Abstract--Kurtosis-based Independent Component Analysis (ICA) weakens in wide, balanced mixtures. We also show that purification--selecting m R sign-consistent sources--restores R-independent contrast Ω(1/m), with a simple data-driven heuristic. Synthetic experiments validate the predicted decay, the T crossover, and contrast recovery. Independent Component Analysis (ICA) recovers statistically independent latent sources from linear mixtures and is identifiable whenever at most one source is Gaussian [1]. Excess kurtosis--the standardized fourth cumulant--is a central contrast function [9], and kurtosis-type nonlinearities remain standard in FastICA.

Low-precision arithmetic is one of the key techniques for reducing deep neural networks computational costs and fitting larger networks into smaller devices.

We study the effect of high-order statistics of data on the learning dynamics of neural networks (NNs) by using a moment-controllable non-Gaussian data model. Considering the expressivity of two-layer neural networks, we first construct the data model as a generative two-layer NN where the activation function is expanded by using Hermite polynomials. This allows us to achieve interpretable control over high-order cumulants such as skewness and kurtosis through the Hermite coefficients while keeping the data model realistic. Using samples generated from the data model, we perform controlled online learning experiments with a two-layer NN. Our results reveal a moment-wise progression in training: networks first capture low-order statistics such as mean and covariance, and progressively learn high-order cumulants. Finally, we pretrain the generative model on the Fashion-MNIST dataset and leverage the generated samples for further experiments. The results of these additional experiments confirm our conclusions and show the utility of the data model in a real-world scenario. Overall, our proposed approach bridges simplified data assumptions and practical data complexity, which offers a principled framework for investigating distributional effects in machine learning and signal processing.

Large language models require significant computational resources for deployment, making quantization essential for practical applications. However, the main obstacle to effective quantization lies in systematic outliers in activations and weights, which cause substantial LLM performance degradation, especially at low-bit settings. While existing transformation-based methods like affine and rotation transformations successfully mitigate outliers, they apply the homogeneous transformation setting, i.e., using the same transformation types across all layers, ignoring the heterogeneous distribution characteristics within LLMs. In this paper, we propose an adaptive transformation selection framework that systematically determines optimal transformations on a per-layer basis. To this end, we first formulate transformation selection as a differentiable optimization problem to achieve the accurate transformation type for each layer. However, searching for optimal layer-wise transformations for every model is computationally expensive. To this end, we establish the connection between weight distribution kurtosis and accurate transformation type. Specifically, we propose an outlier-guided layer selection method using robust $z$-score normalization that achieves comparable performance to differentiable search with significantly reduced overhead. Comprehensive experiments on LLaMA family models demonstrate that our adaptive approach consistently outperforms the widely-used fixed transformation settings. For example, our method achieves an improvement of up to 4.58 perplexity points and a 2.11% gain in average six-task zero-shot accuracy under aggressive W3A3K2V2 quantization settings for the LLaMA-3-8B model compared to the current best existing method, FlatQuant, demonstrating the necessity of heterogeneous transformation selection for optimal LLM quantization.

Existing strategies for finite-armed stochastic bandits mostly depend on a parameter of scale that must be known in advance. Sometimes this is in the form of a bound on the payoffs, or the knowledge of a variance or subgaussian parameter. The notable exceptions are the analysis of Gaussian bandits with unknown mean and variance by Cowan et al. [2015] and of uniform distributions with unknown support [Cowan and Katehakis, 2015]. The results derived in these specialised cases are generalised here to the non-parametric setup, where the learner knows only a bound on the kurtosis of the noise, which is a scale free measure of the extremity of outliers.

Robust validation metrics remain essential in contemporary deep learning, not only to detect overfitting and poor generalization, but also to monitor training dynamics. In the supervised classification setting, we investigate whether interactions between training data and model weights can yield such a metric that both tracks generalization during training and attributes performance to individual training samples. We introduce Gradient-Weight Alignment (GWA), quantifying the coherence between per-sample gradients and model weights. We show that effective learning corresponds to coherent alignment, while misalignment indicates deteriorating generalization. GWA is efficiently computable during training and reflects both sample-specific contributions and dataset-wide learning dynamics. Extensive experiments show that GWA accurately predicts optimal early stopping, enables principled model comparisons, and identifies influential training samples, providing a validation-set-free approach for model analysis directly from the training data.

Current frontier large-language models rely on reasoning to achieve state-of-the-art performance. Many existing interpretability are limited in this area, as standard methods have been designed to study single forward passes of a model rather than the multi-token computational steps that unfold during reasoning. We argue that analyzing reasoning traces at the sentence level is a promising approach to understanding reasoning processes. We introduce a black-box method that measures each sentence's counterfactual importance by repeatedly sampling replacement sentences from the model, filtering for semantically different ones, and continuing the chain of thought from that point onwards to quantify the sentence's impact on the distribution of final answers. We discover that certain sentences can have an outsized impact on the trajectory of the reasoning trace and final answer. We term these sentences \textit{thought anchors}. These are generally planning or uncertainty management sentences, and specialized attention heads consistently attend from subsequent sentences to thought anchors. We further show that examining sentence-sentence causal links within a reasoning trace gives insight into a model's behavior. Such information can be used to predict a problem's difficulty and the extent different question domains involve sequential or diffuse reasoning. As a proof-of-concept, we demonstrate that our techniques together provide a practical toolkit for analyzing reasoning models by conducting a detailed case study of how the model solves a difficult math problem, finding that our techniques yield a consistent picture of the reasoning trace's structure. We provide an open-source tool (thought-anchors.com) for visualizing the outputs of our methods on further problems. The convergence across our methods shows the potential of sentence-level analysis for a deeper understanding of reasoning models.