R is a smooth nonconvex function for all j [ m]. We assume that Assumptions 1, 2 and 3 hold and the following assumption.

R is a smooth nonconvex function for all j [ m]. We assume that Assumptions 1, 2 and 3 hold and the following assumption.

Complex numbers represent the information of the amplitude and phase simultaneously.

A natural way to unlock particular capabilities is to improve this training data.

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets, such as graphs and point clouds.

Feature hashing, also known as the hashing trick, introduced by Weinberger et al. (2009), is one of the key techniques used in scaling-up machine learning algorithms.

This research aims to optimize intricate learning models by implementing quantization and bit-depth optimization techniques. The objective is to significantly cut time complexity while preserving model efficiency, thus addressing the challenge of extended execution times in intricate models. Two medical datasets were utilized as case studies to apply a Logistic Regression (LR) machine learning model. Using efficient quantization and bit depth optimization strategies the input data is downscaled from float64 to float32 and int32. The results demonstrated a significant reduction in time complexity, with only a minimal decrease in model accuracy post-optimization, showcasing the state-of-the-art optimization approach. This comprehensive study concludes that the impact of these optimization techniques varies depending on a set of parameters.

We propose the Modified Mahalanobis Distance Conformal Prediction (MMDCP), a unified framework for multi-class classification and outlier detection under label shift, where the training and test distributions may differ. In such settings, many existing methods construct nonconformity scores based on empirical cumulative or density functions combined with data-splitting strategies. However, these approaches are often computationally expensive due to their heavy reliance on resampling procedures and tend to produce overly conservative prediction sets with unstable coverage, especially in small samples. To address these challenges, MMDCP combines class-specific distance measures with full conformal prediction to construct a score function, thereby producing adaptive prediction sets that effectively capture both inlier and outlier structures. Under mild regularity conditions, we establish convergence rates for the resulting sets and provide the first theoretical characterization of the gap between oracle and empirical conformal $p$-values, which ensures valid coverage and effective control of the class-wise false discovery rate (CW-FDR). We further introduce the Summarized Class-Wise FDR (SCW-FDR), a novel global error metric aggregating false discoveries across classes, and show that it can be effectively controlled within the MMDCP framework. Extensive simulations and two real-data applications support our theoretical findings and demonstrate the advantages of the proposed method.

High-dimensional data often contain low-dimensional signals obscured by structured background noise, which limits the effectiveness of standard PCA. Motivated by contrastive learning, we address the problem of recovering shared signal subspaces from positive pairs, paired observations sharing the same signal but differing in background. Our baseline, PCA+, uses alignment-only contrastive learning and succeeds when background variation is mild, but fails under strong noise or high-dimensional regimes. To address this, we introduce PCA++, a hard uniformity-constrained contrastive PCA that enforces identity covariance on projected features. PCA++ has a closed-form solution via a generalized eigenproblem, remains stable in high dimensions, and provably regularizes against background interference. We provide exact high-dimensional asymptotics in both fixed-aspect-ratio and growing-spike regimes, showing uniformity's role in robust signal recovery. Empirically, PCA++ outperforms standard PCA and alignment-only PCA+ on simulations, corrupted-MNIST, and single-cell transcriptomics, reliably recovering condition-invariant structure. More broadly, we clarify uniformity's role in contrastive learning, showing that explicit feature dispersion defends against structured noise and enhances robustness.

Incomplete multi-view unsupervised feature selection (IMUFS), which aims to identify representative features from unlabeled multi-view data containing missing values, has received growing attention in recent years. Despite their promising performance, existing methods face three key challenges: 1) by focusing solely on the view-missing problem, they are not well-suited to the more prevalent mixed-missing scenario in practice, where some samples lack entire views or only partial features within views; 2) insufficient utilization of consistency and diversity across views limits the effectiveness of feature selection; and 3) the lack of theoretical analysis makes it unclear how feature selection and data imputation interact during the joint learning process. Being aware of these, we propose CLIM-FS, a novel IMUFS method designed to address the mixed-missing problem. Specifically, we integrate the imputation of both missing views and variables into a feature selection model based on nonnegative orthogonal matrix factorization, enabling the joint learning of feature selection and adaptive data imputation. Furthermore, we fully leverage consensus cluster structure and cross-view local geometrical structure to enhance the synergistic learning process. We also provide a theoretical analysis to clarify the underlying collaborative mechanism of CLIM-FS. Experimental results on eight real-world multi-view datasets demonstrate that CLIM-FS outperforms state-of-the-art methods.