Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics
By Tommaso Cai, Anru R. Zhang. 2016
185 citations recorded by the source corpus.
Perturbation bounds for singular spaces, in particular Wedin's $\sin \Theta$ theorem, are a fundamental tool in many fields including high-dimensional statistics, machine learning, and applied mathematics. In this paper, we establish separate perturbation bounds, measured in both spectral and Frobenius $\sin \Theta$ distances, for the left and right singular subspaces. Lower bounds, which show that the individual perturbation bounds are rate-optimal, are also given. The new perturbation bounds are applicable to a wide range of problems. In this paper, we consider in detail applications to low-rank matrix denoising and singular space estimation, high-dimensional clustering, and canonical correlation analysis (CCA). In particular, separate matching upper and lower bounds are obtained for estimating the left and right singular spaces. To the best of our knowledge, this is the first result that gives different optimal rates for the left and right singular spaces under the same perturbation. In addition to these problems, applications to other high-dimensional problems such as community detection in bipartite networks, multidimensional scaling, and cross-covariance matrix estimation are also discussed.
Subfields: Computational Mechanics, Statistics and Probability, Signal Processing
View Paper: https://www.semanticscholar.org/paper/00a8da12f221bb8f22aea1cbc82ed003d75191a1
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Computational Mechanics
In Statistics and Probability
In Signal Processing
Citations in the corpus
- Singular vector and singular subspace distribution for the matrix denoising model (2018, background): “The estimation of singular subspace is important in the recovery of low-rank matrix based on noisy observations (see for instance [17, 19, 27] and reference therein).”
- Statistical Inference for High-Dimensional Matrix-Variate Factor Models (2020, background): “See Cai et al. (2018); Wedin (1972); Cai et al. (2019) and references therein for more discussions on this topic.”
- Concentration bounds on response-based vector embeddings of black-box generative models (2025, unknown): “In settings that are much simpler to ours, such as the ones discussed in Cai and Zhang (2018), Zhang et al. (2022) and Chen et al. (2021), a condition similar to ours is shown to be necessary.”