The spectral norm of random inner-product kernel matrices
By Zhou Fan, Andrea Montanari. 2015
60 citations recorded by the source corpus.
We study an “inner-product kernel” random matrix model, whose empirical spectral distribution was shown by Xiuyuan Cheng and Amit Singer to converge to a deterministic measure in the large n and p limit. We provide an interpretation of this limit measure as the additive free convolution of a semicircle law and a Marchenko–Pastur law. By comparing the tracial moments of this matrix to an additive deformation of a Wigner matrix, we establish that for odd kernel functions, the spectral norm of this matrix converges almost surely to the edge of the limiting spectrum. Our study is motivated by the analysis of a covariance thresholding procedure for the statistical detection and estimation of sparse principal components, and our results characterize the limit of the largest eigenvalue of the thresholded sample covariance matrix in the null setting.
Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/81b1b796d1b9bfefe404ec0a191163ef15426fa8
How researchers see this paper
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In Discrete Mathematics and Combinatorics
In Mathematical Physics
- Lenka Zdeborov'a
- Florent Krzakala
- Robert Tibshirani
- Trevor J. Hastie
- Andrea Montanari
- Saharon Rosset
- Taiji Suzuki
- Murat A. Erdogdu
In Statistics and Probability
- Robert Tibshirani
- Trevor J. Hastie
- Jonathan E. Taylor
- Stefan Wager
- Saharon Rosset
- Andrea Montanari
- Taiji Suzuki
- Sara A. van de Geer
Citations in the corpus
- Surprises in High-Dimensional Ridgeless Least Squares Interpolation (2019, background): “In the context of kernel inner product random matrices, it appears (somewhat implicitly) in [17] and (more explicitly) in [27].”
- Capturing the learning curves of generic features maps for realistic data sets with a teacher-student model (2021, background): “The asymptotic behavior of kernel matrices has indeed been the subject of intense scrutiny [45, 62, 46, 11, 63, 48].”
- Generalization of Two-layer Neural Networks: An Asymptotic Viewpoint (2020, background): “For instance, El Karoui et al. (2010); Cheng and Singer (2013); Fan and Montanari (2019) studied the spectral properties of kernel matrix via Taylor series or Hermite polynomials on the nonlinearity, which in turn explains the generalization of high-dimensional kernel ridgeless interpolators (Liang…”