The Nyström method is a popular low-rank approximation technique for large matrices that arise in kernel methods and convex optimization. Yet, when the data exhibits heavy-tailed spectral decay, the effective dimension of the problem often becomes so large that even the Nyström method may be outside of our computational budget. To address this, we propose Block-Nyström, an algorithm that injects a block-diagonal structure into the Nyström method, thereby significantly reducing its computational cost while recovering strong approximation guarantees. We show that Block-Nyström can be used to construct improved preconditioners for second-order optimization, as well as to efficiently solve kernel ridge regression for statistical learning over Hilbert spaces. Our key technical insight is that, within the same computational budget, combining several smaller Nyström approximations leads to stronger tail estimates of the input spectrum than using one larger approximation. Along the way, we provide a novel recursive preconditioning scheme for efficiently inverting the Block-Nyström matrix, and provide new statistical learning bounds for a broad class of approximate kernel ridge regression solvers.

We present SemiOccam, an image recognition network that leverages semi-supervised learning in a highly efficient manner. Existing works often rely on complex training techniques and architectures, requiring hundreds of GPU hours for training, while their generalization ability with extremely limited labeled data remains to be improved. To address these limitations, we construct a hierarchical mixture density classification mechanism by optimizing mutual information between feature representations and target classes, compressing redundant information while retaining crucial discriminative components. Experimental results demonstrate that our method achieves state-of-the-art performance on three commonly used datasets, with accuracy exceeding 95% on two of them using only 4 labeled samples per class, and its simple architecture keeps training time at the minute level. Notably, this paper reveals a long-overlooked data leakage issue in the STL-10 dataset for semi-supervised learning and removes duplicates to ensure reliable experimental results. We release the deduplicated CleanSTL-10 dataset to facilitate fair and reproducible research. Code available at https://github.com/Shu1L0n9/SemiOccam.

We analyze the extent to which internal representations of language models (LMs) identify and distinguish mentions of named entities, focusing on the many-to-many correspondence between entities and their mentions. We first formulate two problems of entity mentions -- ambiguity and variability -- and propose a framework analogous to clustering quality metrics. Specifically, we quantify through cluster analysis of LM internal representations the extent to which mentions of the same entity cluster together and mentions of different entities remain separated. Our experiments examine five Transformer-based autoregressive models, showing that they effectively identify and distinguish entities with metrics analogous to precision and recall ranging from 0.66 to 0.9. Further analysis reveals that entity-related information is compactly represented in a low-dimensional linear subspace at early LM layers. Additionally, we clarify how the characteristics of entity representations influence word prediction performance. These findings are interpreted through the lens of isomorphism between LM representations and entity-centric knowledge structures in the real world, providing insights into how LMs internally organize and use entity information.

-- Thermal imaging can greatly enhance the application of intelligent unmanned aerial vehicles (UA V) in challenging environments. However, the inherent low resolution of thermal sensors leads to insufficient details and blurred boundaries. Super-resolution (SR) offers a promising solution to address this issue, while most existing SR methods are designed for fixed-scale SR. They are computationally expensive and inflexible in practical applications. T o address above issues, this work proposes a novel any-scale thermal SR method (AnyTSR) for UA V within a single model. Specifically, a new image encoder is proposed to explicitly assign specific feature code to enable more accurate and flexible representation. Additionally, by effectively embedding coordinate offset information into the local feature ensemble, an innovative any-scale upsampler is proposed to better understand spatial relationships and reduce artifacts. Moreover, a novel dataset (UA V-TSR), covering both land and water scenes, is constructed for thermal SR tasks. Experimental results demonstrate that the proposed method consistently outperforms state-of-the-art methods across all scaling factors as well as generates more accurate and detailed high-resolution images.

Concept Bottleneck Models (CBMs) aim to enhance interpretability by structuring predictions around human-understandable concepts. However, unintended information leakage, where predictive signals bypass the concept bottleneck, compromises their transparency. This paper introduces an information-theoretic measure to quantify leakage in CBMs, capturing the extent to which concept embeddings encode additional, unintended information beyond the specified concepts. We validate the measure through controlled synthetic experiments, demonstrating its effectiveness in detecting leakage trends across various configurations. Our findings highlight that feature and concept dimensionality significantly influence leakage, and that classifier choice impacts measurement stability, with XGBoost emerging as the most reliable estimator. Additionally, preliminary investigations indicate that the measure exhibits the anticipated behavior when applied to soft joint CBMs, suggesting its reliability in leakage quantification beyond fully synthetic settings. While this study rigorously evaluates the measure in controlled synthetic experiments, future work can extend its application to real-world datasets.

We propose perturbed proximal algorithms that can provably escape strict saddles for nonsmooth weakly convex functions. The main results are based on a novel characterization of $ε$-approximate local minimum for nonsmooth functions, and recent developments on perturbed gradient methods for escaping saddle points for smooth problems. Specifically, we show that under standard assumptions, the perturbed proximal point, perturbed proximal gradient and perturbed proximal linear algorithms find $ε$-approximate local minimum for nonsmooth weakly convex functions in $O(ε^{-2}\log(d)^4)$ iterations, where $d$ is the dimension of the problem.

In network analysis, various matrix representations have been developed to investigate the structural properties of the corresponding network, among which the non-backtracking (NBT) matrix is one such representation. The NBT matrix is well-known for its relationship with the Ihara zeta function [15] defined as an infinite product over equivalence classes of primitive cycles. It was shown in [15, 36] that, for regular graphs, the reciprocal of the Ihara zeta function can be expressed as a polynomial related to the adjacency matrix. The relation between the zeta function and the determinant of the NBT matrix was elucidated in [12], extended to irregular graphs in [3], and an elementary proof was provided in [35]. The connection between the polynomial and the determinant of the NBT matrix, via the Ihara zeta function, is known as the Ihara's formula.

Temporal causal representation learning is a powerful tool for uncovering complex patterns in observational studies, which are often represented as low-dimensional time series. However, in many real-world applications, data are high-dimensional with varying input lengths and naturally take the form of irregular tensors. To analyze such data, irregular tensor decomposition is critical for extracting meaningful clusters that capture essential information. In this paper, we focus on modeling causal representation learning based on the transformed information. First, we present a novel causal formulation for a set of latent clusters. We then propose CaRTeD, a joint learning framework that integrates temporal causal representation learning with irregular tensor decomposition. Notably, our framework provides a blueprint for downstream tasks using the learned tensor factors, such as modeling latent structures and extracting causal information, and offers a more flexible regularization design to enhance tensor decomposition. Theoretically, we show that our algorithm converges to a stationary point. More importantly, our results fill the gap in theoretical guarantees for the convergence of state-of-the-art irregular tensor decomposition. Experimental results on synthetic and real-world electronic health record (EHR) datasets (MIMIC-III), with extensive benchmarks from both phenotyping and network recovery perspectives, demonstrate that our proposed method outperforms state-of-the-art techniques and enhances the explainability of causal representations.

Regression problems with bounded continuous outcomes frequently arise in real-world statistical and machine learning applications, such as the analysis of rates and proportions. A central challenge in this setting is predicting a response associated with a new covariate value. Most of the existing statistical and machine learning literature has focused either on point prediction of bounded outcomes or on interval prediction based on asymptotic approximations. We develop conformal prediction intervals for bounded outcomes based on transformation models and beta regression. We introduce tailored non-conformity measures based on residuals that are aligned with the underlying models, and account for the inherent heteroscedasticity in regression settings with bounded outcomes. We present a theoretical result on asymptotic marginal and conditional validity in the context of full conformal prediction, which remains valid under model misspecification. For split conformal prediction, we provide an empirical coverage analysis based on a comprehensive simulation study. The simulation study demonstrates that both methods provide valid finite-sample predictive coverage, including settings with model misspecification. Finally, we demonstrate the practical performance of the proposed conformal prediction intervals on real data and compare them with bootstrap-based alternatives.

The amount of quality data in many machine learning tasks is limited to what is available locally to data owners. The set of quality data can be expanded through trading or sharing with external data agents. However, data buyers need quality guarantees before purchasing, as external data may be contaminated or irrelevant to their specific learning task. Previous works primarily rely on distributional assumptions about data from different agents, relegating quality checks to post-hoc steps involving costly data valuation procedures. We propose a distribution-free, contamination-aware data-sharing framework that identifies external data agents whose data is most valuable for model personalization. To achieve this, we introduce novel two-sample testing procedures, grounded in rigorous theoretical foundations for conformal outlier detection, to determine whether an agent's data exceeds a contamination threshold. The proposed tests, termed conformal data contamination tests, remain valid under arbitrary contamination levels while enabling false discovery rate control via the Benjamini-Hochberg procedure. Empirical evaluations across diverse collaborative learning scenarios demonstrate the robustness and effectiveness of our approach. Overall, the conformal data contamination test distinguishes itself as a generic procedure for aggregating data with statistically rigorous quality guarantees.