Kaili Shi | Searcharxiv

arXiv2025

Doubly Free-Boundary Macdonald Processes: Reflection Identities and Jack Asymptotics

We introduce a doubly free-boundary Macdonald process on rail-yard interlacings and develop a reflection calculus for its observables. Boundary Cauchy--Littlewood identities, combined with Negu\c t operators, yield exact multipoint contour formulas for arbitrary \(L/R\) words. Under the Jack scaling \[ q=t^\alpha,\qquad t=e^{-n\beta\epsilon}, \] and piecewise-periodic data, these formulas imply a Laplace-transform law of large numbers and a weak slope-measure limit shape at \(L\)-type columns. For arbitrary piecewise-periodic \(L/R\) backgrounds and finitely many \(L\)-type marked columns, under the stated contour, branch, and normal-convergence hypotheses, the centered height-Laplace observables converge jointly to a Gaussian vector. Its covariance exhibits a boundary--deformation separation: the Jack parameter and the microscopic rail-yard data enter through the one-point spectral factors and the normalization, whereas the two-point interaction is the logarithmic derivative of an annular prime function generated by the two boundary reflections. Thus the deformation changes the spectral map while preserving the annular image geometry of the Schur specialization. For \(\beta=1\), in the all-\(L\) sector and under explicit signed zero--pole and root-localization hypotheses, we characterize regular liquid and frozen points through the nonreal-root structure of the characteristic equation and show that nondegenerate regular interfaces lie on the real double-root locus. The half-space Macdonald-process formulas are recovered in the continuous degeneration \(v\downarrow0\), which forces the right boundary partition to be empty.

math.PR↗

arXiv2021

Stochastic tensor space feature theory with applications to robust machine learning

In this paper we develop a Multilevel Orthogonal Subspace (MOS) Karhunen-Loeve feature theory based on stochastic tensor spaces, for the construction of robust machine learning features. Training data are treated as instances of a random field within a relevant Bochner space. Our key observation is that separate machine learning classes can reside predominantly in mostly distinct subspaces. Using the Karhunen-Loeve expansion and a hierarchical expansion of the first (nominal) class, a MOS is constructed to detect anomalous signal components, treating the second class as an outlier of the first. The projection coefficients of the input data into these subspaces are then used to train a Machine Learning (ML) classifier. These coefficients become new features from which much clearer separation surfaces can arise for the underlying classes. Tests in the blood plasma dataset (Alzheimer's Disease Neuroimaging Initiative) show dramatic increases in accuracy. This contrast to popular ML methods such as Gradient Boosting, RUS Boost, Random Forest and Neural Networks. We show that with a non-invasive blood test, high-accuracy results can be obtained for predicting AD stages such as cognitive normal, mild cognitive impairment and dementia.

stat.ML↗