Principal Component Analysis

Streaming principal component analysis (PCA) seeks to recover a leading spectral subspace in a single pass over a data stream. We give a new analysis of the ubiquitous Oja's algorithm [Oja82] for the most general, gap-free variant of this problem, where no eigengap assumptions are made on the underlying mean matrix, complemented by a nearly-matching lower bound. Prior works achieving near-optimal rates for streaming PCA either required gap assumptions [JJK+16, HNWW21], or were limited to rank-one updates [AZL17, Lia23]. Our proof only uses a second moment bound on the individual stochastic updates, bypassing the almost sure bounds needed by prior near-optimal analyses, and the analogous offline matrix Bernstein bound. We also extend our result to a Rayleigh quotient notion of approximate PCA, addressing an open question of [JJK+16]. As our main application, we give gap-free differentially private PCA guarantees for sub-Gaussian data, settling Conjecture 1.1 of [Bro26] up to logarithmic factors.

Principal component analysis (PCA) is a fundamental tool to reduce the dimensionality of the data in many applications. PCA finds a few signal directions that contain most of the variability of the data by computing the eigenvectors of the sample covariance matrix. In this work, we focus on the spiked covariance model, in which the data vectors are defined by a few orthogonal signals plus an isotropic Gaussian noise, and our goal is to estimate one or more of the leading signals. Our main theoretical finding is that the subspace spanned by several leading eigenvectors of the sample covariance matrix contains significant information about the desired signals long before the individual eigenvectors converge to the population principal components. To prove this, we derive a posteriori bounds for the angle between the subspace spanned by the desired population signals and the subspace obtained from the sample using perturbation theory for singular vectors. This leads to a new algorithm, SuperPCA (SUbsPace subsamplER PCA), which capitalizes on an approximate eigenspace of the sample covariance matrix to find the leading signals far more efficiently and accurately than classical PCA in the high-dimensional, multi-signal setting. SuperPCA exploits only a small number of subsampled coordinates of the data, which can lead to tremendous savings in data acquisition cost, especially when the signals are approximately sparse. For the same number of measurements, SuperPCA can offer a factor $10$ improvement in accuracy compared to the classical PCA method.

In high-dimensional tabular learning, feature screening provides a lightweight, model-agnostic way to remove irrelevant features before model fitting. However, scoring raw values directly can miss nonlinear or distributional structure. We introduce PACE (Plug-and-Play Contextual Embedding), which inserts a frozen tabular foundation model (TFM) column encoder before an existing feature-scoring rule, expanding each feature into a higher-dimensional contextual representation. Across controlled studies, PACE improves raw-space screening of complex nonlinear dependence with only modest additional encoding cost. These gains translate to downstream prediction on TALENT datasets: PACE-DC improves binary AUC by 0.077 and multiclass macro-AUC by 0.064, with a median normalized RMSE improvement of 0.063 across ten learners. Matched random-weight and random-feature controls show that PACE gains from pretrained structure beyond generic dimensional expansion. PACE further achieves favorable performance--time trade-offs against task-fitted selectors and attribution-based methods, positioning pretrained column geometry as a reusable upstream primitive for tabular learning.

Accurate optimization of a supervised spectral objective need not produce an accurate population subspace or a better predictive representation. We investigate these distinctions for Online Kernel Supervised Principal Component Analysis (OKSPCA), which combines a centered cross-moment in finite random-feature coordinates with an Adam-style orthonormal basis update for an established objective. Fixed-map consistency, concentration and perturbation results describe the estimator and its exact subspace; same-target comparisons then assess the practical iterate separately. Across six predictive benchmarks, performance depends on the declared pipeline: replacing the tracker with the exact empirical target leaves the two regression deficits largely unchanged. Direct classification-rank models capture nearly all terminal objective energy on average, but a saved intermediate state exhibits substantial geometric deviation; a controlled sample-size study further separates empirical accuracy from population recovery. In distinct numerical-service workloads, exact on-request computation is faster in the tested classification settings, whereas Adam saves time relative to the tested full thin-SVD service for some dense wider-regression requests, alongside persistent geometric error. These diagnostics limit explanations based solely on terminal optimization accuracy and distinguish numerical cost from quality, rank coverage and freshness; they establish neither practical-tracker convergence nor predictive or deployment benefits from basis availability.

Long-term multivariate time series plays a significant role in many application areas such as power systems, trading, etc. However, their accurate prediction is quite difficult for conventional forecasting methods as they often exhibit high dimensionality and complex relationships. Recent works show that transformer-based approaches are quite effective for long-term forecasting thanks to their attention mechanism. However, in the presence of complex high-dimensional inputs, they show evidence of oversmoothing, limited capacity, and opacity. To this end, this paper introduces SETTer, a transformer-based model that addresses these challenges by incorporating novel techniques for decoupled self-attention and hybrid masking. The proposed techniques enable SETTer to effectively capture the dominant short- and long-term patterns across the temporal and channel dimensions. In addition, we enrich the model layers with simple explainable structures that indicate the discriminative pattern of SETTer. We show that with a single-layer transformer architecture, SETTer can effectively model long-term dependencies in the presence of varying data complexities. Extensive experiments on real-word benchmark datasets for long-term multivariate time series forecasting demonstrate that SETTer outperforms state-of-the-art models in 88% of the scenarios.

We propose a framework for restricted inference on random dot product graphs whose latent positions lie on an unknown low-dimensional support manifold. For general decision problems, we propose semisupervised decision rules that use auxiliary data to learn the support manifold. Specifically, our rules use the Isomap manifold learning procedure to construct a low-dimensional Euclidean representation of the observed graph, in which space an isometrically invariant function maps configurations of points to actions. We study the behavior of the proposed rules as the quantity of auxiliary data sampled from the unknown support manifold increases. We show that, as the auxiliary sample size increases, the risk of the semisupervised rule converges to the risk of an oracle rule that relies on the maximal amount of low-dimensional Euclidean structure that can be extracted from the support manifold. Examples, applications, and simulation studies are deferred to a sequel.

The standard geometric derivation of neural scaling exponents takes the intrinsic dimension of a data manifold as its input. On modular addition in $\mathbb{Z}_p$ that derivation has no input. The exact algebraic solution is an orbit of $\mathbb{Z}_p$ acting by isometries. Transitivity alone makes the ratio statistic underlying the standard dimension estimator a point mass, so the estimator is undefined, and here the two nearest neighbour distances coincide exactly. Breaking the symmetry at scale $ε$ returns a number, but one that tracks $1/ε$ with no scale free plateau. We show that the failure is general, since on any finite orbit of a group acting by isometries the estimator reports the resolution at which the set is probed rather than a dimension. What replaces the power law is exponential in hidden width, $L(h)=L_\infty+A\exp(-c\,h^α)$, with $R^2$ between 0.982 and 0.995 against 0.857 to 0.906 for a power law admitting the same floor and fitted under the same protocol. Where the data supply is sufficient the rate belongs to the regulariser rather than to the group, since weight decay moves $c$ by a factor of 47 while group order moves it by 1.10, a residual below seed to seed resolution, for every fixed $α$ between 0.75 and 2. The critical width falls with group order rather than rising, against capacity counting that assigns a fixed number of neurons to each irreducible representation.

The exploration of high-dimensional spaces remains a challenging problem, in particular in the presence of narrow passages and small clearances. We propose novel sampling-based path-planning methods for high-dimensional spaces combining Rapidly-exploring Random Trees (RRT) and Hit-and-Run (HAR) random walks by decoupling the point being extended from the direction of extension. We also show that RRT and HAR appear as special cases of a generic algorithm coupling the biases used for the point and direction extension, respectively. We further study a sparse-move strategy in which only a fraction p_r of the robots is moved at each step, helping both RRT and the proposed HAR algorithms handle cluttered instances. Tests are presented for two families of models: classical piano mover problems in 3D, and complex molecular systems involving tens of rigid domains moving relatively to one another -- the latter viewed as independent robots exploring the motion space SE(3)N . Within seconds on a standard laptop, our algorithms solve instances with up to 64 robots and 384 degrees of freedom. We conclude by suggesting one of our methods, HARF, as the method of choice for complex multi-robot planning problems, being up to two orders of magnitude faster than the classical RRT moving all robots at each step--when it succeeds at all, and still up to 2.4 fold faster on most instances when both use their best p_r.

Machine learning models process large amounts of data, and Principal Component Analysis (PCA) is a widely used technique to reduce the dimensionality of the data and extract useful features. In practice, datasets often change over time (data drift) and/or arrive one sample at a time (streaming data), making it infeasible to process the entire dataset at once in batch mode. Real-world data also often contains nonlinear patterns, which traditional PCA cannot extract. Kernel PCA addresses this by implicitly mapping samples into a Reproducing Kernel Hilbert Space (RKHS). Raw data also often contains outliers, which can have an outsized effect on the estimated subspace unless the algorithm is made robust. However, existing online robust kernel PCA algorithms are designed to converge to a subspace that is assumed to be fixed, and gradient-descent-based updates lose their effectiveness at tracking further changes once this initial alignment is achieved. This paper introduces a rotation-based update mechanism, which updates the subspace estimate by rotating it toward each new incoming feature vector in Reproducing Kernel Hilbert Space, rather than relying on gradient descent alone. We present two complementary rotation strategies, and show that the extent of rotation can be moderated by a robust influence function to mitigate the effect of outliers. Through experiments on synthetic streaming data with a known ground-truth subspace, we show that per-sample rotations converge faster than gradient descent alone, demonstrating an effective mechanism for dynamically tracking a nonlinear subspace in streaming data.

What did an ancestral bird species sound like? Existing ancestral state reconstruction methods can infer low-dimensional traits such as morphological characters at internal nodes of a phylogenetic tree, but no one has tried to produce rich perceptual signals such as audio. Some of the challenges include inferred representations that are either too low-dimensional to decode or lie in non-generative feature spaces, so no method to date can produce ancestral audio. We introduce the first framework that generates plausible ancestral vocalizations. Our pipeline encodes bird recordings into a VAE latent space, learns a low-dimensional trait projection aligned with phylogenetic distances, performs ancestral inference in this trait space, and recovers decodable latents through an anchored inverse lift before emitting novel waveforms for each ancestral node. Because the entire pipeline stays within a decodable latent space, every internal node receives a genuinely new audio output representing plausible intermediate ancestral sounds unavailable to retrieval-based alternatives. Experiments on two phylogenetically distant bird clades, 21-species Tyrannidae and 19-species Paridae, show that our method is the only approach that simultaneously achieves genuine generation, phylogenetic consistency, and naturalistic audio quality across both datasets.

The Gaussian kernel is a widely used similarity measure underlying kernel methods such as kernel PCA and spectral clustering, but computing Gaussian kernel distances for many pairs of points can be expensive. Using Random Fourier Features (RFF), Chen and Phillips [ALT 2017] showed that for points in a $d$-dimensional Euclidean ball in ${\mathbb R}^N$, $t=Ω((d/\varepsilon^2)\log(dR/\varepsilon))$ features suffice to preserve all pairwise Gaussian kernel distances within a $(1\pm\varepsilon)$ factor with high probability. We establish a uniform relative-error embedding theorem for the more general setting of an arbitrary positive-reach submanifold $\mathcal M\subset{\mathbb R}^N$ of intrinsic dimension $d$. We show that $t=O((d/\varepsilon^2)\log(\operatorname{vol}(\mathcal M)^2N^{2d}/(\operatorname{vol}(B_1^d(0))^2\operatorname{rch}(\mathcal M)^{2d}\varepsilon^{2d+1}δ)))$, or approximately $O((d^2/\varepsilon^2)(\log N+\log(1/(\varepsilonδ))))$, RFFs suffice, with probability $1-δ$, to preserve the Gaussian kernel distance between every pair of manifold points up to relative error $\varepsilon$. Thus the bound depends only logarithmically on the ambient dimension and on manifold parameters such as volume and reach, while retaining the $1/\varepsilon^2$ Euclidean rate. We also prove a topological consequence: under the same RFF embedding, persistent homology is preserved in the sense that weighted Cech and Rips filtrations built from Gaussian kernel power distance are $(1\pm\varepsilon_\star)$-interleaved, where $\varepsilon_\star$ accounts for both distance distortion and kernel-weight approximation.

This paper introduces a novel framework for Regularized Multivariate Functional Principal Component Analysis (ReMFPCA) via Functional Singular Value Decomposition (SVD). The proposed method extends existing MFPCA approaches by incorporating a generalized functional SVD within a Hilbert space framework, enabling simultaneous regularization of both functional principal components (PCs) and their associated PC scores. A key innovation of this framework is the inclusion of a sparsity penalty on the PC scores, which enhances interpretability by filtering out irrelevant subject-specific variations. This dual-penalization strategy represents a significant advancement beyond existing covariance-based eigen decomposition methods, which penalize only the functional PCs. Two power algorithm implementations, sequential and joint, are proposed, together with a cross-validation approach based on iterative regression for optimal smoothing parameter selection. Comprehensive simulation studies and real data applications demonstrate that the proposed framework substantially improves the extraction of informative and interpretable components, offering methodological and practical benefits for analyzing multivariate functional data across diverse domains.

Reconfigurable robots can change their contact geometry when a fixed body cannot negotiate an obstacle. A variable-geometry truss (VGT) distributes this shape change through a load-bearing structure, but coupling it to a mobile base creates a high-dimensional coordination problem. GeoTrussRover combines an electrically actuated VGT, a wheeled base, and contact-semantic morphology planning and control. We solve one source traversal and extract four contact-semantic primitives that describe coordination among 21 members. Physics-constrained projection adapts them to unseen step heights with the same contact topology. When every phase remains feasible, adaptation does not recompute the complete motion. If one phase violates the new physical constraints, only that phase is recomputed. A full-space QP then tracks the adapted motion and corrects member and wheel errors. For transfer from 0.10m to 0.075m, the method reduces objective-function evaluations by 63.7% relative to full recomputation. Contact-phase feasibility analysis covers step heights from 0.10 to 0.46m, or 1.08 to 4.97 wheel radii, with the upper value near the theoretical feasible boundary. The electric prototype traverses 2.11 wheel radii. The resulting low-dimensional representation stores task coordination in a hyper-redundant, load-bearing morphology and reuses it during locomotion.

This feature article provides an overview of the theoretical foundations for coVariance neural networks (VNNs), i.e., graph neural networks (GNNs) operating on covariance matrices as graphs. Covariance matrices are ubiquitous across domains, and hence, the deployment of GNNs often leverages graphs of pairwise statistical dependencies. Existing theoretical contributions on GNNs consider abstract graph representations and cannot accommodate the data-driven nuances associated with covariance matrices. This tutorial brings into focus various novel theoretical insights via mathematical analyses of VNNs that have broad signal processing implications, including: (i) a conceptual equivalence between VNNs and principal component analysis (PCA)-based information processing; (ii) refined stability bounds on predictive outcomes in the presence of finite sample-induced covariance matrix perturbations; and (iii) refined characterization of transferability of VNNs across multiscale datasets. The theoretical insights discussed herein provide the underlying principles and justification towards adopting VNNs over workhorse PCA-based learning pipelines, in applications where covariance matrices are useful descriptors of data structure. We also convey how impact of these foundational advances permeates to \textit{principled} designs and applications of learning methods across broad domains where covariance matrices emerge. Notably, we elucidate the conceptual insights facilitated by VNNs to the specific task of characterizing brain age gap for neurodegenerative conditions using neuroimaging datasets, a timely problem in computational neuroscience. Broader impacts to other application domains are discussed as well.

This paper addresses the issue of supervised classification in the context of hyperspectral satellite images. It deals with two fundamental aspects: dimensionality reduction of data and the selection of appropriate supervised classification techniques. Firstly, we delve into dimensionality reduction, a critical step in simplifying the management of hyperspectral data. The reduction aims to decrease complexity in terms of memory and computing time. We examine two commonly used methods: Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA). Subsequently, we explore the selection of the most suitable supervised classification algorithms for hyperspectral images. We compare the performance of three methods: K-Nearest Neighbors (KNN), Support Vector Machines (SVM), and Random Forest (RF) using real hyperspectral data. The results highlight that the combination of PCA and RF yields the highest overall accuracy and Kappa coefficient.

The small-sample-size (SSS) problem remains a fundamental challenge in machine learning when labeled data are scarce due to cost, accessibility, or ethical constraints. While numerous approaches have been proposed, existing methods often struggle to maintain stable and discriminative representations under high-dimensional and limited-data conditions. Kernelized Linear Principal Component Discriminant Analysis (KLPCDA), a recently proposed modular framework, integrates variance preservation, inter-class separability, and intra-class compactness within a unified kernel space. Although its formulation has shown promising initial results, a systematic understanding of how its components interact across diverse SSS scenarios remains lacking. In this paper, we present a systematic cross-domain study of KLPCDA to characterize the interaction mechanisms among its core objectives. We analyze the behavior of its seven variants across multiple real-world SSS tasks, including hyperspectral image classification, mechanical fault diagnosis, medical diagnosis, and face recognition. Through extensive experiments and ablation studies, we investigate how different objective combinations influence performance under varying conditions such as noise, class imbalance, and high dimensionality. Our analysis reveals consistent patterns in the interaction of the three core objectives variance, between-class, and within-class terms, providing a unified and interpretable understanding of their roles in stabilizing representations and enhancing discrimination in SSS settings. Based on these findings, we further derive practical guidelines for selecting appropriate KLPCDA variants under different data characteristics. Experimental results demonstrate that KLPCDA achieves strong and robust performance across domains, while maintaining low computational complexity suitable for resource-constrained environments.

Balanced k-shot sampling draws exactly k labeled examples per class. We show that it induces an exact, provable degeneracy in a family of small-sample discriminant estimators. Under balanced sampling, the within-class scatter operator of Kernelized Linear Principal Component Discriminant Analysis (KLPCDA) is not merely rank-deficient but exactly a scaled orthogonal projector. We derive the consequences in closed form: two of KLPCDA's seven variants have every signal eigenvalue exactly equal, so their eigenvector selection criterion is provably indifferent rather than ill-conditioned, and a third has a provably void objective. This follows from the estimators' construction, not any dataset; we confirm it on frozen sentence embeddings and, separately, on residual-stream activations from a decoder-only generative model. An in-formula tie-break repairs the two repairable variants, with recovery gated by class count: the residual subspace constraint costs 5x more on few-class than many-class datasets (p=0.000001). We then evaluate the repaired framework on few-shot text classification on frozen LLM embeddings (n much smaller than d, up to 4096), across four datasets, three embedding sizes, and three trained baselines (SetFit, LoRA, in-context learning). A properly cross-validated logistic-regression probe still beats every KLPCDA variant on three of four datasets, at every embedding size; guidance carried from pixel, vibration-signal, and gene-expression data does not directly generalize to this feature space. Three independent geometric separability metrics fail to explain why one high-dimensional decoder-based embedding model underperforms smaller bidirectional encoders, ruling out anisotropy; the gap is substantially an estimation-efficiency effect, not a permanent ceiling, closing by more than 80% when the support set grows from k<=10 to k=30-50 (p=0.00195, both many-class datasets).

Many scientific graphs attach several variables to each node, so a single scalar edge weight cannot describe direction-dependent interactions. We model each edge by a symmetric positive-definite (SPD) matrix and infer a posterior over matrix-valued graph geometries, which we call the Bayesian matrix-valued graph (BMVG). We ask how these interactions reconfigure across contexts: how large the change is and which multivariate directions strengthen or weaken. The geodesic distance induced by the affine-invariant Riemannian metric (AIRM) quantifies deformation magnitude and generalized eigenvalues resolve its signed directions.Against fused graphical lasso, Bayesian multiple-GGM, and common principal components, BMVG is competitive on global precision recovery while retaining identifiable matrix-valued edge structure and accurately recovering edge-level deformation directions. In controlled known-truth experiments, it resolves structural change with increasing sample size, including orientation changes that leave ordinary eigenvalues unchanged. In one year of Bay Area weather data, the geometry of 12-hour change reconfigures spatial coupling about as much as whole seasons differ. In TCGA-BRCA, estrogen-receptor (ER)-associated reconfiguration concentrates on specific gene-module pairs and persists under graph-scaffold sparsification and removal of subgroup mean differences. These results establish posterior matrix-valued edge geometry as a unified framework for quantifying and interpreting context-dependent multivariate reconfiguration.

Time series data is very common in many real-world applications and in numerous domains, with increasing interest for automated information extraction using machine learning. One of these subfields is time series clustering, which consists in identifying clusters among a set of time series in an unsupervised fashion. Most time series clustering algorithms suffer from the same balancing act: they trade clustering performance for faster runtimes or vice versa. We present a novel time series clustering algorithm that we call CLUES-WEASEL, which stands for CLustering with the UnsupervisEd Second version of Word ExtrAction for time SEries cLassification. CLUES-WEASEL extracts features using the unsupervised version of the transformation step of WEASEL 2.0, which is a time series classification algorithm, then reduces these features using principal component analysis, and finally performs clustering with the $k$-means algorithm using these reduced extracted features. Through extensive experiments, we prove that CLUES-WEASEL is significantly better than any other existing time series clustering algorithm while being (much) faster than any state-of-the-art one. We also show that the architecture of CLUES-WEASEL can work well with other time series feature extraction algorithms. Our findings highlight the relevance of CLUES-WEASEL for time series clustering.

Long-term electricity-consumption profiles exhibit several simultaneous periodic structures, including daily, weekly, and annual cycles. This work introduces Calendar-Structured Sparse Principal Component Analysis (Calendar-SPCA), a structured representation-learning method that incorporates this known multi-periodic geometry directly into a low-dimensional factorization. The method represents the feature domain as the Cartesian product of cyclic calendar axes and combines an L1 loading penalty with graph total variation, producing sparse, locally coherent, and directly interpretable latent factors. In this study, Calendar-SPCA is applied to the interpretable analysis of long-term electricity-consumption profiles and evaluated on two independent public smart-meter datasets, GoiEner and Low Carbon London, with different population sizes and temporal resolutions. A factorial experiment characterizes the effects of sparsity and calendar coherence and examines robustness across sample size, latent dimensionality, and repeated fits. At rank 15, Calendar-SPCA retains 96.92% and 82.90% of the explained variance of rank-matched principal component analysis (PCA) in GoiEner and Low Carbon London, respectively, with mean loading sparsities of 61.95% and 81.50%. Comparisons with classical sparse PCA and sparse PCA with total variation (SPCA-TV) show that Calendar-SPCA organizes latent factors into interpretable structures over the daily, weekly, and annual calendar axes while preserving substantial low-rank information.

Existing coded-computing designs do not explicitly exploit the intrinsic structure of the input data. In communication systems, statistical structure and redundancy are often removed through source coding (or compression) before channel coding is applied. This principle, however, does not transfer directly to coded computation. In many computational tasks, particularly in machine learning, the structure of the data is precisely what the computation seeks to exploit to infer outputs or learn meaningful patterns. Consequently, coded-computing schemes should preserve and leverage this structure in their code design, rather than ignoring or eliminating it through source coding. This observation motivates a different perspective on code construction. In many channel-coding schemes, such as Reed-Solomon codes, coded symbols are generated by evaluating a low-dimensional algebraic representation at selected points. In contrast, many high-dimensional datasets naturally concentrate near low-dimensional manifolds. In this paper, we exploit this intrinsic geometry by designing coded samples that follow the natural manifold of the data, rather than imposing an artificial low-dimensional structure unrelated to the data distribution. Inspired by graph-based manifold learning, we propose a manifold-aware encoding strategy for general coded computing (GCC). Experiments on neural network inference and high-dimensional polynomial evaluation demonstrate that the proposed strategy consistently and significantly reduces the mean squared recovery error under straggling compared with standard GCC.

This paper studies the problem of computing the stochastic probability (shortest code length) of the encoded vectors containing cluster structure using Normalized Maximum Likelihood (NML) model. This is of great theoretical and practical importance in data clustering based on Minimum Description Length (MDL) principle, such as for estimating the best number of clusters and best cluster structure for the data. Straightforward computation of the shortest code length of the vector containing cluster structure based on the NML model requires polynomial time with respect to the size of the vector and number of clusters. We show that this is a tractable problem by introducing a recursion formula for the efficient computation of normalizing constant from the NML model. The time complexity of the new formula is linear opposed to previous polynomial time with respect to the size of the vector and number of clusters.

We present TSExplorer, a cross-platform tool for interactive annotation and exploration of time-series data. The tool enables users to inspect high-dimensional datasets through multiple complementary 2D visualizations derived from high-dimensional feature representations. TSExplorer is designed as a general-purpose research tool supporting a wide range of workflows, including exploratory data analysis, annotation of unlabeled or partially-labeled datasets, comparison of feature representations, and post-hoc inspection and refinement of existing labels with interactive visual feedback.

Autoencoders are widely used for nonlinear dimensionality reduction and manifold learning. While most common implementations rely on both nonlinear encoders and decoders, we investigate the specific role of the encoder and the extent to which it can be constrained to be linear without reducing accuracy. We conduct a comparative study on four autoencoder architectures: standard fully nonlinear autoencoders (AE), linear-encoder autoencoders (Lenc-AE), linear-decoder autoencoders (Ldec-AE), and fully linear autoencoders (LAE), evaluated on synthetic manifolds, computational mechanics data sets, and real-world image data sets including MNIST. We demonstrate that imposing a linear encoder preserves most of the representational capacity of the autoencoder, provided the decoder remains nonlinear. In particular, Lenc-AE consistently outperforms both Ldec-AE and LAE, and achieves reconstruction quality comparable to fully nonlinear AE, while offering advantages in terms of parsimony and interpretability of the latent representation. These results suggest that the nonlinear decoder is the critical component for manifold learning, rather than the encoder. A geometric interpretation of this finding is developed, which identifies the precise conditions under which a linear encoder is sufficient, and the specific manifold configurations that expose its limitations.

A token-embedding table holds a hub of short rows near its origin, and we show that this cluster biases what nearest-neighbor intrinsic-dimension (ID) estimators report. Because of the concentration of measure, a token is closer to the central cluster than to any other token, so its first two neighbors are both hub rows at nearly the same distance. As a result, the ID estimators such as TwoNN return a dimension far above the real ID. Measured one token at a time, dimension is a heavy-tailed distribution. Measured on the full vocabulary, it grows with the model's parameter count. However, when we remove the hub, the heavy tail disappears and the measured dimension collapses to a narrow range for eleven models, from GPT-2 to models such as K3 and GLM-4.7. The hub acts as a switch: a few hundred rows are enough to fully inflate the estimate. We reproduced an experiment stating that the intrinsic dimension (ID) of Pythia's token-embedding table grows with the parameter count, from $27$ to $122$ between 160M and 12B parameters. We show that this result disappears when the hub is removed: the table then reads $10$ to $17$ at every size. The hub contains a subset of the population that under-trained-token detectors flag, but on Pythia the hub that we detected and removed as a whole was updated during training: what seem to characterize these rows is simply their length, not an absence of updates. Finally, we show that normalizing the rows instead of removing them gives the same lower reading.

How much matrix rank is required to preserve every bounded value output of normalized softmax attention? We study the unrestricted maximum-row-$\ell_1$ approximation rank $r_\varepsilon(A)$, exactly the least rank achieving uniform error over all bounded vector-valued values. Row softmax exposes the intrinsic interaction $C=P_m(\log A)P_N$, whereas invertible $Q/K$ gauges leave $A$ fixed while changing the Euclidean geometry of a chosen query/key factorization. We replace that coordinate-dependent description by a projective residual $q(C-T)$ and an attained factor-radius size $κ(T)$. For every rank-$r$ retained interaction with $τ(T)<\varepsilon$, we prove $r_\varepsilon(A)\le \min\left\{ N,\; C_r\left( 1+\frac{κ(T)} {(\varepsilon-τ(T))^2} \right)^{r/2} \right\},$ with the same unknown dimension constant as the underlying weighted Gibbs-row cover. The profile is gauge invariant, termwise no worse than native retained-subspace bounds at the same declared dimension, and has a worst-case sharp $r/2$ size exponent at fixed $r$ and $\varepsilon$. We then measure $r_\varepsilon(A)$ directly on learned attention using 9,978 certified brackets across BERT, GPT-2, Qwen2.5, and two ViT checkpoints; where certificates do not close, the optimum remains interval-valued. A pre-specified 2,302-cell held-out study further shows that the historical native-coordinate geometry block contains coarse, mostly head-level information but no detectable incremental information beyond a strong calibrated baseline. The new intrinsic descriptor is not evaluated in that study. Together, the theory and measurements distinguish an operator-intrinsic complexity control from a stronger empirical explanation that the learned-head evidence does not support.

Understanding how neural networks learn and organize features is central to understanding their behavior. Much existing theory of feature learning has focused on the emergence of a global low-dimensional representation. We show that this picture is incomplete. In regression problems with clustered data, we demonstrate that multilayer perceptrons (MLPs) naturally develop monosemantic specialized neurons: individual neurons become strongly aligned with a specific predictive feature relevant to a particular region of the input space. Rather than learning a single global low-dimensional representation, MLPs learn a collection of local low-dimensional representations. We show that this ability to specialize gives MLPs a provable data-efficiency advantage over feature-learning methods based on a global low-dimensional representation.

Classification of edible oils in processed foods is important for food quality, fraud prevention, and regulatory compliance. This study develops a Mutually Exclusive, Collectively Exhaustive framework integrating spectral organization, interpretable classification, Physics-Informed Artificial Intelligence (PI-AI), and Frugal AI-based feature reduction. Five edible oils were analyzed in pure form and within a fried-potato-chip matrix using t-SNE, K-means clustering, Decision Trees, and Non-Negative Least Squares (NNLS)-based spectral decomposition. Unsupervised analyses showed stronger class organization and separability in pure oils, while food-matrix effects caused substantial spectral overlap. Decision Trees achieved 100% classification accuracy for pure oils using only four Raman variables from 1866 spectral features. These variables represented only 0.21% of the available spectral information while retaining perfect test-set performance. Two variables associated with lipid unsaturation (about 1650 cm-1) and hydrocarbon-chain organization (about 1127 cm-1) remained important after NNLS matrix correction. Their combined contribution increased from 50% in pure oils to about 62% and 89% in paper-subtracted and paper-plus-potato-subtracted datasets, respectively. NNLS-based PI-AI improved food-matrix classification by separating oil signatures from paper and potato contributions. Optimized post-pruned models achieved nearly 80% test accuracy using only five and four Raman variables, respectively. The four-feature representation reduced the data footprint by 99.44% without loss of accuracy. These findings demonstrate that Raman-based oil identification can use compact, physically meaningful, and interpretable spectral representations, supporting Frugal AI, Edge AI, portable sensing, and embedded food-quality monitoring.

To avoid missing important variables and their connections in networks, more and more variables are included in network analysis. Here we show that in a setting with many more parameters than observations (high-dimensional) it is possible to get a conservative (i.e., low false positive rate) estimate of the neighbourhood for each node (which connections are in the network). A neighbourhood is often estimated with a linear model, and this leads to two interesting cases: (i) If the true model is linear, then neighbourhood selection work reasonably well, and (ii) if the true model is nonlinear, then neighbourhood selection requires a penalty for the high dimensions. Here we show the impact of the ridge parameter on the mean squared error, and how this leads to low test variance and hence to neighbourhoods with large numbers of edges. We connect these insights with results from machine learning, where the so-called double descent (when more parameters are included than observations, the mean squared error goes down a second time) has put the traditional view on model selection upside down. Essentially, for adequate neighbourhood selection in models with a large number of parameters, the volume of the model space needs to be included in the penalty. Most neighbourhood selection methods (e.g., Lasso, AIC, BIC) lead to spurious edges (high false positive rate), but we prove that in the high-dimensional setting, minimum description length leads to correct neighbourhood selection or smaller (low false positive rates) in both cases when either the model is correctly or incorrectly assumed linear

In this work, we introduce DMDIntel which uses dynamic mode decomposition (DMD) to make the predictions made by LLMs in a classification task interpretable. It develops an input attribution pipeline, that first decomposes the hidden states of an LLM into prominent patterns, also known as modes, and then associates ranks to the input tokens based on the projection values on those modes. Rigorous experiments across three datasets and three model families consistently show that the ranked attribution of input tokens obtained using DMDIntel by far outperforms state-of-the-art techniques such as principal component analysis, integrated gradients and SHAP.

This article presents with DYSANOS the first generative market model for smooth SANOS option surfaces for all strikes and expiries which are free of static arbitrage. Our model is designed to generate entire paths of daily spot and option prices for years in the future. We present a robust and useful if somewhat simplistic baseline hidden state generative model in the form of an AR(1) model. We discuss model setup, data pipeline, and training and investigate numerical resence of dynamic arbitrage. We illustrate model performance on Option Metrics' IvyDB S&P Index data from 2020 to~2025 and compare it to a pure implied-vol PCA model.

Feature selection is one of the most important and fundamental tasks in data mining, tackled by a family of methods with an established set of evaluation techniques to measure the quality of a specific method. Most of the methods commonly used for the unsupervised evaluation of feature selection algorithms suffer from critical design flaws which question their unsupervised nature. In this paper, we provide a critical discussion on the established allegedly unsupervised evaluation techniques, and shed light on the reasons why they are not truly unsupervised but, at best, supervised evaluation under an unsupervised downstream task. We also propose a novel, truly unsupervised evaluation framework to measure the quality of the feature selection algorithms without any form of information about the labels. The proposed framework utilizes unsupervised Principal Component Analysis, and optimal transport to measure the quality of the feature selection methods in a truly unsupervised manner.

In this paper, we test different methods for multivariate detection of outliers in the data of transmitted heat energy in the selected substation of local District Heating System, by also considering outside ambient temperature, namely Z-score (univariate, as a benchmark), Mahalanobis distances, Principal Component Analysis (PCA), Isolation Forest and Hotelling's T-squared test. The overall research aims at uncovering irregular plant operation, with a wider objective of identifying the opportunities for reducing the consumption of gas in central heating plants as well as the CO2 emission. The proposed approach considers specific domain circumstances, such as irrelevance of zero transmit-ted energy timepoints as indication of off-grid plant. The outcomes of the different methods are discussed with domain experts. It was concluded that PCA, Isolation Forest and Hotelling method provide relevant results. Finally, we adopt the ensemble method (selection based on the agreement of all three methods on the detected outliers) as the final approach.

Determining the complexity, or Intrinsic Dimension (ID), of data is fundamental to efficient and interpretable representation learning. This is particularly challenging in multi-modal settings when trying to learn disentangled representations for shared and private information. Existing techniques leave a critical gap: they are often static, uni-modal, or in the case of contrastive methods, adapt only to the shared ID implicitly. We introduce Fidelity-Guided Rank Optimization (FiGuRO), a framework for approximating the ID of uni- and multi-modal data under constraints of model capacity and hyperparameters. FiGuRO learns the dimensions of low-rank projections using truncated singular value decomposition and an algorithm that determines when to reduce or increase dimension and in which latent space. Disentanglement of shared and private information arises as an emergent property of this optimization, eliminating the need for complex auxiliary loss functions. We demonstrate that FiGuRO outperforms existing ID estimation techniques and is more robust to hyperparameter changes. Across simulations and real-world data, FiGuRO captures distinct ID scales and varying subspace ratios, and decomposes shared and private information successfully. Furthermore, we show that FiGuRO can be applied to modern uni-modal pretrained models, enabling efficient, post-hoc disentanglement of multi-modal representations.

How many directions does a neural representation use to encode a concept? A common answer repeatedly erases probe directions and reports the stopping count or cumulative removed rank. We show that both quantities can change under an information-preserving invertible reparameterization, so neither is intrinsically a concept dimension. We distinguish model-defined population quantities (generating dimension, sufficient linear dimension, and minimum guarding rank) from procedure-defined quantities such as stopping count and cumulative edit rank. In a population Gaussian construction, an invertible shear preserves the prediction problem and all three quantities, yet changes the cumulative Euclidean erasure count from one to two. The separation holds for Moore--Penrose ordinary least squares and every finite nonnegative ridge weight. For a two-output full-QR procedure matching our motivating video analysis, cumulative edit rank similarly changes from two to the ambient dimension four. Conversely, the complete cumulative metric-QR trajectory is affine-equivariant when its positive-definite metric, probe, regularizer, and tie-breaking are transported consistently; exact covariance is one corollary, not a canonical semantic metric. In a known-rank finite-sample Adam/QR calibration, identity mixing stops after one accepted update in all 20 large-sample runs, whereas each tested shear $a\in\{.5,.75,1,1.25,2\}$ accepts at least two updates in all 20 runs. Controlled reparameterizations of frozen V-JEPA2 features preserve rank-zero predictions yet alter later Euclidean trajectories under practical optimization. These visual contact experiments are stress tests, not estimates of contact dimension. Iterative erasure therefore returns a procedure-relative estimand jointly determined by representation geometry and the full measurement procedure, not a semantic dimension by itself.

In this work we present a method to accelerate the optimization of learning high dimensional functions using deep neural network (DNN). This optimization procedure introduces contextual features into the first layer of a DNN. The parameters of DNN are optimized via standard gradient descent while keeping the input-feature basis fixed. After optimization of the DNN parameters, the feature layer is provided a chance to update and change before DNN optimization resumes. The feature layer has two types of functions: those that can be evaluated quickly in a matrix-free way on the domain (i.e. rank-1 features) and more complex features that must first be decomposed using tensor network (TN) decomposition strategies (tensor features). In particular, we study the effect of adding features which distill pretrained DNN into TNs using a discretize and decompose strategy. To efficiently decompose high-dimensional functions constructed from discretized DNN, we leverage a randomized tensor decomposition strategy. Using randomization, we are able to reduce the storage cost of decomposing high dimensional functions by at least 8 orders of magnitude. Using this approach, we are able to efficiently train models between 5 and 40 dimensions.

Omics datasets, particularly single-cell RNA sequencing data, are high-dimensional, sparse, noisy, and dominated by zero values, making faithful low-dimensional representation challenging. Existing dimensionality-reduction methods may distort local neighbourhoods, global organization, or the cohesion of meaningful populations, with similar limitations arising in genealogical data. We introduce Contrastive Manifold Approximation and Projection (CosMAP), a graph-based unsupervised dimensionality-reduction method for producing faithful and interpretable embeddings. CosMAP extends the graph-based framework of UMAP by combining cosine-similarity neighbourhoods with temperature-normalized contrastive affinities, which are optimized in the embedding space using an attractive--repulsive objective. It further employs a two-phase refinement strategy: an intermediate higher-dimensional representation is first learned and then used to reconstruct the neighbourhood graph and initialize the final low-dimensional embedding. We evaluate CosMAP on MNIST and USPS handwritten-digit datasets, mouse retina and cortex single-cell RNA-sequencing datasets, and a large genealogical kinship dataset derived from BALSAC-CARTaGENE. Compared with state-of-the-art dimensionality-reduction methods, CosMAP produces more coherent visual representations, improves neighbourhood preservation, and provides clearer global organization of digit classes, biological cell populations, and regional genealogical patterns. These results indicate that CosMAP offers a robust framework for exploratory analysis of complex, sparse, high-dimensional data. The implementation is publicly available at https://github.com/FenosoaRandrianjatovo/CosMAP-dr.

Despite the success of convolutional neural networks in image classification tasks and their general application in multi-modal models, their susceptibility to out-of-distribution and adversarial attack samples raises concerns regarding trustworthiness and safety. Among the approaches to tackle such issues, detection methods that analyze the model's intermediate activations to estimate a confidence score are a promising family that evaluates the decision process, relying on a dimensionality reduction step to enable efficient downstream processing of the high-dimensional activations. However, when considering convolutional layers, the dimensionality reduction methods in the literature either lack a mechanism to control the compression/information-loss trade-off or yield large representations. In this paper, we carefully analyze two state-of-the-art detection methods and their dimensionality reductions for convolutional layers and develop a novel reduction method with a controllable high-compression level. We extend these two state-of-the-art detection methods, enabling the usage of any dimensionality reduction, and evaluate their performance on out-of-distribution and adversarial attack detection. Results show that the detection methods with the proposed dimensionality reduction consistently perform better than, or comparable to, the strongest alternative. Furthermore, the proposed method is shown to reduce computation and memory footprints, given that it has the highest compression among the compared methods.

Vision foundation models are increasingly used as reusable encoders in medical image computing, yet their high-dimensional spatial embeddings are difficult to inspect beyond downstream task performance or global dimensionality reduction. We propose position-prompted PCA (P3CA), an encoder-agnostic method for local probing of channel-rich spatial tensors. Given a user-selected spatial prompt, P3CA estimates the feature normalization and dominant covariance directions within that region, then applies the resulting projection to the full tensor to visualize where locally informative directions are expressed. This produces a region-conditioned representation lens without modifying the encoder, retraining, or requiring task-specific labels. We implement P3CA in EmbedVision, an interactive 3D Slicer-based workflow, and evaluate it across natural images, colorectal pathology foundation-model embeddings, and spatial transcriptomic tensors. Across these settings, prompted projections reveal local structure suppressed by global PCA, improve prompt-matched pathology discrimination from frozen three-dimensional projections, and support comparison between learned and measured spatial representations.

Face recognition systems face two distinct, commonly-separated failure modes: spoofing, where an impostor presents a photograph or video of an authorized user, and disguise, where a legitimate user is rejected because their appearance differs from their enrolled template due to accessories, facial hair, illumination, or pose. This paper proposes and compares five combined feature-extraction and classification pipelines that address both problems within a single framework: PM (PCA and Minimum Euclidean Distance, MED), LPM (Local Binary Patterns with PCA and MED), HPM (Histogram of Oriented Gradients with PCA and MED), SM (Speeded-Up Robust Features with MED), and HM (Harris corner features with MED). Each pipeline follows a common two-phase process comprising pre-processing, feature extraction, feature filtering, and classification. The methods were trained on 115 subjects drawn from the FEI, Disguised Faces Database, and NUAA databases and evaluated on six test conditions covering mixed appearances, frontal faces, dark illumination, left- and right-turned poses, and photo-spoofing attempts. The HOG-based pipeline (HPM) achieved the most consistent performance across conditions, with 94.59% accuracy on mixed-appearance disguise, 81.5-93.2% across pose and illumination variants, and 91.67% on spoofing, while the LBP-based pipeline (LPM) achieved the second-highest spoofing-detection accuracy (93.2%), behind PM (96.67%), but weaker robustness to pose change. These results reveal a measurable trade-off between spoof sensitivity and disguise robustness among classical feature representations, motivating the deep-learning and cross-database extensions discussed in the concluding sections.

Hyperdimensional computing (HDC) is an approach from the cognitive science literature for solving information processing tasks using data represented as high-dimensional random vectors. The technique has a rigorous mathematical backing, and is easy to implement in energy-efficient and highly parallel hardware like FPGAs and "processing-in-memory" architectures. The effectiveness of HDC in machine learning largely depends on how raw data is mapped to high-dimensional space. In this work, we propose NysHD, a new method for constructing this mapping that is based on the Nyström method from the literature on kernel approximation. Our approach provides a simple recipe to turn any user-defined positive-semidefinite similarity function into an equivalent mapping in HDC. There is a vast literature on the design of such functions for learning problems. Our approach provides a mechanism to import them into the HDC setting, expanding the types of problems that can be tackled using HDC. Empirical evaluation against existing HDC encoding methods shows that NysHD can achieve, on average, 11% and 17% better classification accuracy on graph and string datasets respectively.

We introduce the Intrinsic Hybrid Latent Diffusion Model (ILDM), a generative framework that integrates probabilistic dimensionality reduction with geometry-aware diffusion on unknown manifolds. While diffusion models (DMs) have achieved state-of-the-art results in high-dimensional data synthesis, they rely on large training datasets and ignore intrinsic geometric structure. Latent diffusion models (LDMs) address the high dimensionality by learning a latent space, but they typically impose a Euclidean structure, failing to capture the underlying manifold geometry, especially problematic in data-sparse regimes. ILDM addresses these limitations by interpreting the latent space as a chart of an unknown Riemannian manifold, with geometry and uncertainty quantified through a probabilistic decoder. The forward process is a hybrid diffusion that switches between Riemannian and Euclidean dynamics based on local uncertainty, where the Riemannian component is governed by a probabilistic metric tensor derived from the decoder. To learn the generative dynamics, we introduce an approximate denoising score matching method tailored to the hybrid diffusion setting, enabling a backward process defined by hybrid Langevin dynamics. Experiments on COIL-100, MNIST, and cardiac MRI datasets demonstrate that ILDM significantly improves generation quality, achieving lower FID and LPIPS scores compared to standard diffusion and latent diffusion models.

Long-context large language models (LLMs) are increasingly deployed in real-world applications, yet self-attention remains a major efficiency bottleneck -- especially during decoding -- due to the necessity of repeatedly processing ever-growing key-value (KV) caches. Existing sparse attention reduce computation by attending to fewer KV pairs, but often suffer from substantial accuracy degradation, require additional training, or rely on expensive hashing. In this work, we present BinaryPC, a training-free, data-aware hashing-based sparse attention for long-context LLMs. BinaryPC constructs compact binary hash codes and corresponding hash function by computing binary principal components of data. Unlike Locality-Sensitive Hashing (LSH) with data-independent random projections or learned non-linear hashing methods, BinaryPC constructs binary codes that explicitly preserve the structural information of data without requiring gradient-based training. Comprehensive experiments across multiple model families and long-context benchmarks show that BinaryPC preserves accuracy relative to full attention while achieving superior performance among sparse and hashing-based baselines. On modern GPUs, BinaryPC improves end-to-end decoding throughput by 3.56$\times$ over the FlashAttention kernel. Our code is available at https://github.com/yudaohai666/BPC.

Existing low rank KV cache methods preserve either model weights or key variance, neither of which directly reflects the attention scores used during inference. We derive the expected attention score distortion caused by rank r key compression and show that it yields a covariance weighted low rank objective. Under a margin condition, controlling this distortion also improves top k recall. The optimal rank r solution has a closed form asymmetric factorization obtained from the SVD of the covariance weighted query key operator. This motivates SAKI, a training free KV cache index that directly preserves attention scores rather than key reconstruction quality. Across LLaMA 3.1 8B, Qwen 2.5 7B, Mistral 7B v0.1, and Llama 3.2 3B, SAKI outperforms key PCA at every tested rank. At rank 32, it removes 13 to 30 percent of PCA's remaining top 64 recall error, including improvements from 0.748 to 0.799 on LLaMA 3.1 8B and from 0.786 to 0.850 on Qwen 2.5 7B. It improves 68 to 89 percent of attention heads per model, with the largest gains in deeper layers. Predicted score MSE reductions closely match empirical measurements, with a Pearson correlation of 0.997, while ablation studies confirm that the gains arise from optimizing the attention score objective rather than covariance weighting alone. Analysis of the scoring operator further explains why weight only, invariant subspace, and key reconstruction methods can be suboptimal. SAKI uses random-matrix theory to separate genuine covariance signal from autocorrelated sampling noise, matching PCA with only 512 calibration tokens and adding value exactly where PCA sees no reliable signal.

Recent studies have reported clearly identifiable dynamical changes in the high-frequency range of EEG signals recorded during specific stimuli, such as visual or auditory inputs, or in cases of brain disorders like epileptic seizures. In this study, we utilized Dynamic Mode Decomposition (DMD) to extract consistent and persistent dynamical changes in the high-frequency band from the signals of neurologically relevant EEG channels. High-frequency DMD modes were employed as features, composing a feature table. Through post-processing, a random distribution test was performed, revealing that approximately 70% of the samples exhibited consistent high-frequency dynamics within the signal of a specific channel. Furthermore, classification experiments confirmed that the PCA components of the feature table that passed the test formed a consistent pattern that distinguished the alcohol-dependent group from the control group.

Sequential subset search -- forward selection with floating backtracking and its descendants -- remains the quality reference in feature selection, but every member of the family sweeps the full pool of remaining candidate features at each step, which excludes it from very-high-dimensional problems; there, only individual-feature ranking remains practical, and it models feature interplay weakly or not at all. We introduce a budgeted sampled step operator pair that replaces the full sweeps by a fixed number of candidate evaluations per step. Candidates are drawn by temperature-controlled softmax sampling from dependency-aware per-feature statistics learned online from every criterion evaluation the search performs, guarded by a uniform exploration floor; per-step cost becomes independent of dimensionality. Substituting the operators turns any sequential method into its stochastic counterpart, defining the Stochastic Sequential Search (SSS) family; we study the stochastic counterpart of floating search, sSFFS. On 500-dimensional madelon, sSFFS retains at least 97% of the full-SFFS criterion value at every subset size at about a quarter of its evaluations, while uniform sampling at the same budget collapses on madelon's synergistic features. On 5,000-dimensional gisette, far beyond full-SFFS reach, sSFFS exceeds the saturated criterion level of DAF and BIF ranking at matched budgets; holdout validation shows that at 500 training samples the binding constraint beyond the sequential frontier becomes the criterion, not the search. On 10,105-dimensional reuters, under a trustworthy multinomial filter criterion, sSFFS dominates BIF and DAF on the search objective and on holdout accuracy at every subset size, in about two minutes of single-core evaluation work. A verified standalone implementation accompanies the paper.

Quality metrics play a crucial role in the proper use of dimensionality reduction projections for visual analysis of high-dimensional data. They quantify the degree of distortion of a projection compared to the high-dimensional data and provide a reliable indication of how confident users can be in the structures they see in the resulting layouts. However, most popular metrics focus on capturing direct relationships between points (e.g., distances or neighborhoods) while neglecting distortions in empty areas of the layout, even though these often compose visually relevant features of a 2D layout. In this paper, we introduce the Gap Index (GI), a quality metric for 2D projections that captures visual distortion by measuring spatial distortion in empty areas of a projection. It does so by decomposing the space into empty triangles, which are then compared to their high-dimensional counterparts to compute the deformation. This per-triangle deformation can be aggregated into a single scalar value or overlaid on a projection to visualize regional distortion patterns. Results show that, contrary to popular quality metrics, the GI is sensitive to small structural deformations that have high visual impact. It is also fast to compute and interpretable.

We extend a recently introduced Entropy-Optimal Manifold Clustering (EOMC) to allow for a joint simultaneous identification of subsets and subspaces of relevant features in nonstationary and nonlinear regression problems. It is shown that the proposed extension - that we coin as Entropy-Optimal Manifold Regression (EOMR) - allows a robust learning with linearly-scaling iteration and memory complexities. EOMR is compared to the most complete set of state-of-the-art tools from the Artificial Intelligence (AI) and Machine Learning (ML) that is available to the author, on the very challenging problems from chaotic and fluid dynamics: (i) on predicting the Lorenz-96 systems dynamics in strongly- and very-strongly chaotic regimes (with forcing parameter being $F=8$ and $F=12$, respectively); and, (ii) on a data from the Hasegawa-Wakatani model on the edge of the tokamak plasma. It is demonstrated that the proposed benchmarks (i) and (ii), indeed, are the very challenging problems for the state of the art ML and AI tools - since both the general-purpose gradient boosted random forests and deep neuronal networks, as well as transformer-based AI tools like TabPFN v.03 (more spezialised for large-dimensional small data learning problems) - result in orders of magnitude inferior root mean squared prediction errors, and orders of magnitude larger model complexities, when compared to the EOMR. For a Hasegawa-Wakatani example, EOMR distills a very simple entropy-optimal and skilful description of the leading Essential Orthogonal Function (EOF) dynamics, given by linear, causal and weakly-stationary autoregressive process described by just 8 parameters.

Matrix factorisation is a fundamental tool for exploiting low-dimensional structure in high-dimensional data, with applications such as data compression, denoising, structure discovery, interpretable representation learning, and dimensionality reduction. Compared to conventional two-factor models, matrix tri-factorisation provides greater modelling flexibility, while sparsity constraints often improve both interpretability and recovery performance. Although the role of sparsity has been extensively studied for two-factor matrix factorisation, rigorous theoretical guarantees for general real-valued matrix tri-factorisation remain largely unexplored. To address this gap, we establish, to the best of our knowledge, the first rigorous theoretical study for sparsity-induced identifiability in general real-valued matrix tri-factorisation. Our analysis is enabled by a novel decomposition strategy that transforms the original problem into two coupled auxiliary factorisation problems, while preserving the structural information necessary to the recovery of the original factor matrices from the observations. Building upon this decomposition, we derive recovery guarantees and structural consistency results that characterise how coefficient sparsity influences the sufficient recovery conditions, convergence behaviour, spectral approximation error, high-probability bounds, and structure preservation. Comprehensive Monte Carlo experiments validate the proposed theory and demonstrate close agreement between the theoretical results and empirical observations.

Dimensionality Reduction (DR) is a fundamental tool for high-dimensional data exploration, reducing the complexity of latent spaces of machine learning models, and assisting in the explanation of complex opaque models. However, non-linear DR techniques often function as opaque transformations themselves, making it challenging to understand how individual features influence instance positioning in the reduced space. This lack of transparency complicates the analysis and interpretation of structural patterns, hindering the ability to reason about the organization of high-dimensional data based on the projected layout. In order to address this challenge, dimensionality reduction explanation methods have shown promise in improving the understanding of the observed groups and cluster structures. Unfortunately, existing DR explanation approaches tend to suffer from limitations such as multiple attributions per feature and restricted applicability to specific dimensionality reduction methods, which hinder their use. In this work, we propose FADEx, a novel local per-instance feature attribution method that leverages local linear approximation via first-order Taylor expansion and Singular Value Decomposition to provide explanations. FADEx computes the local linear models via weighted least squares, eliminating the need for out-of-sample data mapping, making it agnostic to the DR method, while simultaneously providing local feature attributions and distortion analysis. Through qualitative and quantitative evaluations, comparisons with existing methods, and case studies, we demonstrate FADEx's effectiveness and versatility in providing explanations and analytical resources for analyzing the behavior of DR methods. The results indicate FADEx yields robust and reliable explanations, outperforming existing approaches in several aspects.

Minimax-optimal rates for multivariate distribution estimation are known to suffer from the curse of dimensionality. We propose a sparse Bayesian network approach in which each conditional probability is estimated using sparsity-aware conditional mean methods. The resulting estimator, \textit{BAyesian Network Distribution regression} (BAND), handles mixed data types in high-dimensional time series and achieves polynomial total variation convergence rates while allowing the feature dimension to grow polynomially with the sample size. These rates are substantially faster than the classical optimal rates for multivariate histogram density estimators that lack sparsity. Empirical evaluations show that BAND performs competitively for data sampling and confidence region forecasting against a range of state-of-the-art benchmarks.

Latent low-dimensional structure in datasets of natural and engineered systems enables their sparse sensing, or full-state reconstruction from historical data and very few carefully chosen localized measurements. Depending on the reconstruction algorithm, sensor locations, and measurement noise, the reconstruction risk curves demonstrate a diversity of patterns including a dramatic peak in error known as double descent in Machine Learning literature. Here we explore those scenarios under a unified Data-Noise Averaging theory. Qualitatively, we formulate sufficient criteria for double descent to emerge through a catastrophic amplification of a pathological signal in reconstruction. Quantitatively, we predict the detailed risk curves at a fraction of computational cost, trace reconstruction instability to individual sensors and their combinations, and provide regularization mechanisms to mitigate the instability. We demonstrate results for both static reconstruction of Sea Surface Temperature patterns and time integration of a reduced order model of a PDE.

It is common for two-dimensional embeddings of high-dimensional data to be read far beyond what they can support. Distances in and between clusters, the meaning behind empty spaces, and the amount of structure hidden at each point are generally invisible in the output of methods such as t-SNE and UMAP. This is because the information that could support the meaning of these properties is discarded during the optimisation process. Here, we present FloDR, a dimensionality reduction method that embeds data through an invertible normalising flow. While FloDR only uses the first two output coordinates to create a two-dimensional embedding, it retains the remaining coordinates rather than discarding them. In addition to the embedding, an exact inverse and an exact density are properties of a trained mapping, which enable diagnostic visualisations that are computed from the exact inverse of the model that drew the layout rather than from an approximate one. Specifically, we draw two fields, the conditional spread, which measures how much of the original data remains undetermined at each embedding position in input units, and the hidden contrast, which measures how much information about a labelled contrast the two plotted coordinates discard. Both fields are rendered with a prespecified test against a held out portion of the input data and a bootstrap confidence. A field that fails the test is reported as refused.

Single-cell ribonucleic acid sequencing (scRNA-seq) is a foundational technology for precision-medicine workflows that contribute to United Nations Sustainable Development Goal 3 on Good Health and Well-being, and unsupervised clustering is the analytical step that turns raw expression matrices into interpretable cell populations. Practitioners therefore face a recurring engineering decision: is an additional deep representation stage worth its compute and tuning cost, or do classical principal component analysis (PCA) pipelines already suffice? We address this question with a diagnostic benchmark of nine clustering pipelines on ten real datasets (90-5,685 cells, 19,046-41,480 genes, 4-11 cell types), augmented by a partial scVI V2 specialized comparison on seven datasets. The protocol integrates Optuna hyperparameter search, repeated-run robustness, Friedman/Wilcoxon-Holm/TOST testing, and Sobol total-order sensitivity analysis. The contrastive autoencoder achieved the highest mean Adjusted Rand Index (0.7872), but Holm-corrected tests did not establish dominance over the strongest baselines. Per-dataset analysis reveals three reproducible regimes: probabilistic variational autoencoder (VAE) variants help on the smallest datasets, deep autoencoders win on mid-scale data with multi-batch or many-type structure, and classical PCA pipelines remain competitive when linear projection already captures the dominant variation. Sobol indices identify learning rate ($S_T=0.70$) and latent dimensionality ($S_T=0.56$) as the dominant variance contributors, indicating where limited tuning budgets should be allocated. The contribution is therefore a dataset-aware and compute-conscious decision framework for biomedical AI pipelines supporting sustainable healthcare analytics, rather than a universal superiority claim.

Integrating heterogeneous industrial knowledge, including factual relations and decision constraints, remains a core challenge in industrial information systems. Machining process planning exemplifies this problem because engineers must select operations by combining material properties, feature characteristics, and quality requirements. Existing methods rely mainly on similarity retrieval or classification, without a unified ranking objective or standardized evaluation. We propose PCA-GAT, which formulates machining process plan recommendation as a knowledge graph enhanced collaborative filtering problem. Bayesian Personalized Ranking provides the learning objective, while Recall@K and NDCG@K define evaluation. The knowledge graph supplies semantic structure when collaborative signals are sparse. Four domain constraints, material compatibility, precision requirements, feature applicability, and operation sequencing, are introduced as attention biases during graph propagation. Type-specific weights learn their importance, and an adaptive gate adjusts their influence using local context. On a real aerospace dataset with 115 parts and 507 plans, PCA-GAT achieves Recall@1 = 0.9087 and strong cold-start robustness, with about half the degradation of the strongest baseline under severe sparsity. Ablation studies show that knowledge graph enrichment is essential, constraints add value, and ungated constraint injection can hurt performance. The learned weights identify material-operation compatibility as the dominant factor, consistent with domain expertise. Results on three public benchmarks show no degradation when constraints are absent, supporting generalization beyond manufacturing. This study establishes a standardized recommendation protocol for engineering process planning and benchmarks seven methods across three categories, showing that knowledge representation is the main bottleneck.

Neural operators provide data-driven mappings for modeling dynamical systems. Extending them to families of systems typically requires explicit conditioning variables such as physical parameters, geometries, or boundary conditions. In many real-world settings, these quantities are unobserved. Here, we formulate neural operator discovery (NOD) as the problem of learning both shared solution operators and system-specific variation directly from heterogeneous trajectories without access to labeled governing factors. We introduce a factorized latent-conditioning formulation that jointly learns a neural operator and a low-dimensional latent representation through factorized prediction, trajectory-decoupled sampling, and dimension selection. Across diverse systems, the learned latent representation captures the intrinsic dimensionality of system variation and organizes system instances in a smooth and approximately invertible latent structure aligned with the underlying governing factors. This organization enables generalization to previously unseen system instances, including zero-shot extrapolation across regimes and stable long-horizon prediction. These results establish an interpretable paradigm for operator learning in the absence of explicit factor supervision.

We present Variance-Preserving Orthogonal Selection (VPOS), an unsupervised feature-selection method that performs sequential orthogonal deflation in the variance-weighted principal component analysis (PCA) loading space $\mathbf{V}_d\mathbfΛ_d^{1/2}$. After each feature is selected, its loading direction is projected out of all remaining candidates, so subsequent selections cover complementary directions of the rank-$d$ covariance approximation while returning original variables. We establish rank-reduction guarantees and a determinant-growth interpretation, and distinguish VPOS from greedy selection on raw data, unweighted eigenvector pivoting, Principal Feature Analysis (PFA), and Principal Variable Selection (PVS). Experiments enforce $k\leq d$, tune method-specific parameters on validation observations, and evaluate on unseen outer folds. Across seven labelled benchmarks, VPOS improves held-out normalised reconstruction error over matched PCA without deflation on every dataset, with reductions of 1--78%. It obtains the lowest mean reconstruction error on Wine, Breast Cancer, and MNIST and is within 1.7% of the lowest error on CIFAR-10 and HighDim. On CIFAR-10, VPOS is approximately 24$\times$ faster than the closely related PVS baseline while incurring a 1.7% reconstruction gap. These results establish VPOS as an efficient covariance-coverage method, particularly when correlated high-dimensional data must be represented by a small set of identifiable original variables.

Domain Generalization (DG) aims to learn representations robust to distribution shifts. Recent geometric alignment methods, such as CPCANet, extract domain-invariant structures through batch-wise Common Principal Component Analysis (CPCA). However, CPCANet suffers from rank-deficient covariance estimation due to the small-sample-size issue in mini-batch training. To address this limitation, we propose Projection Pursuit CPCANet (PP-CPCANet), a covariance-free framework that learns a global orthogonal basis on the Stiefel manifold and jointly optimizes it with network parameters via the Cayley transform. We further introduce a symmetry-breaking detached-median PP dispersion objective to extract common principal components (CPCs) with dense and robust optimization signals. Experiments on four DG benchmarks show that PP-CPCANet achieves SOTA performance while maintaining stable training.

We show that, up to isotropic scaling, the Gaussian RBF reproducing kernel Hilbert space (RKHS) is asymptotically isometric to Euclidean space in the large bandwidth limit. This strongly suggests that kernel-based constructions reliant on metric properties of the RKHS will yield results for Gaussian RBF kernels that similarly approach those of linear kernels for large bandwidths. The asymptotic behavior of Gaussian CKA can be understood in this light. We further consider kernel PCA, showing that Gaussian RBF eigenvalues, eigenprojections, and principal components all converge to those of classical (linear) PCA as bandwidth $σ\rightarrow \infty$. For a given data representation, both the RKHS feature embeddings and the orthogonal PCA eigenframes of the two kernel types differ asymptotically by a geometric similarity transformation, up to a residual of size $O \left (\fracρσ \right )^2$, where $ρ$ is a measure of geometric eccentricity of the representation, equal to the ratio of maximum to median pairwise distance between data examples. Experiments over a diverse collection of data sets demonstrate that $ρ$ provides a simple and reliable predictor of dataset-specific convergence behavior in the top principal directions.

Fine-tuning has been widely used to adapt large language models (LLMs) for domain-specific tasks. Parameter efficient fine-tuning (PEFT) methods such as low-rank adaptation (LoRA) are frequently used to reduce computational costs. PortLLM is a training-free and data-free scheme used to adapt LLMs after continual pretraining. Although the initial PortLLM results show that LoRA patches exhibit short-term temporal portability, the long-term performance of PortLLM across several updates of continual pretraining remains underexplored. Furthermore, the intriguing effectiveness of PortLLM is not well understood from a theoretical standpoint. We address these two open questions by (1) performing an extensive empirical study of the long-term temporal portability of PortLLM patches across 10 continual pretraining steps using base models Mistral, Gemma, and Qwen; and (2) offering two theoretical analyses to explain our observation that the simple PortLLM method achieves competitive performance. We find empirically that the portability persists across longer time duration, indicating that repeated fine-tuning is not required when the base model is periodically updated. We find theoretically that near-orthogonality of high-dimensional vectors is a key justification for temporal portability. Our analyses also demonstrate a geometric perspective of the loss landscape in facilitating the theoretical comparison of different adaptation options.