We investigate the least squares linear regression problem with random partial Discrete Fourier Transform (DFT) matrices, providing a rigorous analysis of the model's generalization error. By leveraging tools from random matrix theory, we derive exact non-asymptotic bounds for the risk of the Moore-Penrose estimator, which hold for finite-dimensional problems and reveal the precise dependence on key parameters such as the sample size, dimension, and noise variance. Then we obtain a characterization of the double descent phenomenon in the linear regression context, demonstrating how the risk evolves when the number of parameters $p$ and the number of samples $n$ tend to infinity, with $p/n$ fixed. The analysis relies on applications of the Stieltjes transform for random Fourier matrices, enabling a precise description of the spectral properties of these matrices and their impact on regression performance. To validate our theoretical findings, we present several numerical examples that illustrate the double descent curves. These simulations align closely with our derived bounds, confirming their predictive power in both under-parameterized and over-parameterized regimes.
In the era of high-dimensional data, the classical assumption that the number of observations n vastly exceeds the number of variables p is frequently violated. When p and n grow proportionally (p/n → c > 0), the sample covariance matrix becomes severely distorted by sampling noise. Its eigenvalues are systematically b...
Innocent Nsabimana· International Journal For Mu...· 0 citations
Stochastic approximation provides a general framework for online estimation and optimization. Statistical inference based on the resulting estimates requires understanding their fluctuations around the target. For Polyak--Ruppert averaging, functional central limit theorems describe the normalized cumulative estimation...
This paper investigates the asymptotic behavior of the out-of-sample prediction risk of the high-dimensional ridgeless least-squares estimator when the feature dimension $p$ and the sample size $n$ grow proportionally. We consider a generalized spiked population covariance model with multiple latent factors, where the...
Approximate Bayesian computation (ABC) replaces likelihood evaluation by simulation and comparison of observed and synthetic data. We establish minimax-rate guarantees for nonparametric ABC under random-series priors with simulable finite-dimensional coordinates. The contraction theorem uses local prior mass, bounds on...
Resolvent Monte Carlo estimates eigenvalues of large matrices by sampling Markov chains and reading the target value off a truncated resolvent quotient, trading exact arithmetic for a stochastic error that the almost-optimal sampling scheme is designed to suppress. This paper studies when that error vanishes outright....
Tsvetelina Kostadinov, I. Dimov· Mathematics· 0 citations
We first prove spectral convergence of the random feature method (RFM) for multidimensional targets in Sobolev, Gevrey, ultra-analytic, and bandlimited classes. The analysis establishes general high-probability approximation estimates in the interpolation scale generated by a kernel integral operator. On a single event...
P. Ming, Hao Yu· 1 citation
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