Graph contrastive learning attracts/disperses node representations for similar/dissimilar node pairs under some notion of similarity.

We thank all the reviewers for the responses and detailed comments. The first difference between the earlier SRM versus our SSTL lies in defining the shared space. Empirical studies in [3] also showed that the original forms of SRM and HA ( i.e., the Y es, this subject ordering can matter, but this is fairly standard -- i.e., The revised version will explicitly summarize the entire training and performance processes. Reviewer 1: Thank you for your insightful comments. Instead, we said that'scatter matrices The revision will address all of those comments.

In modern gear manufacturing, stringent Noise, Vibration, and Harshness (NVH) requirements demand high-precision finishing operations such as power honing. Conventional quality control strategies rely on post-process inspections and Statistical Process Control (SPC), which fail to capture transient machining anomalies and cannot ensure real-time defect detection. This study proposes a novel, data-driven framework for in-process monitoring of gear power honing using vibration signal analysis and machine learning. Our proposed methodology involves continuous data acquisition via accelerometers, followed by time-frequency signal analysis. We investigate and compare the efficacy of three subspace learning methods for features extraction: (1) Principal Component Analysis (PCA) for dimensionality reduction; (2) a two-stage framework combining PCA with Linear Discriminant Analysis (LDA) for enhanced class separation; and (3) Uncorrelated Multilinear Discriminant Analysis with Regularization (R-UMLDA), adapted for tensor data, which enforces feature decorrelation and includes regularization for small sample sizes. These extracted features are then fed into a Support Vector Machine (SVM) classifier to predict four distinct gear quality categories, established through rigorous geometrical inspections and test bench results of assembled gearboxes. The models are trained and validated on an experimental dataset collected in an industrial context during gear power-honing operations, with gears classified into four different quality categories. The proposed framework achieves high classification accuracy (up to 100%) in an industrial setting. The approach offers interpretable spectral features that correlate with process dynamics, enabling practical integration into real-time monitoring and predictive maintenance systems.

We thank all the reviewers for the responses and detailed comments. The first difference between the earlier SRM versus our SSTL lies in defining the shared space. Empirical studies in [3] also showed that the original forms of SRM and HA ( i.e., the Y es, this subject ordering can matter, but this is fairly standard -- i.e., The revised version will explicitly summarize the entire training and performance processes. Reviewer 1: Thank you for your insightful comments. Instead, we said that'scatter matrices The revision will address all of those comments.

We consider the problem of estimating a regularization parameter, or a shrinkage coefficient $α\in (0,1)$ for Regularized Tyler's M-estimator (RTME). In particular, we propose to estimate an optimal shrinkage coefficient by setting $α$ as the solution to a suitably chosen objective function; namely the leave-one-out cross-validated (LOOCV) log-likelihood loss. Since LOOCV is computationally prohibitive even for moderate sample size $n$, we propose a computationally efficient approximation for the LOOCV log-likelihood loss that eliminates the need for invoking the RTME procedure $n$ times for each sample left out during the LOOCV procedure. This approximation yields an $O(n)$ reduction in the running time complexity for the LOOCV procedure, which results in a significant speedup for computing the LOOCV estimate. We demonstrate the efficiency and accuracy of the proposed approach on synthetic high-dimensional data sampled from heavy-tailed elliptical distributions, as well as on real high-dimensional datasets for object recognition, face recognition, and handwritten digit's recognition. Our experiments show that the proposed approach is efficient and consistently more accurate than other methods in the literature for shrinkage coefficient estimation.

Graph contrastive learning attracts/disperses node representations for similar/dissimilar node pairs under some notion of similarity. It may be combined with a low-dimensional embedding of nodes to preserve intrinsic and structural properties of a graph. COLES, a recent graph contrastive method combines traditional graph embedding and negative sampling into one framework. COLES in fact minimizes the trace difference between the within-class scatter matrix encapsulating the graph connectivity and the total scatter matrix encapsulating negative sampling. In this paper, we propose a more essential framework for graph embedding, called Generalized Laplacian EigeNmaps (GLEN), which learns a graph representation by maximizing the rank difference between the total scatter matrix and the within-class scatter matrix, resulting in the minimum class separation guarantee.

Over the years there has been ongoing interest in detecting authorship of a text based on statistical properties of the text, such as by using occurrence rates of noncontextual words. In previous work, these techniques have been used, for example, to determine authorship of all of \emph{The Federalist Papers}. Such methods may be useful in more modern times to detect fake or AI authorship. Progress in statistical natural language parsers introduces the possibility of using grammatical structure to detect authorship. In this paper we explore a new possibility for detecting authorship using grammatical structural information extracted using a statistical natural language parser. This paper provides a proof of concept, testing author classification based on grammatical structure on a set of "proof texts," The Federalist Papers and Sanditon which have been as test cases in previous authorship detection studies. Several features extracted from the statistical natural language parser were explored: all subtrees of some depth from any level; rooted subtrees of some depth, part of speech, and part of speech by level in the parse tree. It was found to be helpful to project the features into a lower dimensional space. Statistical experiments on these documents demonstrate that information from a statistical parser can, in fact, assist in distinguishing authors.

Most signal processing and statistical applications heavily rely on specific data distribution models. The Gaussian distributions, although being the most common choice, are inadequate in most real world scenarios as they fail to account for data coming from heavy-tailed populations or contaminated by outliers. Such problems call for the use of Robust Statistics. The robust models and estimators are usually based on elliptical populations, making the latter ubiquitous in all methods of robust statistics. To determine whether such tools are applicable in any specific case, goodness-of-fit (GoF) tests are used to verify the ellipticity hypothesis. Ellipticity GoF tests are usually hard to analyze and often their statistical power is not particularly strong. In this work, assuming the true covariance matrix is unknown we design and rigorously analyze a robust GoF test consistent against all alternatives to ellipticity on the unit sphere. The proposed test is based on Tyler's estimator and is formulated in terms of easily computable statistics of the data. For its rigorous analysis, we develop a novel framework based on the exchangeable random variables calculus introduced by de Finetti. Our findings are supported by numerical simulations comparing them to other popular GoF tests and demonstrating the significantly higher statistical power of the suggested technique.

Linear Discriminant Analysis (LDA) is a well-known scheme for feature extraction and dimension reduction. It has been used widely in many ap- plications involving high-dimensional data, such as face recognition and image retrieval. An intrinsic limitation of classical LDA is the so-called singularity problem, that is, it fails when all scatter matrices are singu- lar. A well-known approach to deal with the singularity problem is to apply an intermediate dimension reduction stage using Principal Com- ponent Analysis (PCA) before LDA. The algorithm, called PCA LDA, is used widely in face recognition.