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.

We study attention mechanisms through the lens of a canonical unsupervised problem: principal component analysis (PCA). We show that, when trained on Gaussian data, both softmax and linear attention layers learn parameters that align with the principal eigenvectors of the covariance matrix, thereby establishing a direct and explicit connection with PCA. Our analysis covers both finite and infinite prompt regimes. In the infinite-prompt limit, we prove convergence to globally optimal solutions aligned with the leading spectral direction, while in the finiteprompt setting we show that the same behavior emerges up to sampling effects. We further extend the analysis to an in-context setting with spiked Wishart covariances, where attention successfully recovers the underlying signal direction. These results demonstrate that attention inherently performs PCA-like computations under unsupervised objectives, providing a theoretical foundation for its representation-learning capabilities.

Existing clustering methods for functional data often prioritize partitioning accuracy over interpretability, making it challenging to extract meaningful insights when the data-generating process follows a specific underlying structure and an ordinal relationship among clusters is suspected. This work introduces K-Models, a novel framework that integrates ordinal constraints and estimates key underlying elements of the random process generating the observed functional profiles, improving both interpretability and structure identification. The proposed method is evaluated through simulations and real-world applications. In particular, it is tested on Region of Interest (ROI) curves, which represent reaction profiles from a reflectometric sensor monitoring biomolecular interactions, such as antigen-antibody binding. These curves represent changes in reflected light intensity over time at multiple measurement spots with immobilized antigens during analyte exposure, capturing the binding dynamics of the system. The goal is to identify intrinsic signal patterns solely from the observed dynamics, making this dataset an ideal benchmark for assessing the added interpretability of the proposed approach. By incorporating structural assumptions into the clustering process, K-Models enhances interpretability while maintaining performance comparable to state-of-the-art techniques, providing a valuable tool for analyzing functional data with an underlying ordinal structure.

We consider streaming principal component analysis when the stochastic datagenerating model is subject to perturbations. While existing models assume a fixed covariance, we adopt a robust perspective where the covariance matrix belongs to a temporal uncertainty set. Under this setting, we provide fundamental limits on convergence of any algorithm recovering principal components. We analyze the convergence of the noisy power method and Oja's algorithm, both studied for the stationary data generating model, and argue that the noisy power method is rate-optimal in our setting. Finally, we demonstrate the validity of our analysis through numerical experiments on synthetic and real-world dataset.

To understand neural network behavior, recent works quantitatively compare different networks' learned representations using canonical correlation analysis (CCA), centered kernel alignment (CKA), and other dissimilarity measures. Unfortunately, these widely used measures often disagree on fundamental observations, such as whether deep networks differing only in random initialization learn similar representations. These disagreements raise the question: which, if any, of these dissimilarity measures should we believe? We provide a framework to ground this question through a concrete test: measures should have sensitivity to changes that affect functional behavior, and specificity against changes that do not. We quantify this through a variety of functional behaviors including probing accuracy and robustness to distribution shift, and examine changes such as varying random initialization and deleting principal components. We find that current metrics exhibit different weaknesses, note that a classical baseline performs surprisingly well, and highlight settings where all metrics appear to fail, thus providing a challenge set for further improvement.

To understand neural network behavior, recent works quantitatively compare different networks' learned representations using canonical correlation analysis (CCA), centered kernel alignment (CKA), and other dissimilarity measures. Unfortunately, these widely used measures often disagree on fundamental observations, such as whether deep networks differing only in random initialization learn similar representations. These disagreements raise the question: which, if any, of these dissimilarity measures should we believe? We provide a framework to ground this question through a concrete test: measures should have sensitivity to changes that affect functional behavior, and specificity against changes that do not. We quantify this through a variety of functional behaviors including probing accuracy and robustness to distribution shift, and examine changes such as varying random initialization and deleting principal components. We find that current metrics exhibit different weaknesses, note that a classical baseline performs surprisingly well, and highlight settings where all metrics appear to fail, thus providing a challenge set for further improvement.

EVEN supervised learning is subject to the famous NoFree Lunch Theorems [1]-[3], which say that, in combinatorial optimization, there is no universal algorithm that works better than its competitors for every objective function [4]-[6]. Indeed, David Wolpert has recently proven that, on average, cross-validation performs as well as anti-crossvalidation (choosing among a set of candidate algorithms based on which has the worst out-of-sample behavior) for supervised learning. Still, he acknowledges that "it is hard to imagine any scientist who would not prefer to use [crossvalidation] to using anti-cross-validation" [7]. On the other hand, unsupervised learning has seldom been studied from the perspective of the NFLTs. This may be because the adjective "unsupervised" suggests that no human input is needed, which is misleading as many unsupervised tasks are combinatorial optimization problems that depend on the choice of the objective function. For instance, it is well known that, among the eigenvectors of the covariance matrix, Principal Components Analysis selects those with the largest variances [8]. However, mode-hunting techniques that rely on spectral manipulation aim at the opposite objective: selecting the eigenvectors of the covariance matrix with the smallest variances [9], [10]. Therefore, unlike in supervised learning, where it is difficult to identify reasons to optimize with respect to anti-cross-validation, in unsupervised learning there are strong reasons to reduce dimensionality for variance minimization. D. A. D ıaz-Pach on and T. Liu are with the Division of Biostatistics, University of Miami, Miami, FL, 33136 USA (e-mail: ddiaz3@miami.edu,

Since its inception in 1982, Oja's algorithm has become an established method for streaming principle component analysis (PCA).