Principal Component Analysis

We present a simple and efficient algorithm for robust approximate message passing (AMP) in the spiked matrix setting. In particular, let $\varepsilon$ be a sufficiently small constant, and suppose that $X \in \mathbb R^{n \times n}$ is a Gaussian matrix with a planted rank-$1$ spike, and $E \in \mathbb R^{n \times n}$ is an adversarially chosen matrix supported on an $\varepsilon n \times \varepsilon n$ principal minor. Let $v_{\mathrm{AMP}}(X)$ be the output of an AMP iteration on the uncorrupted matrix $X$. We give a procedure that, given access only to the corrupted matrix $Y = X + E$, computes a vector $v_{\mathrm{ALG}}(Y)$ which is $\tilde{O}(\sqrt{\varepsilon})$-close to $v_{\mathrm{AMP}}(X)$, for any of a class of AMP iterations which includes sparse Principal Component Analysis (PCA), non-negative PCA, and $\mathbb Z_2$ synchronization. Our algorithm consists of a spectral pre-processing step combined with a robust spectral initialization procedure; given these inputs, we prove that (perhaps surprisingly) AMP is robust out-of-the-box.

Compositional data must be analysed through log-ratios: scale invariance, the defining axiom of the field, leaves no alternative. The centred log-ratio divides by the geometric mean of every part, so a single contaminated component shifts every centred-log-ratio coordinate at once, displacing the log-ratio vector by a fixed amount that no choice of coordinates can reduce. We develop a theory of cellwise contamination on the simplex around this observation. A scale-invariant contamination model built from multiplicative perturbation combines with a propagation theorem showing that corruption of a single raw part induces a rank-one shift of the log-ratio vector, with direction determined by the contrast matrix. The resulting perturbation pattern is not equivalent to any independent cellwise contamination model in log-ratio coordinates -- so standard Euclidean cellwise methods applied to log-ratios are ill-posed under the simplex contamination mechanism. For estimators whose Euclidean cellwise breakdown is witnessed by a column-concentrated configuration -- a class including MCD, $S$-, $τ$-, and coordinate-wise $M$-estimators of location and scatter -- the cellwise breakdown value on the simplex is reduced by the factor $(D-1)/D$ relative to its Euclidean counterpart, a reduction that is tight and arises purely from the normalisation mismatch between $nD$ raw cells and $n(D-1)$ ilr cells. The cellwise influence function for the variation matrix carries a diagnostic fingerprint: contamination of a single part inflates exactly one row and column, identifying the responsible component. These results form the theoretical foundation for cellwise-robust methods on the simplex; a companion paper develops a cellwise-robust PCA estimator that exploits the propagation geometry and demonstrates it on simulated and geochemical data.

This book provides a compact, derivation-oriented introduction to the mathematical foundations of modern generative artificial intelligence. Rather than surveying every recent architecture or implementation detail, it develops a coherent route through the ideas connecting major families of generative models, from PCA, probabilistic PCA, variational autoencoders, and diffusion models to normalising flows, autoregressive factorisations, GANs, Wasserstein GANs, and energy-based models. The aim is to make the structure of generative modelling more accessible without removing the mathematical substance needed to understand how these models are derived and related. The book is intended as a foundation-building primer for mathematically curious researchers, practitioners, and students.

Principal component analysis (PCA) is traditionally implemented through a covariance or kernel matrix, leading-eigenvector extraction, and hard rank-$k$ projection. These steps can be computationally costly in high-dimensional and quantum-data settings, sensitive to small eigengaps, and unnecessary when downstream tasks only require principal-subspace scores. Such score-based objectives are important in applications such as anomaly detection, spectral-energy profiling, and other postselection tasks. To address these needs, we introduce a measurement-based soft PCA framework replacing the hard top-$k$ projector with an entropy-regularized Fermi--Dirac filter. This filter is the unique optimizer of an entropy-regularized variational formulation of PCA and converges to the classical PCA projector in the zero-temperature limit. This filter has a direct interpretation as a quantum measurement, which naturally suggests a quantum approach. For centered covariance operators represented by quantum feature states, a single fixed circuit, together with threshold calibration, accesses all optimal filters for different rank budgets or retained-variance levels without rank-dependent circuit updates or eigenvector recovery. For new inputs, the same calibrated quantum circuit yields soft principal subspace scores, spectral energy profiles, and postselected filtered states. The required centering of both training and test data is performed coherently inside the quantum protocol, which is particularly important for quantum data where no classical feature vectors or centered Gram matrix are directly available. By reframing PCA as a calibrated measurement task, this framework bypasses the need for iterative eigenvector extraction and achieves a dimension-independent sample complexity $O(η^{-2})$ for normalized fractional-rank or retained variance scoring at additive accuracy $η$.

We propose KO-PDE-IDENT, a data-driven framework for identifying parsimonious partial differential equations (PDEs) with false discovery rate (FDR) control. PDE discovery from noisy observations is often hindered by extreme multicollinearity among candidate terms, which causes typical sparse-regression methods to select spurious terms. To address this problem, KO-PDE-IDENT initially mines a support set of potential candidate terms via model-X knockoff filters with finite-sample FDR control, then refines and ranks the surviving PDE alternatives. The framework integrates three components. First, knockoff feature statistics are constructed by coupling $\ell_{0}$-constrained adaptive best-subset selection with SHapley Additive exPlanations (SHAP), yielding an effective and computationally efficient difference statistic. Second, a recursive feature elimination (RFE) procedure removes terms whose marginal contributions are dispensable and assesses statistical necessity through knockoff-perturbed hypothesis testing. Third, the final model selection is formulated as a multi-criteria decision-making (MCDM) problem, where the optimal governing equation is the alternative that best balances a wide range of criteria such as predictive accuracy, model complexity and coefficient uncertainty. We evaluate KO-PDE-IDENT on five canonical PDEs under severe noise corruption. Empirical results show that our framework can exactly recover the true PDE structure, eliminating false discoveries while retaining all true underlying terms, with low coefficient estimation error.

Geometric deep learning organises neural architectures around the symmetries of their data domain, with the choice of symmetry group serving as a geometric prior that determines what representations can be learned. Metric-Aware Principal Component Analysis (MAPCA) parameterises principal component analysis by a positive-definite metric matrix, with a canonical subfamily interpolating between standard PCA and output whitening and a diagonal-metric point recovering Invariant PCA (IPCA). This paper positions MAPCA within the geometric deep learning framework. The metric is read as the geometric prior; the orthogonal group preserving it is the symmetry group it induces; MAPCA solutions are equivariant under this group with the resulting spectrum invariant; and MAPCA's defining constraint is the linear analogue of the Schur-type weight constraints used in equivariant networks. Across six axes - domain, symmetry group, equivariance, invariance, architectural primitive, and geometric prior - we construct a precise dictionary between MAPCA and geometric deep learning. The technical anchor is a uniqueness theorem characterising IPCA as the unique linear data-derived metric in the MAPCA family that is equivariant under arbitrary diagonal rescaling and projects onto the fixed-point set of the action, equivalent under normalisation to the variance-maximisation criterion in its precise form. The paper closes with three bridges: kernel PCA as the nonlinear extension, spectral graph methods as MAPCA on graphs, and a deep MAPCA construction extending the positioning into deep equivariant networks

Removing noise is difficult, but adding noise is easy. In this work, we show how to eliminate mean-shift noisy components from PCA by deliberately introducing knockoff mean-shift perturbation. Standard PCA is highly sensitive to shifts in the sample mean: a small fraction of samples from a shifted distribution can cause large deviations in the leading principal components. In high-dimensional regimes, existing Robust PCA approaches cannot handle the mean-shift contamination structure inherent in the mixture model. Using tools from Random Matrix Theory, we prove that the mean-shift spikes are spectrally separable from the stable eigenvalues of the original covariance. Furthermore, the original eigenspace remains asymptotically invariant to the contamination, independent of the mixture weight. Exploiting this spectral stability, we propose a simple, two-stage PCA algorithm by adding knockoff mean that identifies and removes the mean-shift component using only standard PCA operations.

Unsupervised feature selection is commonly formulated as a multiobjective optimisation problem that jointly optimises subset quality and subset size. Yet the behaviour of this formulation depends critically on the choice of evaluation objective, the direction of subset-size regularisation, and the initialisation strategy. We study these factors in a controlled setting using a synthetic dataset with known informative, redundant, and irrelevant feature types. Six formulations are compared by combining three evaluation objectives: accuracy, silhouette score, and PCA reconstruction loss with subset-size minimisation or maximisation. The results show that formulation strongly affects both search dynamics and the quality of the resulting Pareto front. Silhouette-based formulations exhibit a strong bias toward trivial low-cardinality solutions and remain weak proxies for predictive performance. In contrast, the proposed PCA loss objective produces compact subsets with test accuracy comparable to subsets obtained by directly optimising supervised accuracy. Overall, the study shows that objective design is central to effective multiobjective unsupervised feature selection.

Distributed principal component analysis (PCA) produces node-level estimates of both a mean vector and a principal subspace. Robustly aggregating these heterogeneous objects requires a relative scale between mean error and subspace error. We study a scale-calibrated median-of-means estimator for this problem using the product geometry of Euclidean space and the Grassmann manifold. A node-level PCA expansion shows that the mean component has the usual linear influence, whereas the subspace component is an eigengap-weighted covariance perturbation. We prove a local reduction showing that the proposed product-manifold median-of-means estimator is asymptotically equivalent to a scaled spatial median of node influence errors. This yields fixed-node non-Gaussian limits, growing-node Gaussian limits with finite-block bias, and an explicit scale-dependent covariance formula. We propose robust block-scale and inference-optimal calibration rules, establish high-probability median-of-means bounds, characterize factorwise bad-node influence, and prove node-bootstrap validity. Simulations and large-scale single-cell RNA-seq data show that scale calibration adapts to eigengap-driven subspace uncertainty and provides a robust distributed PCA summary.

High-dimensional biological data often exhibit a severe mismatch between feature dimensionality and sample size, making reliable classification difficult in extremely small-data regimes. In these settings, kernel methods can lose discriminative power when latent compression fails to preserve class-separating structure. We study this problem in fine-grained plant phenomics and propose a hybrid workflow that compresses 1280-dimensional deep image embeddings into a 64-dimensional PCA space and then restructures them into an 11-dimensional supervised latent space using Linear Discriminant Analysis (LDA), followed by GPU-accelerated Quantum Kernel Alignment (QKA) on NVIDIA L40S hardware. Empirically, supervised latent restructuring substantially improves the geometric separability of the compressed representation, increasing the Silhouette coefficient from 0.003 in the raw embedding space and -0.006 in PCA-64 to 0.197 in the supervised LDA-11 space. However, downstream classical evaluation reveals a clear compression trade-off: Linear SVM and XGBoost improve in the restructured latent space, whereas RBF-SVM and Random Forest degrade under the same 11-dimensional bottleneck. Under a constrained optimization budget, QKA in this regime remains challenging, indicating that latent geometry alone is not sufficient for strong trainable quantum performance. These findings position representation geometry as a central design variable in small-data quantum learning and expose the practical difficulty of recovering nonlinear discriminative structure from aggressively compressed biological representations.

Optical navigation is a critical component for lunar orbiter and lander missions. Image-based crater identification has emerged as a promising technology for optical navigation due to the abundance of craters on the lunar surface and the availability of extensive crater catalogs. Moreover, due to the relative morphological homogeneity among lunar craters, template matching has been identified as a promising approach for identification. In this paper, we propose EigenCrater, an automated crater template generation method based on principal component analysis of crater digital elevation maps (DEMs). We demonstrate superior detection and position estimation performance relative to hand-picked templates on simulated lunar imagery.

Electroencephalogram (EEG) signals are highly susceptible to artifacts, resulting in a low signal-to-noise ratio which makes extraction of meaningful neural information challenging. Artifact Subspace Reconstruction (ASR) is one of the most widely used artifact filtering techniques in EEG-based BCI applications, owing to its real-time applicability. ASR reconstructs artifact-free signals by operating in Principal Component (PC) space within sliding windows. However, ASR performance is critically sensitive to its threshold parameter - an incorrect threshold risks removing task-relevant neural features alongside artifacts. Furthermore, since PCs are linear combinations of all channels, subspace reconstruction in PC space may alter the underlying data structure, potentially discarding essential neural information. To address these limitations, we propose nASR, a novel end-to-end trainable Keras layer that jointly optimizes artifact rejection and downstream decoding. nASR introduces two trainable threshold parameters: K, which governs artifact detection in PC variance space, and L, which quantifies eigen-spread to pinpoint the primary artifact--contributing channels, enabling selective channel-level reconstruction that preserves clean channel information. An ablation study comprising five model variants (m01 - m05), evaluated across two subjects from the BCI Competition IV Dataset 1, confirms that nASR variants consistently outperform traditional ASR on test classification metrics, while achieving a 6-8x reduction in inference time, making nASR a strong candidate for real-time BCI applications demanding both low latency and high decoding performance.

Predictive maintenance in complex systems is often complicated by the heterogeneity and redundancy of monitored variables,which can obscure fault-relevant information and reduce model interpretability. This work proposes a semantic feature segmentation framework that decomposes the monitored feature space into a canonical component,expected to retain the dominant predictive information, and a residual component containing structurally peripheral signals. The segmentation is defined through domain informed criteria and sets up monitoring variables into functional groups reflecting operational mechanisms such as throughput,latency,pressure,network activity,and structural state. To evaluate the effectiveness of this decomposition, we adopt a predictive perspective in which expected predictive risk is used as an operational proxy for task-relevant information. Experimental results obtained through time-aware cross-validation show that the canonical space consistently achieves lower predictive risk than the residual space across multiple temporal configurations, indicating that the semantic segmentation concentrates the most relevant information for fault anticipation. In addition, the canonical segments exhibit significantly stronger intra-segment coherence than inter-segment dependence, and this structural organization remains stable after redundancy reduction. When compared with the full feature space and with a Principal Component Analysis (PCA) representation, the canonical space carries out comparable predictive performance and furthermore preserves the semantic meaning of the original variables. These findings suggest that semantic feature segmentation provides an interpretable and information-preserving decomposition of monitoring signals, enabling competitive predictive performance without sacrificing the operational interpretability required in predictive maintenance applications.

We address shortcomings of principal component analysis (PCA) for visualizing high-dimensional data lying on a nonlinear low-dimensional manifold via two-dimensional scatterplots, focusing on a fossil teeth dataset from the early mammalian insectivore Kuehneotherium. While the PCA scatterplot reported by Jolliffe and Cadima (Philosophical Transactions of the Royal Society A, 2016) shows clustering in the region where PC2 < 0, our analysis based on t-SNE and persistent homology (PH) reveals a ring-like structure with no evident clustering and intrinsic dimensionality equal to one. We further propose a generative probabilistic-geometric model in which the data are sampled uniformly from a unit circle. Under this model, pairwise cosine distances follow an arcsine distribution, in qualitative agreement with the observed U-shaped distribution, thereby independently supporting the analysis based on t-SNE and persistent homology.

Symbolic-control drum generation requires preserving explicit event timing and dynamics while synthesizing acoustically plausible waveforms. We present Sec2Drum-DAC, a conditional latent-diffusion model for symbolic-to-audio drum rendering. The model conditions on event features sampled in physical time at codec-frame locations and predicts standardized principal-component coordinates of frozen DAC summed-codebook embeddings rather than waveform samples. In the evaluated DAC configuration, 72 principal components capture the observed training-frame summed-latent subspace under the stated SVD threshold, yielding a compact continuous denoising target with a deterministic reconstruction path to the 1024-dimensional DAC latent space before waveform decoding. Across 1,733 held-out four-beat windows, PCA diffusion improves paired spectral and transient metrics over deterministic PCA regression and a symbolic rendering baseline, while direct regression remains stronger on phase-sensitive waveform L1. Auxiliary RVQ cross-entropy improves short-step diffusion on mel error, onset-flux cosine, and waveform L1, with the most favorable trade-offs occurring at 6-25 denoising steps depending on the metric.

The power method is one of the most fundamental tools for extracting top principal components from data through low-rank matrix approximation. Yet, when the target rank is large, the cost of matrix multiplication associated with this procedure becomes a major bottleneck. We develop an algorithmic and theoretical framework for accelerating the power method using fast sketching, which is a popular paradigm in randomized linear algebra. Our framework leads to simple and provably efficient methods for singular value decomposition, low-rank factorization, and Nyström approximation, which attain strong numerical performance on benchmark problems. The key novelty in our analysis is the use of regularized spectral approximation, a property of fast sketching methods which proves more flexible in generalizing power method guarantees than traditional arguments.

Domain Generalization (DG) aims to learn representations that remain robust under out-of-distribution (OOD) shifts and generalize effectively to unseen target domains. While recent invariant learning strategies and architectural advances have achieved strong performance, explicitly discovering a structured domain-invariant subspace through second-order statistics remains underexplored. In this work, we propose CPCANet, a novel framework grounded in Common Principal Component Analysis (CPCA), which unrolls the iterative Flury-Gautschi (FG) algorithm into fully differentiable neural layers. This approach integrates the statistical properties of CPCA into an end-to-end trainable framework, enforcing the discovery of a shared subspace across diverse domains while preserving interpretability. Experiments on four standard DG benchmarks demonstrate that CPCANet achieves state-of-the-art (SOTA) performance in zero-shot transfer. Moreover, CPCANet is architecture-agnostic and requires no dataset-specific tuning, providing a simple and efficient approach to learning robust representations under distribution shift. Code is available at https://github.com/wish44165/CPCANet.

Federated techniques such as federated learning and federated analysis have emerged as a powerful paradigm for enabling multi-center research on sensitive clinical data while preserving patient privacy. In this study, we introduce a simulation framework that emulates a real-world federated research project focused on the analysis of multiple sclerosis (MS) patient data. The project comprises two components: an image segmentation task and a clinical data analysis task, where federated variants of survival analysis and Principal Component Analysis (PCA) are employed. To capture the complexity and heterogeneity of real clinical datasets, we construct a federation of high-fidelity synthetic cohorts designed to mirror MS-related clinical and demographic characteristics, while the imaging component leverages publicly available real-world datasets. Our simulation replicates key elements of authentic federated workflows, including distributed data governance, site-specific preprocessing, model training across isolated nodes, and the secure aggregation of analytical outputs. This framework provides a realistic testbed for developing, evaluating, and benchmarking federated learning methods in the context of MS research.

In this work, a data-driven framework based on Phase-Field simulations data is proposed to highlight the capabilities of neural networks to ensure accurate low dimensionality reduction of simulated microstructural images and to provide time-series analysis. The dataset was indeed constructed from high-fidelity Phase-Field simulations. Analyses demonstrated that the association of auto-encoder neural networks and principal component analyses leads to ensure efficient and significant dimensionality reduction: 1/196 of reduction ratio with more than 80% of accuracy. These findings give insight to apply analyses on data from the latent dimension. Application of Long Short Term Memory (LSTM) neural networks showed the possibility of making next frame predictions; that makes possible the acceleration of Phase-Field simulation without the need of high computing resources. We discussed the application of such a framework on various areas of research. Different methods are proposed from the conducted analyses, in order to ensure dimensionality reduction, including auto-encoders, principal component analysis and Artificial Neural Networks, and time-series analysis, including LSTM and Gated Recurrent Unit (GRU).

We present Sparse Backdoor, a supply-chain attack that plants a provably undetectable backdoor in pre-trained image classifiers, including convolutional networks and Vision Transformers. The attack injects a structured sparse perturbation along a randomly chosen direction into a small subset of columns at each fully connected layer, propagating a trigger signal to an adversary-chosen target class, and masks the perturbation with an independent isotropic Gaussian dither. The dither serves a single technical purpose: it induces a clean reference distribution anchored at the pre-trained weights, against which undetectability can be formalized. Under a mild margin condition on the pre-trained classifier, we show that the dithered reference is functionally equivalent to the original classifier. We prove that distinguishing the backdoor-injected model from this reference is at least as hard as Sparse PCA detection, which is computationally infeasible under standard hardness assumptions. The guarantee holds against any probabilistic polynomial-time distinguisher with white-box access to the parameters.

The paper studies the local geometry of embedding clouds induced by \emph{controlled local classes of semantically close sentences}. The central question is how controlled paraphrase-like semantic variation is organized in sentence embedding space and whether this local structure can be explicitly modeled by low-degree fitted carriers. We introduce a local geometric modeling scheme based on affine, quadratic, and cubic fitted models. We also use a surface-based latent probing procedure that constructs synthetic latent points in a reduced local PCA space with respect to the fitted carrier. The procedure is intended as an offline method for representation-space analysis, local manifold modeling, and geometry-aware latent probing. Generated latent points are evaluated using criteria that measure consistency with the fitted surface, preservation of neighborhood structure, agreement with the empirical distribution, stability of Hessian-based second-order shape descriptors, and stability of fitted-model coefficients. Experiments on controlled sets of semantically close sentences show that nonlinear local models describe embedding clouds more accurately than affine models. Surface-based generation provides strong fitted-geometry fidelity, including surface consistency, Hessian-based shape consistency, and coefficient consistency. Downstream experiments show that geometric validity of synthetic latent points does not automatically translate into improved classification performance. The results support explicit local geometric modeling of sentence embedding space and highlight the need to distinguish geometric validity from discriminative utility. As a resource contribution, we introduce \textbf{CoPaGE-300K}, a controlled template-based dataset of semantically close sentence variants with slot-level annotations and precomputed sentence embeddings.

We provide evidence of quantum kernel advantage under noiseless simulation in binary insurance classification on MIMIC-CXR chest radiographs using quantum support vector machines (QSVM) with frozen embeddings from three medical foundation models (MedSigLIP-448, RAD-DINO, ViT-patch32). We propose a two-tier fair comparison framework in which both classifiers receive identical PCA-q features. At Tier 1 (untuned QSVM vs. untuned linear SVM, C = 1 both sides), QSVM wins minority-class F1 in all 18 tested configurations (17 at p < 0.001, 1 at p < 0.01). The classical linear kernel collapses to majority-class prediction on 90-100% of seeds at every qubit count, while QSVM maintains non-trivial recall. At q = 11 (MedSigLIP-448 plateau center), QSVM achieves mean F1 = 0.343 vs. classical F1 = 0.050 (F1 gain = +0.293, p < 0.001) without hyperparameter tuning. Under Tier 2 (untuned QSVM vs. C-tuned RBF SVM), QSVM wins all seven tested configurations (mean gain +0.068, max +0.112). Eigenspectrum analysis reveals quantum kernel effective rank reaches 69.80 at q = 11, far exceeding linear kernel rank, while classical collapse remains C-invariant. A full qubit sweep reveals architecture-dependent concentration onset across models. Code: https://github.com/sebasmos/qml-medimage

Quantitative analysis of the kinematic chain in sports motion is essential for performance evaluation and injury prevention. Conventional methods such as the kinematic-sequence (KS) and continuous relative phase (CRP) are confined to adjacent joint pairs and lack a unified framework for whole-body coordination, while segmental power-flow analysis requires force plates and inertial parameters that restrict it to laboratory environments. We apply Complex Hilbert Principal Component Analysis (CHPCA) separately to each motion phase (backswing and downswing) on markerless 3D pose estimation data, extracting the dominant whole-body phase pattern as a single complex eigenvector. The pipeline further includes a fully automatic signal-based phase segmentation (no priors on strike count or rest location) and an extension to 1,079 body-surface mesh vertices, so that the kinematic chain is represented as a continuous phase field across the body. On 14 hammer-striking trials of a single subject, the framework reveals (i) a trunk-anchored global phase architecture, (ii) a functional asymmetry between preparation and execution phases quantified by Mode-1 contribution (45.5% vs. 70.5%) and inter-trial Spearman consistency (0.38 vs. 0.58), and (iii) a consistent reorganisation across both skeletal joints and mesh vertices ($p < 10^{-10}$ on 1,079 vertices). As a methodological consistency check, pairwise phase differences from the Mode-1 eigenvector are compared against CRP on all 190 joint pairs by a permutation test ($ρ= 0.473$, $p = 0.0005$). A correspondence analysis between Mode-1 amplitude and kinetic-energy mobilisation variance further shows a strong positive correlation in the downswing ($ρ\approx 0.71$ on both skeleton and mesh) and no correlation in the backswing, indicating that the proposed framework bridges kinematic and kinetic descriptions of coordination through phase structure.

Statistical modeling of popular music presents a unique challenge due to the complexity of song structures, which cannot be easily analyzed using conventional statistical tools. However, recent advances in data science have shown that converting non-standard data objects into real vector-valued embeddings enables meaningful statistical analysis. In this work, we demonstrate an approach based on logistic principal component analysis to construct embeddings from global song features, allowing for standard multivariate analysis. We apply this method to a corpus of Lennon and McCartney songs from 1962-1966, using embeddings derived from chords, melodic notes, chord and pitch transitions, and melodic contours. Our analysis explores how these song embeddings cluster by Beatles album, how songwriting styles evolved over time, and whether Lennon and McCartney's compositions exhibited convergence or divergence. This embedding-based approach offers a powerful framework for statistically examining musical structure and stylistic development in popular music.

Dimensionality reduction is a critical preprocessing step for clustering high-dimensional data, yet comprehensive evaluation of its impact across diverse methods and data types remains limited. In this study, we systematically assess the influence of five dimensionality reduction techniques - Principal Component Analysis (PCA), Kernel Principal Component Analysis (Kernel PCA), Variational Autoencoder (VAE), Isometric Mapping (Isomap), and Multidimensional Scaling (MDS) - on the performance of four popular clustering algorithms - k-means, Agglomerative Hierarchical Clustering (AHC), Gaussian Mixture Models (GMM), and Ordering Points to Identify the Clustering Structure (OPTICS). We evaluate clustering quality using the Adjusted Rand Index (ARI), comparing results without and with dimensionality reduction at different reduction levels recommended in the literature (i.e., k-1, where k is the number of clusters, and 25% and 50% of the original number of dimensions). Our findings underscore the importance of a careful selection of the dimensionality reduction technique and the dimensionality reduction level that should be tailored to intrinsic data geometry and clustering algorithm(s) under consideration.

Principal Component Analysis (PCA) is a fundamental tool for representation learning, but its global linear formulation fails to capture the structure of data supported on curved manifolds. In contrast, manifold learning methods model nonlinearity but often sacrifice the spectral structure and stability of PCA. We propose \emph{Geodesic Tangent Space Aggregation PCA (GTSA-PCA)}, a geometric extension of PCA that integrates curvature awareness and geodesic consistency within a unified spectral framework. Our approach replaces the global covariance operator with curvature-weighted local covariance operators defined over a $k$-nearest neighbor graph, yielding local tangent subspaces that adapt to the manifold while suppressing high-curvature distortions. We then introduce a geodesic alignment operator that combines intrinsic graph distances with subspace affinities to globally synchronize these local representations. The resulting operator admits a spectral decomposition whose leading components define a geometry-aware embedding. We further incorporate semi-supervised information to guide the alignment, improving discriminative structure with minimal supervision. Experiments on real datasets show consistent improvements over PCA, Kernel PCA, Supervised PCA and strong graph-based baselines such as UMAP, particularly in small sample size and high-curvature regimes. Our results position GTSA-PCA as a principled bridge between statistical and geometric approaches to dimensionality reduction.

The present study aims to investigate a cluster cleaning algorithm that is both computationally simple and capable of solving the PU classification when the SCAR condition is unsatisfied. A secondary objective of this study is to determine the robustness of the LassoJoint method to perturbations of the SCAR condition. In the first step of our algorithm, we obtain cleaning labels from 2-means clustering. Subsequently, we perform logistic regression on the cleaned data, assigning positive labels from the cleaning algorithm with additional true positive observations. The remaining observations are assigned the negative label. The proposed algorithm is evaluated by comparing 11 real data sets from machine learning repositories and a synthetic set. The findings obtained from this study demonstrate the efficacy of the clustering algorithm in scenarios where the SCAR condition is violated and further underscore the moderate robustness of the LassoJoint algorithm in this context.

Supervised No Free Lunch Theorems (NFLTs) are well studied, yet unsupervised NFLTs remain underexplored. For elliptical distributions, we prove that there exist two equally optimal, scientifically meaningful bump-hunting strategies that are exact opposites, with no universal winner. Specifically, peeling $k$ orthogonal dimensions from $\mathbb{R}^d$ ($d \ge k$), retaining an inter-quantile region of probability $1-α$ per peeled dimension, maximizes total variance and Frobenius norm when the $k$ smallest principal components (called pettiest components) are selected, and minimizes them when the selected dimensions are the $k$ leading principal components. These optima inspire PRIM-based bump-hunting algorithms either by minimizing variance or by minimizing volume, thereby motivating an NFLT. We test our results on the Fashion-MNIST database, showing that peeling the largest principal components captures multiplicity, while peeling the smallest principal components isolates popular styles.

Low-Rank Adaptation (LoRA) methods have emerged as crucial techniques for adapting large pre-trained models to downstream tasks under computational and memory constraints. However, they face a fundamental challenge in balancing task-specific performance gains against catastrophic forgetting of pre-trained knowledge, where existing methods provide inconsistent recommendations. This paper presents a comprehensive analysis of the performance-forgetting trade-offs inherent in low-rank adaptation using principal components of weight matrices as initialization. Our investigation reveals that fine-tuning intermediate components leads to better balance and robustness to high learning rates than first (PiSSA) and last (MiLoRA) components in existing work. Building on these findings, we provide practical guidelines for initialization of LoRA methods to balance the performance-forgetting trade-off. In a thorough empirical study on a variety of computer vision and NLP tasks we confirm that these guidelines achieve high accuracy and reduced forgetting.

A common approach to perform PCA on probability measures is to embed them into a Hilbert space where standard functional PCA techniques apply. While convergence rates for estimating the embedding of a single measure from $m$ samples are well understood, the literature has not addressed the setting involving multiple measures. In this paper, we study PCA in a double asymptotic regime where $n$ probability measures are observed, each through $m$ samples. We derive convergence rates of the form $n^{-1/2} + m^{-α}$ for the empirical covariance operator and the PCA excess risk, where $α>0$ depends on the chosen embedding. This characterizes the relationship between the number $n$ of measures and the number $m$ of samples per measure, revealing a sparse (small $m$) to dense (large $m$) transition in the convergence behavior. Moreover, we prove that the dense-regime rate is minimax optimal for the empirical covariance error. Our numerical experiments validate these theoretical rates and demonstrate that appropriate subsampling preserves PCA accuracy while reducing computational cost.

Multivariate data often exhibit complex dependencies that violate the assumption of isotropic residual noise. For such cases, we introduce Graph Regularized PCA (GR-PCA). It is a graph-based regularization of PCA that incorporates the dependency structure of the data features by learning a sparse precision graph and biasing loadings toward the low-frequency Fourier modes of the corresponding graph Laplacian. Consequently, high-frequency signals are suppressed, while graph-coherent low-frequency ones are preserved, yielding interpretable principal components aligned with conditional relationships. We evaluate GR-PCA on synthetic data spanning diverse graph topologies, signal-to-noise ratios, and sparsity levels. Compared to mainstream alternatives, it concentrates variance on the intended support, produces loadings with lower graph-Laplacian energy, and remains competitive in out-of-sample reconstruction. When high-frequency signals are present, the graph Laplacian penalty prevents overfitting, reducing the reconstruction accuracy but improving structural fidelity. The advantage over PCA is most pronounced when high-frequency signals are graph-correlated, whereas PCA remains competitive when such signals are nearly rotationally invariant. The procedure is simple to implement, modular with respect to the precision estimator, and scalable, providing a practical route to structure-aware dimensionality reduction that improves structural fidelity without sacrificing predictive performance.

PCA can be used for rotation invariant features, describing a shape with its $p_{ab}=E[(x_i-E[x_a])(x_b-E[x_b])]$ covariance matrix approximating shape by ellipsoid, allowing for rotation invariants like its traces of powers. However, real shapes are usually much more complicated, hence there is proposed its extension to e.g. $p_{abc}=E[(x_a-E[x_a])(x_b-E[x_b])(x_c-E[x_c])]$ order-3 or higher tensors describing central moments, or polynomial times Gaussian allowing decodable shape descriptors of arbitrarily high accuracy, and their analogous rotation invariants. Its practical applications could be rotation-invariant features to include shape modulo rotation e.g. for molecular shape descriptors, or for up to rotation object recognition in 2D images/3D scans maybe also for 3D scene understanding, or shape similarity metric allowing inexpensive comparison of objects modulo rotation avoiding costly optimization over rotations.

Estimating the tangent spaces of a data manifold is a fundamental problem in geometric data analysis. The standard approach, Local Principal Component Analysis (LPCA), struggles in high-noise setting due to a critical trade-off in choosing the neighborhood size. Selecting an optimal size requires prior knowledge of the geometric and noise characteristics of the data that are often unavailable. In this paper, we propose a spectral method, Laplacian Eigenvector Gradient Orthogonalization (LEGO), that utilizes the global structure of the data to guide local tangent space estimation. Instead of relying solely on local neighborhoods, LEGO estimates the tangent space at each data point by orthogonalizing the gradients of low-frequency eigenvectors of the graph Laplacian. We provide two theoretical justifications of our method. First, a differential geometric analysis on the tubular neighborhood of a manifold shows that gradients of the low-frequency Neumann eigenfunctions of the tube align closely with the manifold's tangent bundle, while an eigenfunction with high gradient in directions orthogonal to the manifold lie deeper in the spectrum. Second, a random matrix theoretic analysis also demonstrates that low-frequency eigenvectors are robust to sub-Gaussian noise. These results allow us to derive the asymptotic scaling and stability of the estimated eigenvector gradients. Numerical experiments demonstrate that LEGO yields tangent space estimates that are significantly more robust to noise than those from LPCA, resulting in marked improvements in downstream tasks such as manifold learning, boundary detection, and local intrinsic dimension estimation.

Approximate Nearest-Neighbor Search (ANNS) pipelines for high-dimensional neural embeddings spend the bulk of their query time in candidate verification, making it the primary bottleneck in the search process. In this paper, we present PANORAMA, a state-of-the-art refinement technique that accelerates verification by exploiting the inherent spectral decay of these embeddings. Using PCA to compact signal energy, PANORAMA evaluates candidate distances incrementally, computing at each step a strict lower bound on the full-vector distance and dynamically pruning candidates the moment this bound exceeds the running k-th nearest neighbor distance. While PCA's concentration of variance facilitates pruning, it breaks the uniform-variance assumption required by Product Quantization (PQ); we resolve this with a variance-shaping step that redistributes energy across subvectors, rendering accretive refinement compatible with quantized indexes. Optimized for modern memory hierarchies via vectorized bulk-pruning and cache-conscious data layouts, PANORAMA has been upstreamed into the FAISS library across major index families (IVFPQ/Flat, HNSW, and Refine). PANORAMA achieves higher QPS at any target recall with a cost that provably scales inversely with dataset spectral decay, delivering end-to-end speedups of up to 28.9x and outperforming probabilistic methods across all recall bands.

Out-of-Distribution (OoD) detection is vital for the reliability of deep neural networks, the key of which lies in effectively characterizing the disparities between OoD and In-Distribution (InD) data. In this work, such disparities are exploited through a fresh perspective of non-linear feature subspace. That is, a discriminative non-linear subspace is learned from InD features to capture representative patterns of InD, while informative patterns of OoD features cannot be well captured in such a subspace due to their different distribution. Grounded on this perspective, we exploit the deviations of InD and OoD features in such a non-linear subspace for effective OoD detection. To be specific, we leverage the framework of Kernel Principal Component Analysis (KPCA) to attain the discriminative non-linear subspace and deploy the reconstruction error on such subspace to distinguish InD and OoD data. Two challenges emerge: (i) the learning of an effective non-linear subspace, i.e., the selection of kernel function in KPCA, and (ii) the computation of the kernel matrix with large-scale InD data. For the former, we reveal two vital non-linear patterns that closely relate to the InD-OoD disparity, leading to the establishment of a Cosine-Gaussian kernel for constructing the subspace. For the latter, we introduce two techniques to approximate the Cosine-Gaussian kernel with significantly cheap computations. In particular, our approximation is further tailored by incorporating the InD data confidence, which is demonstrated to promote the learning of discriminative subspaces for OoD data. Our study presents new insights into the non-linear feature subspace for OoD detection and contributes practical explorations on the associated kernel design and efficient computations, yielding a KPCA detection method with distinctively improved efficacy and efficiency.

The problem of using proximity (similarity or dissimilarity) data for the purpose of "adding a point to a vector diagram" was first studied by J.C. Gower in 1968. Since then, a number of methods -- mostly kernel methods -- have been proposed for solving what has come to be called the problem of out-of-sample embedding. We survey the various kernel methods that we have encountered and show that each can be derived from one or the other of two competing strategies: projection or restricted reconstruction. Projection can be analogized to a well-known formula for adding a point to a principal component analysis. Restricted reconstruction poses a different challenge: how to best approximate redoing the entire multivariate analysis while holding fixed the vector diagram that was previously obtained. This strategy results in a nonlinear optimization problem that can be simplified to a unidimensional search. Various circumstances may warrant either projection or restricted reconstruction.

Brownian motion is a compact mathematical language for continuous-time uncertainty in biostatistics. This tutorial develops the process from construction and path properties to tools that recur in applied biomedical work: the Markov and strong Markov properties, the Karhunen-Loeve expansion, functional principal component analysis (Functional PCA), reflection principles, local time, stochastic differential equations (SDEs), Brownian bridges, and empirical-process limits. The applications emphasize longitudinal biomarkers, degradation modelling, first-passage endpoints, dynamic frailty, group-sequential monitoring, calibration diagnostics, recurrent-event processes, electronic health records, and wearable streams. A short cross-domain section uses literary and historical archives to make Brownian-bridge thinking concrete without shifting the paper away from biostatistics, and includes a reproducible chapter-level experiment on Frankenstein. The Black-Merton-Scholes model is included as a solved SDE template, not as a finance application in its own right. The aim is to connect rigorous probability with modelling decisions faced by biostatisticians when biological processes evolve between noisy observation times.

Accurate reconstruction of long-duration neural recordings is challenging because local field potentials (LFPs) are high-resolution, multichannel, transient, and variable across subjects. We present PCA-DMD, a scalable operator-theoretic framework that segments LFP recordings into overlapping windows, projects them into a compact PCA space, learns linear Koopman evolution in the latent space, and reconstructs continuous signals through inverse projection and overlap-add aggregation. On 200,000-sample hippocampal recordings, PCA-DMD outperformed Classical DMD, SpDMD, MrDMD, and HODMD, achieving KLD=0.0761 and HD=0.0847. In all-pair cross-subject zero-shot generalization at 300,000 samples, correlations were 0.9504-0.9800, with HD=0.0010-0.0072 and KLD=0.0005-0.0022, without target-subject fine-tuning. The prediction showed close one-step agreement on temporally held-out LFP segments across the unseen interval and multiple channels. Scalability analysis from 400,000 to 900,000 samples showed stable zero-shot reconstruction, with mean correlation remaining about 0.965-0.968 while computational cost increased predictably. External validation on an independent 93-channel Allen Neuropixels recording yielded mean and median channel-wise correlations of 0.7427 and 0.7990, respectively. Koopman spectral and mode analyses revealed dominant eigenvalues concentrated near the unit circle. PCA-DMD therefore provides an interpretable, generalizable, and computationally scalable framework for reconstructing high-dimensional neural dynamics.