measure spectral concentration

This work addresses the gap between algorithmic prototypes and efficient implementations in scientific research by proposing a lightweight approach to translate statistical and machine learning algorithms—such as kernel ridge regression and stochastic gradient descent matrix factorization—from mathematical formulations into readable, high-performance C++ code. Leveraging the Eigen template library for core linear algebra operations—including kernel matrix construction, regularized solvers, and vectorized updates—the implementation seamlessly integrates into the Python ecosystem via pybind11, enabling efficient interoperability with NumPy arrays. The project provides concise, reproducible code examples that encapsulate common computational patterns in research, significantly lowering the barrier for researchers to adopt C++ for high-performance development while balancing performance, readability, and usability.

This study addresses the underexplored challenge of effectively visualizing bivariate distributions on graph edges under the spatial constraints of adjacency matrix layouts. The work proposes a novel approach that encodes edge-wise bivariate distributions using two statistical summaries: central tendency and dispersion. Through a preregistered crowdsourced experiment, the authors systematically evaluate four compact encoding designs—bivariate color mapping, embedded bar charts, and two superimposed mark types combining area or angle with color. Results demonstrate that the area-based superimposed marks and embedded bar charts yield the best overall performance, while angle-based encodings show moderate but inconsistent accuracy, and bivariate color mappings perform significantly worse. This research provides empirical evidence and practical guidance for designing visualizations of bivariate edge data in graph structures.

This work addresses the lack of a unified theoretical understanding of existing knowledge transfer mechanisms—such as knowledge distillation and weak-to-strong generalization—by proposing a cohesive spectral framework within high-dimensional linear regression. By analyzing the dynamics of stochastic gradient descent (SGD), the study introduces two key mechanisms: “spectral horizon expansion” and “spectral denoising.” These mechanisms jointly elucidate how diverse transfer paradigms leverage implicit regularization and heterogeneous spectral learning rates to enhance the model’s capacity to learn high-frequency signals while suppressing optimization-induced noise. The paper establishes the first unified spectral analysis framework for knowledge transfer, revealing common principles underlying the effectiveness of various transfer strategies.

This work addresses the bias in heavy-tailedness estimation arising from aspect-ratio disparities in weight matrices during spectral analysis of deep neural networks. We propose Fixed-Aspect-Ratio Matrix Sampling (FARMS), a method that mitigates this bias by randomly sampling submatrices with fixed aspect ratios, modeling their empirical spectral density (ESD), and fitting α-stable distributions to estimate the tail index. FARMS is the first framework to systematically eliminate the intrinsic aspect-ratio-induced bias in spectral statistics. It exhibits strong cross-architecture and cross-task robustness, significantly improving model diagnostics and layer-wise hyperparameter allocation. Extensive validation across computer vision (CV), scientific machine learning (SciML), and large language model (LLM) pruning tasks confirms its effectiveness: when applied to LLaMA-7B pruning, FARMS reduces perplexity by 17.3%, outperforming state-of-the-art methods.

In multi-view subspace learning, theoretical guarantees for distinguishing shared and individual signal subspaces from high-dimensional noisy data remain lacking. This paper establishes, for the first time, necessary and sufficient conditions for subspace separability and develops a rigorous theoretical framework based on spectral perturbation analysis of projection matrix products. Integrating rotational bootstrap with random matrix theory, we propose a parameter-free, interpretable method that automatically partitions subspaces into three categories—shared, individual, and noise—without manual tuning. Leveraging principal angle analysis and diagnostic visualization, our approach enhances estimation robustness. Extensive simulations demonstrate substantial improvements in estimation accuracy for both joint and individual subspaces over state-of-the-art methods. On real-world multi-omics colorectal cancer and murine nutritional genomics datasets, downstream classification and prediction performance is significantly enhanced.

Traditional graph representation learning treats the latent dimension as a fixed hyperparameter, which inadequately captures the model’s true capacity and suffers from non-identifiability due to rotational and scaling ambiguities in latent factors. This work proposes Spectra, a method that uses the spectral distribution of normalized positive-definite kernels as the fundamental analytical unit. By leveraging Shannon effective rank as a dynamic measure to quantify and control model capacity, Spectra reinterprets capacity not as an external hyperparameter but as an intrinsic property of the model itself. Through spectral prefix extraction, trace-normalized kernel matrices, and bisection-based optimization, Spectra uncovers performance–capacity trade-offs across diverse real-world networks, achieves competitive results against strong baselines in link prediction, and enables the generation of aligned low-dimensional views at varying capacities from a single trained model.

Existing tools struggle to provide unified analysis of high-dimensional, spatially resolved multimodal spectral imaging data, often requiring analysts to switch between software and manually integrate results. This work proposes the first modality-agnostic visualization system that supports end-to-end integrated analysis, incorporating derivative-based preprocessing, hierarchical and landmark-guided embeddings, interactive and automated segmentation with cross-view shared state, spectral similarity search, and multimodal co-registration. By enabling synergistic exploration of multiscale embeddings and joint multimodal representations, the system successfully reproduces tissue compartmentalization, identifies pigment constituents, and fuses molecular and elemental imaging across three real-world chemical analysis cases. This integrated approach substantially reduces software switching and enhances expert analytical efficiency and insight generation.

This work addresses the limitations of existing explainable AI methods in interpreting spectral machine learning models, which often overlook chemically meaningful contiguous spectral regions. To bridge this gap, the authors propose SMX, a novel framework that incorporates chemical prior knowledge into model interpretation by leveraging expert-defined spectral intervals to deliver global, post-hoc, and model-agnostic explanations. SMX integrates principal component analysis (PCA), logical predicates, and perturbation analysis, and employs a directed weighted graph to aggregate region-wise importance scores. Furthermore, it introduces a threshold-based spectral reconstruction technique to intuitively visualize explanations directly in the original spectral domain. Experimental results across eight real-world datasets—comprising six X-ray fluorescence (XRF) and two gamma-ray spectra—as well as one synthetic benchmark demonstrate that SMX significantly outperforms current approaches in both explanatory fidelity and interpretability.

Current AI evaluation benchmarks often report numerous scores without verifying their informational independence, leading to redundancy and potential misinterpretation. This work proposes Effective Dimensionality (ED) as a diagnostic metric to quantify the informational breadth of benchmarks, enabling the first rapid estimation of an upper bound on the number of independent evaluation dimensions under population-level conditions. Leveraging 22 benchmarks across 8 domains and over 8,400 model evaluations, we develop a four-step diagnostic pipeline and reference atlas. ED is computed via the participation ratio of centered score spectra and validated through null models, reliability and saturation analyses, and controlled experiments. Our findings reveal substantial redundancy in widely used benchmarks: the Open LLM Leaderboard effectively captures only about 1.7 independent dimensions, while BBH and MMLU-Pro are highly interchangeable (ρ = 0.96), with benchmark informational breadth varying by more than twentyfold.

This work addresses the limitations of conventional interpretability methods in capturing the key dynamic directions governing grokking. The authors propose a novel approach that analyzes the spectral margin of training dynamics to uncover low-dimensional functional patterns emerging in the input domain, whose structure is dictated by the algebraic symmetries of the task. This framework transcends existing paradigms rooted in local structures of parameter or feature spaces. By integrating spectral analysis, Fourier and discrete logarithm basis transformations, multi-task training, and variance concentration metrics—and cross-validating findings via head attribution and sparse autoencoders—the method achieves up to a 5.9× improvement in directional concentration across tasks such as modular addition, multiplication, subtraction, and $x^2 + y^2$. Notably, it reveals for the first time that functional patterns in multi-task settings exhibit compositional inheritance.