Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2009.03979

A Distance-preserving Matrix Sketch

Leland Wilkinson·Hengrui Luo
View original source ↗Read source document ↗Export metadata ↓

Abstract

Visualizing very large matrices involves many formidable problems. Various popular solutions to these problems involve sampling, clustering, projection, or feature selection to reduce the size and complexity of the original task. An important aspect of these methods is how to preserve relative distances between points in the higher-dimensional space after reducing rows and columns to fit in a lower dimensional space. This aspect is important because conclusions based on faulty visual reasoning can be harmful. Judging dissimilar points as similar or similar points as dissimilar on the basis of a visualization can lead to false conclusions. To ameliorate this bias and to make visualizations of very large datasets feasible, we introduce two new algorithms that respectively select a subset of rows and columns of a rectangular matrix. This selection is designed to preserve relative distances as closely as possible. We compare our matrix sketch to more traditional alternatives on a variety of artificial and real datasets.

Explore related subjects

cs.HCcs.LGstat.ML

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Leland Wilkinson, Hengrui Luo. 2021-11-19. A Distance-preserving Matrix Sketch. https://doi.org/10.1080/10618600.2022.2050246

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

ADSEL: Adaptive Dual Self-Expression Learning for EEG Feature Selection via Incomplete Multi-Dimensional Emotion Labels

EEG based multi-dimension emotion recognition has attracted substantial research interest in affective computing. However, the high dimensionality of EEG features, coupled with limited sample sizes, frequently leads to classifier overfitting and high computational complexity. Feature selection constitutes a critical strategy for mitigating these challenges. However, most existing EEG feature selection methods assume complete multi-dimensional emotion labels. In practice, open acquisition environment and the inherent subjectivity of emotion perception often result in incomplete label data, which can compromise model generalization. Additionally, existing feature selection methods for handling incomplete multi-dimensional labels primarily focus on correlations among various dimensions during label recovery, neglecting the correlation between samples in the label space and their interaction with various dimensions. To address these issues, we propose a novel incomplete multi-dimensional emotion feature selection framework integrating Adaptive Dual Self-Expression Learning (ADSEL) with least squares regression. ADSEL could establish a bidirectional pathway between sample-level and dimension-level self-expression learning processes within the label space. It could facilitate the cross-sharing of learned information between these processes, enabling the simultaneous exploitation of effective information across both samples and dimensions for label reconstruction. Consequently, ADSEL could enhance label recovery accuracy and effectively identifies the optimal EEG feature subset for multi-dimensional emotion recognition. ADSEL was evaluated against fourteen state-of-the-art feature selection methods on three public EEG datasets with multi-dimensional emotion labels. Experimental results demonstrate that ADSEL could achieve superior performance under conditions of partial label absence.

cs.HC↗
arXiv2026

A Human-AI Collaborative Workflow for Mathematical Discovery: A Case Study in Grover-Compatible Riemannian Optimization

We investigate how large language models can be used as research tools in scientific computing while preserving mathematical rigor. We propose a human-in-the-loop workflow for interactive theorem proving and discovery with LLMs. Human experts retain control over problem formulation and assumptions, while the model searches for proofs or contradictions, proposes candidate properties and theorems, and helps construct structures and parameters that satisfy explicit constraints, supported by numerical experiments and simple verification checks. Experts treat these outputs as raw material, further refine them, and organize the results into precise statements and rigorous proofs. We instantiate this workflow in a main case study on the connection between manifold optimization and Grover's quantum search algorithm, where the pipeline identifies invariant subspaces and explores Grover-compatible retractions. The main case study uses the corresponding Grover-compatible convergence analysis, including an $O(\sqrt{N} \log(1/\varepsilon))$ PL-based bound established in the companion mathematical work, to illustrate the refinement stage of the workflow. Prompt records and reusable templates for implementing the workflow are provided. We further include a multi-oracle case study, document representative failed and corrected routes arising from this setting, and provide a structured failure-mode analysis.

cs.HC↗
arXiv2026

Learning Password Best Practices Through In-Task Instruction

Users often make security- and privacy-relevant decisions without a clear understanding of the rules that govern safe behavior. We introduce pedagogical friction, a design approach that inserts brief, instructional interactions at the moment of action. We evaluate this approach in the context of password creation, a familiar task with clear quality criteria. We conducted a randomized study with 128 participants across four interface conditions that varied the depth and interactivity of guidance. We assessed three outcomes: (1) rule compliance in a subsequent password task without guidance, (2) accuracy on survey questions tied to password rules, and (3) behavior-knowledge alignment, which captures whether participants who correctly followed a rule also recognized it on the survey. Across the guided conditions, participants corrected most rule violations in the follow-up task and showed high behavior-knowledge alignment. Survey results suggested clearer advantages for some rule types, especially symbol related questions. These results position pedagogical friction as a lightweight intervention for security- and privacy-critical interfaces.

cs.HC↗