arXiv subjects

Kensuke Aishima

Publications and source records attributed to Kensuke Aishima.

3 recordsLinked to original sources

arXiv2026

Asymptotic analysis of the Gaussian kernel matrix for partially noisy data in high dimensions

The Gaussian kernel is one of the most important kernels, applicable to many research fields, including scientific computing and data science. In this paper, we present asymptotic analysis of the Gaussian kernel matrix in high dimension under a statistical model of noisy data. The main result is a nice combination of Karoui's asymptotic analysis with procedures of constrained low rank matrix approximations. More specifically, Karouli clarified an important asymptotic structure of the Gaussian kernel matrix, leading to strong consistency of the eigenvectors, though the eigenvalues are inconsistent. This paper focuses on the above results and presents a consistent estimator with the use of the smallest eigenvalue, whenever the target kernel matrix tends to low rank in the asymptotic regime. Importantly, asymptotic analysis is given under a statistical model representing partial noise. Although a naive estimator is inconsistent, applying an optimization method for low rank approximations with constraints, we overcome the difficulty caused by the inconsistency, resulting in a new estimator with strong consistency in rank deficient cases.

math.ST↗

arXiv2023

Strong consistency of an estimator by the truncated singular value decomposition for an errors-in-variables regression model with collinearity

In this paper, we prove strong consistency of an estimator by the truncated singular value decomposition for a multivariate errors-in-variables linear regression model with collinearity. This result is an extension of Gleser's proof of the strong consistency of total least squares solutions to the case with modern rank constraints. While the usual discussion of consistency in the absence of solution uniqueness deals with the minimal norm solution, the contribution of this study is to develop a theory that shows the strong consistency of a set of solutions. The proof is based on properties of orthogonal projections, specifically properties of the Rayleigh-Ritz procedure for computing eigenvalues. This makes it suitable for targeting problems where some row vectors of the matrices do not contain noise. Therefore, this paper gives a proof for the regression model with the above condition on the row vectors, resulting in a natural generalization of the strong consistency for the standard TLS estimator.

math.ST↗

arXiv2023

Consistent estimation with the use of orthogonal projections for a linear regression model with errors in the variables

In this paper, we construct an estimator of an errors-in-variables linear regression model. The regression model leads to a constrained total least squares problems with row and column constraints. Although this problem can be numerically solved, it is unknown whether the solution has consistency in the statistical sense. The proposed estimator can be constructed by the use of orthogonal projections and their properties, its strong consistency is naturally proved. Moreover, our asymptotic analysis proves the strong consistency of the total least squares solution of the problem with row and column constraints.

math.NA↗