arXiv subjects

Róisín Luo

Publications and source records attributed to Róisín Luo.

9 recordsLinked to original sources

arXiv2026

ZetaGPT: A Reference Implementation of Positional--Encoding--Free State--Space--Attention Language Models

Transformer-based language models rely on self-attention, whose computation is permutation-equivariant and therefore lacks an intrinsic mechanism for representing token order. Existing architectures address this limitation by explicitly incorporating positional information through learned positional embeddings or hand-crafted positional encodings, such as rotary positional encoding (RoPE), treating positional information as an architecturally acquired capability rather than an inherent property of the model. Motivated by the pursuit of positional-encoding-free architectures, this work explores a language model architecture that integrates causal state-space equations to implicitly encode positional information before attention computation. Specifically, each model block applies a causal state-space equation before self-attention, allowing recurrent state dynamics to encode sequential information into token representations. Consequently, subsequent attention layers operate on position-aware representations without requiring explicit positional encodings while retaining the expressive modeling capacity of self-attention. We present \textsc{ZetaGPT}, a compact hybrid language model designed for research, rapid prototyping, algorithm verification, and educational applications. In addition to the proposed architecture, \textsc{ZetaGPT} provides a fully open-source, end-to-end training pipeline encompassing dataset construction, tokenizer training, pretraining, supervised fine-tuning, reinforcement learning from human feedback (RLHF), and chain-of-thought (CoT) reasoning via pure reinforcement learning. To the best of our knowledge, \textsc{ZetaGPT} is the first open-source small language model without explicit positional encoding and establishes a compact, reproducible reference implementation for the development and empirical study of positional-encoding-free language models.

cs.CL↗

arXiv2026

Lipschitz Continuity in Deep Learning: A Systematic Review of Theoretical Foundations, Estimation Methods, Regularization Approaches, and Certifiable Robustness

Lipschitz continuity is a fundamental property of neural networks that characterizes their sensitivity to input perturbations. It plays a pivotal role in deep learning, governing \textbf{robustness}, \textbf{generalization} and \textbf{optimization dynamics}. Despite its importance, research on Lipschitz continuity is scattered across various domains, lacking a unified perspective. This paper addresses this gap by providing a systematic review of Lipschitz continuity in deep learning. We explore its \textbf{theoretical foundations}, \textbf{estimation methods}, \textbf{regularization approaches}, and \textbf{certifiable robustness}. By reviewing existing research through the lens of Lipschitz continuity, this survey serves as a comprehensive reference for researchers and practitioners seeking a deeper understanding of Lipschitz continuity and its implications in deep learning.

stat.ML↗

arXiv2026

Principles of Lipschitz continuity in neural networks

Deep learning has achieved remarkable success across a wide range of domains, significantly expanding the frontiers of what is achievable in artificial intelligence. Yet, despite these advances, critical challenges remain -- most notably, ensuring robustness to small input perturbations and generalization to out-of-distribution data. These critical challenges underscore the need to understand the underlying fundamental principles that govern robustness and generalization. Among the theoretical tools available, Lipschitz continuity plays a pivotal role in governing the fundamental properties of neural networks related to robustness and generalization. It quantifies the worst-case sensitivity of network's outputs to small input perturbations. While its importance is widely acknowledged, prior research has predominantly focused on empirical regularization approaches based on Lipschitz constraints, leaving the underlying principles less explored. This thesis seeks to advance a principled understanding of the principles of Lipschitz continuity in neural networks within the paradigm of machine learning, examined from two complementary perspectives: an internal perspective -- focusing on the temporal evolution of Lipschitz continuity in neural networks during training (i.e., training dynamics); and an external perspective -- investigating how Lipschitz continuity modulates the behavior of neural networks with respect to features in the input data, particularly its role in governing frequency signal propagation (i.e., modulation of frequency signal propagation).

cs.LG↗

arXiv2026

A Stochastic--Geometric Theory of Scaling Laws in Grokking

Delayed generalization (\ie~grokking) refers to the phenomenon in which a neural network fits its training data early in training but only begins to generalize after a prolonged delay, often through an abrupt transition. Despite extensive empirical study, its underlying mechanism remains poorly understood. In this work, we first theoretically characterize a shell--core topological configuration of the reachable solution space induced by Adam's optimization dynamics with weight-shrinkage regularization, supported by empirical evidence. This optimization-induced topological configuration gives rise to grokking. In model's parameter space, random initialization solutions concentrate on a thin outer spherical shell, enclosing another spherical shell of memorization solutions, which in turn contains a core corresponding to the generalization solutions. Leveraging stopping-time theory, we then analyze the geometry of this topological configuration and the solution transition time at which optimization trajectories escape the memorization manifold and first reach the boundary of the generalization manifold. Our theoretical analysis derives grokking scaling laws for the learning rate, batch size, and $\ell_2$ regularization coefficient, which are further validated through experiments and shown to recover results from prior literature.

stat.ML↗

arXiv2026

Interpreting Global Perturbation Robustness of Image Models using Axiomatic Spectral Importance Decomposition

Perturbation robustness evaluates the vulnerabilities of models, arising from a variety of perturbations, such as data corruptions and adversarial attacks. Understanding the mechanisms of perturbation robustness is critical for global interpretability. We present a model-agnostic, global mechanistic interpretability method to interpret the perturbation robustness of image models. This research is motivated by two key aspects. First, previous global interpretability works, in tandem with robustness benchmarks, e.g. mean corruption error (mCE), are not designed to directly interpret the mechanisms of perturbation robustness within image models. Second, we notice that the spectral signal-to-noise ratios (SNR) of perturbed natural images exponentially decay over the frequency. This power-law-like decay implies that: Low-frequency signals are generally more robust than high-frequency signals -- yet high classification accuracy can not be achieved by low-frequency signals alone. By applying Shapley value theory, our method axiomatically quantifies the predictive powers of robust features and non-robust features within an information theory framework. Our method, dubbed as \textbf{I-ASIDE} (\textbf{I}mage \textbf{A}xiomatic \textbf{S}pectral \textbf{I}mportance \textbf{D}ecomposition \textbf{E}xplanation), provides a unique insight into model robustness mechanisms. We conduct extensive experiments over a variety of vision models pre-trained on ImageNet to show that \textbf{I-ASIDE} can not only \textbf{measure} the perturbation robustness but also \textbf{provide interpretations} of its mechanisms.

cs.AI↗

arXiv2026

Reclaiming Residual Knowledge: A Novel Paradigm to Low-Bit Quantization

This paper explores a novel paradigm in low-bit (i.e. 4-bits or lower) quantization, differing from existing state-of-the-art methods, by framing optimal quantization as an architecture search problem within convolutional neural networks (ConvNets). Our framework, dubbed \textbf{CoRa} (Optimal Quantization Residual \textbf{Co}nvolutional Operator Low-\textbf{Ra}nk Adaptation), is motivated by two key aspects. Firstly, quantization residual knowledge, i.e. the lost information between floating-point weights and quantized weights, has long been neglected by the research community. Reclaiming the critical residual knowledge, with an infinitesimal extra parameter cost, can reverse performance degradation without training. Secondly, state-of-the-art quantization frameworks search for optimal quantized weights to address the performance degradation. Yet, the vast search spaces in weight optimization pose a challenge for the efficient optimization in large models. For example, state-of-the-art BRECQ necessitates $2 \times 10^4$ iterations to quantize models. Fundamentally differing from existing methods, \textbf{CoRa} searches for the optimal architectures of low-rank adapters, reclaiming critical quantization residual knowledge, within the search spaces smaller compared to the weight spaces, by many orders of magnitude. The low-rank adapters approximate the quantization residual weights, discarded in previous methods. We evaluate our approach over multiple pre-trained ConvNets on ImageNet. \textbf{CoRa} achieves comparable performance against both state-of-the-art quantization-aware training and post-training quantization baselines, in $4$-bit and $3$-bit quantization, by using less than $250$ iterations on a small calibration set with $1600$ images. Thus, \textbf{CoRa} establishes a new state-of-the-art in terms of the optimization efficiency in low-bit quantization.

cs.CV↗

arXiv2025

Optimization-Induced Dynamics of Lipschitz Continuity in Neural Networks

Lipschitz continuity characterizes the worst-case sensitivity of neural networks to small input perturbations; yet its dynamics (i.e. temporal evolution) during training remains under-explored. We present a rigorous mathematical framework to model the temporal evolution of Lipschitz continuity during training with stochastic gradient descent (SGD). This framework leverages a system of stochastic differential equations (SDEs) to capture both deterministic and stochastic forces. Our theoretical analysis identifies three principal factors driving the evolution: (i) the projection of gradient flows, induced by the optimization dynamics, onto the operator-norm Jacobian of parameter matrices; (ii) the projection of gradient noise, arising from the randomness in mini-batch sampling, onto the operator-norm Jacobian; and (iii) the projection of the gradient noise onto the operator-norm Hessian of parameter matrices. Furthermore, our theoretical framework sheds light on such as how noisy supervision, parameter initialization, batch size, and mini-batch sampling trajectories, among other factors, shape the evolution of the Lipschitz continuity of neural networks. Our experimental results demonstrate strong agreement between the theoretical implications and the observed behaviors.

cs.LG↗

arXiv2025

Higher-Order Singular-Value Derivatives of Rectangular Real Matrices

We present a theoretical framework for deriving the general $n$-th order Fréchet derivatives of singular values in real rectangular matrices, by leveraging reduced resolvent operators from Kato's analytic perturbation theory for self-adjoint operators. Deriving closed-form expressions for higher-order derivatives of singular values is notoriously challenging through standard matrix-analysis techniques. To overcome this, we treat a real rectangular matrix as a compact operator on a finite-dimensional Hilbert space, and embed the rectangular matrix into a block self-adjoint operator so that non-symmetric perturbations are captured. Applying Kato's asymptotic eigenvalue expansion to this construction, we obtain a general, closed-form expression for the infinitesimal $n$-th order spectral variations. Specializing to $n=2$ and deploying on a Kronecker-product representation with matrix convention yield the Hessian of a singular value, not found in literature. By bridging abstract operator-theoretic perturbation theory with matrices, our framework equips researchers with a practical toolkit for higher-order spectral sensitivity studies in random matrix applications (e.g., adversarial perturbation in deep learning).

stat.ML↗

arXiv2025

Sampling Matters in Explanations: Towards Trustworthy Attribution Analysis Building Block in Visual Models through Maximizing Explanation Certainty

Image attribution analysis seeks to highlight the feature representations learned by visual models such that the highlighted feature maps can reflect the pixel-wise importance of inputs. Gradient integration is a building block in the attribution analysis by integrating the gradients from multiple derived samples to highlight the semantic features relevant to inferences. Such a building block often combines with other information from visual models such as activation or attention maps to form ultimate explanations. Yet, our theoretical analysis demonstrates that the extent to the alignment of the sample distribution in gradient integration with respect to natural image distribution gives a lower bound of explanation certainty. Prior works add noise into images as samples and the noise distributions can lead to low explanation certainty. Counter-intuitively, our experiment shows that extra information can saturate neural networks. To this end, building trustworthy attribution analysis needs to settle the sample distribution misalignment problem. Instead of adding extra information into input images, we present a semi-optimal sampling approach by suppressing features from inputs. The sample distribution by suppressing features is approximately identical to the distribution of natural images. Our extensive quantitative evaluation on large scale dataset ImageNet affirms that our approach is effective and able to yield more satisfactory explanations against state-of-the-art baselines throughout all experimental models.

cs.CV↗