) (2021)

## Abstract (arXiv)

Efficient and accurate low-rank approximation (LRA) methods are of great significance for large-scale data analysis. Randomized tensor decompositions have emerged as powerful tools to meet this need, but most existing methods perform poorly in the presence of noise interference. Inspired by the remarkable performance of randomized block Krylov iteration (rBKI) in reducing the effect of tail singular values, this work designs an rBKI-based Tucker decomposition (rBKI-TK) for accurate approximation, together with a hierarchical tensor ring decomposition based on rBKI-TK for efficient compression of large-scale data. Besides, the error bound between the deterministic LRA and the randomized LRA is studied. Numerical experiences demonstrate the efficiency, accuracy and scalability of the proposed methods in both data compression and denoising.