We discuss the problem of unique determination of the finite free discrete Schrödinger operator from its spectrum, also known as the Ambarzumian problem, with various boundary conditions, namely any real constant boundary condition at zero and Floquet boundary conditions of any angle. Then we prove the following Ambarzumian-type mixed inverse spectral problem: diagonal entries except the first and second ones and a set of two consecutive eigenvalues uniquely determine the finite free discrete Schrödinger operator.

Background: High-dimensional genomic data studies are often found to exhibit strong correlations, which results in instability and inconsistency in the estimates obtained using commonly used regularization approaches including both the Lasso and MCP, and related methods. Result: In this paper, we perform a comparative study of regularization approaches for variable selection under different correlation structures, and propose a two-stage procedure named rPGBS to address the issue of stable variable selection in various strong correlation settings. This approach involves repeatedly running of a two-stage hierarchical approach consisting of a random pseudo-group clustering and bi-level variable selection. Conclusion: Both the simulation studies and high-dimensional genomic data analysis have demonstrated the advantage of the proposed rPGBS method over most commonly used regularization methods. In particular, the rPGBS results in more stable selection of variables across a variety of correlation settings, as compared to recent work addressing variable selection with strong correlations. Moreover, the rPGBS is computationally efficient across various settings.

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