Regularization

Whole-body Positron Emission Tomography (PET) registration is essential for multi-parametric tumor characterization and assessment of metastatic disease progression. In deep learning-based deformable registration, the dense displacement field (DDF) regularizer is crucial for stabilizing optimization and preventing unrealistic deformations in large 3D volumes. A key challenge in whole-body deformable registration is anatomical heterogeneity, rigid structures (e.g., bones) should undergo stronger regularization, whereas soft tissues require more flexible deformation and weaker constraints. In this work, we propose a simple yet effective CT-guided spatially-varying regularization strategy for whole-body cross-tracer deformable PET registration. The key idea is to use the paired CT volume from the PET/CT acquisition to construct a voxel-wise regularization map for the DDF, replacing the conventional single global regularization weight. This yields anatomy-adaptive regularization strength across rigid and soft tissues. The proposed method is evaluated on a real clinical cross-tracer PET/CT dataset of 296 patients involving 18F-PSMA and 18F-FDG, showing that the proposed method achieves statistically significant improvements over weakly-supervised registration baseline in both whole-body registration performance and organ-wise alignment.

Client contribution estimation in Federated Learning is necessary for identifying clients' importance and for providing fair rewards. Current methods often rely on server-side validation data or self-reported client information, which can compromise privacy or be susceptible to manipulation. We introduce a data-free signal based on the matrix von Neumann (spectral) entropy of the final-layer updates, which measures the diversity of the information contributed. We instantiate two practical schemes: (i) SpectralFed, which uses normalized entropy as aggregation weights, and (ii) SpectralFuse, which fuses entropy with class-specific alignment via a rank-adaptive Kalman filter for per-round stability. Across CIFAR-10/100 and the naturally partitioned FEMNIST and FedISIC benchmarks, entropy-derived scores show a consistently high correlation with standalone client accuracy under diverse non-IID regimes - without validation data or client metadata. We compare our results with data-free contribution estimation baselines and show that spectral entropy serves as a useful indicator of client contribution.

We propose a neural regularization method that refines the noisy 3D semantic field produced by lifting multi-view inconsistent 2D features, in order to obtain an accurate and robust 3D semantic Gaussian Splatting. The 2D features extracted from vision foundation models suffer from multi-view inconsistency due to a lack of cross-view constraints. Lifting these inconsistent features directly into 3D Gaussians results in a noisy semantic field, which degrades the performance of downstream tasks. Previous methods either focus on obtaining consistent multi-view features in the preprocessing stage or aim to mitigate noise through improved optimization strategies, often at the cost of increased preprocessing time or expensive computational overhead. In contrast, we introduce a variance-aware conditional MLP that operates directly on the 3D Gaussians, leveraging their geometric and appearance attributes to correct semantic errors in 3D space. Experiments on different datasets show that our method enhances the accuracy of lifted semantics, providing an efficient and effective approach to robust 3D semantic Gaussian Splatting.

Bayesian Optimal Experimental Design (BOED) provides a rigorous framework for decision-making tasks in which data acquisition is often the critical bottleneck, especially in resource-constrained settings. Traditionally, BOED typically selects designs by maximizing expected information gain (EIG), commonly defined through the Kullback-Leibler (KL) divergence. However, classical evaluation of EIG often involves challenging nested expectations, and even advanced variational methods leave the underlying log-density-ratio objective unchanged. As a result, support mismatch, tail underestimation, and rare-event sensitivity remain intrinsic concerns for KL-based BOED. To address these fundamental bottlenecks, we introduce an IPM-based BOED framework that replaces density-based divergences with integral probability metrics (IPMs), including the Wasserstein distance, Maximum Mean Discrepancy, and Energy Distance, resulting in a highly flexible plug-and-play BOED framework. We establish theoretical guarantees showing that IPM-based utilities provide stronger geometry-aware stability under surrogate-model error and prior misspecification than classical EIG-based utilities. We also validate the proposed framework empirically, demonstrating that IPM-based designs yield highly concentrated credible sets. Furthermore, by extending the same sample-based BOED template in a plug-and-play manner to geometry-aware discrepancies beyond the IPM class, illustrated by a neural optimal transport estimator, we achieve accurate optimal designs in high-dimensional settings where conventional nested Monte Carlo estimators and advanced variational methods fail.

Neural networks exhibit a remarkable degree of representational convergence across diverse architectures, training objectives, and even data modalities. This convergence is predictive of alignment with brain representation. A recent hypothesis suggests this arises from learning the underlying structure in the environment in similar ways. However, it is unclear how individual stimuli elicit convergent representations across networks. An image can be perceived in multiple ways and expressed differently using words. Here, we introduce a methodology based on the Generalized Procrustes Algorithm to measure intra-modal representational convergence at the single-stimulus level. We applied this to vision models with distinct training objectives, selecting stimuli based on their degree of alignment (intra-modal dispersion). Crucially, we found that this intra-modal dispersion strongly modulates alignment between vision and language models (cross-modal convergence). Specifically, stimuli with low intra-modal dispersion (high agreement among vision models) elicited significantly higher cross-modal alignment than those with high dispersion, by up to a factor of two (e.g., in pairings of DINOv2 with language models). This effect was robust to stimulus selection criteria and generalized across different pairings of vision and language models. Measuring convergence at the single-stimulus level provides a path toward understanding the sources of convergence and divergence across modalities, and between neural networks and human neural representations.

When approximating an intractable density via variational inference (VI) the variational family is typically chosen as a simple parametric family that very likely does not contain the target. This raises the question: Under which conditions can we recover characteristics of the target despite misspecification? In this work, we extend previous theoretical results on robust VI with location-scale families under target symmetries in two substantial ways: (1) We open them up to a wider range of divergences by providing sufficient conditions for exact recovery of the target mean and correlation matrix when using the forward Kullback-Leibler divergence and $α$-divergences. (2) By doing so, we find that we can drop the restrictive assumption of a log-concave target made in previous work, allowing us to give guarantees for a wider range of targets, including multi-modal ones. In our experiments, we show how our guarantees can serve as guidelines for the choice of the variational family and $α$-value and we illustrate on a diverse set of examples how and why optimization can fail in the absence of our sufficient conditions.

Trajectory Inference (TI) seeks to recover latent dynamical processes from snapshot data, where only independent samples from time-indexed marginals are observed. In applications such as single-cell genomics, destructive measurements make path-space laws non-identifiable from finitely many marginals, leaving held-out marginal prediction as the dominant but limited evaluation protocol. We introduce a general framework for estimating the Kullback-Leibler divergence (KL) divergence between probability measures on function space, yielding a tractable, data-driven estimator that is scalable to realistic snapshot datasets. We validate the accuracy of our estimator on a benchmark suite, where the estimated functional KL closely matches the analytic KL. Applying this framework to synthetic and real scRNA-seq datasets, we show that current evaluation metrics often give inconsistent assessments, whereas path-space KL enables a coherent comparison of trajectory inference methods and exposes discrepancies in inferred dynamics, especially in regions with sparse or missing data. These results support functional KL as a principled criterion for evaluating trajectory inference under partial observability.

We present a systematic empirical study of transformer compression through over 40 experiments on GPT-2 (124M parameters) and Mistral 7B (7.24B parameters). Our analysis covers spectral compression, block-level function replacement, rotation-based quantization, activation geometry, and adaptive early exit. We identify five structural properties relevant to compression. (1) Variance is not importance: high-variance activation directions are approximately 96 percent uncorrelated with predictive directions (measured via CCA), and projecting onto these subspaces preserves over 90 percent of variance while degrading perplexity. (2) Block linearity is conditional: transformer blocks are approximately linear (R^2 ~ 0.95 on GPT-2, 0.93 on Mistral block 31) only under the correct upstream distribution; modifying earlier blocks induces distribution shift that degrades downstream approximations. (3) The reconstruction wall: approaches that factor weights into quantized components amplify errors through cross-terms, making direct quantization strictly superior. (4) Linearity increases with depth: Mistral 7B exhibits a progression from R^2 = 0.17 (block 0) to R^2 = 0.93 (block 31), indicating a division between nonlinear feature construction and linear refinement. (5) Approximately 30 percent of tokens are computationally easy, confirmed via exit heads and KL divergence sensitivity. We demonstrate that single-block linear replacement achieves 34x compression with a 1.71 perplexity increase on the final block of Mistral 7B, while multi-block replacement fails due to residual error accumulation and distribution shift. These findings suggest fundamental limits to static post-training compression and motivate adaptive, per-token computation as a more effective direction.

Dropout is a widely used regularization technique in deep learning, but its effects are typically realized through stochastic masking rather than explicit optimization objectives. We propose a deterministic formulation that expresses dropout as an additive regularizer directly incorporated into the training loss. The framework derives explicit regularization terms for Transformer architectures, covering attention query, key, value, and feed-forward components with independently controllable strengths. This formulation removes reliance on stochastic perturbations while providing clearer and fine-grained control over regularization strength. Experiments across image classification, temporal action detection, and audio classification show that explicit dropout matches or outperforms conventional implicit methods, with consistent gains when applied to attention and feed-forward network layers. Ablation studies demonstrate stable performance and controllable regularization through regularization coefficients and dropout rates. Overall, explicit dropout offers a practical and interpretable alternative to stochastic regularization while maintaining architectural flexibility across diverse tasks.

In this paper, it is shown, for the first time, that centralized performance is achievable in decentralized learning without sharing the local datasets. Specifically, when clients adopt an empirical risk minimization with relative-entropy regularization (ERM-RER) learning framework and a forward-backward communication between clients is established, it suffices to share the locally obtained Gibbs measures to achieve the same performance as that of a centralized ERM-RER with access to all the datasets. The core idea is that the Gibbs measure produced by client~$k$ is used, as reference measure, by client~$k+1$. This effectively establishes a principled way to encode prior information through a reference measure. In particular, achieving centralized performance in the decentralized setting requires a specific scaling of the regularization factors with the local sample sizes. Overall, this result opens the door to novel decentralized learning paradigms that shift the collaboration strategy from sharing data to sharing the local inductive bias via the reference measures over the set of models.

Self-play fine-tuning enables large language models to improve beyond supervised fine-tuning without additional human annotations by contrasting annotated responses with self-generated ones. Many existing methods rely on a fixed divergence regime. SPIN is closely related to a KL-based regime, SPACE to a Jensen-Shannon-style objective via noise contrastive estimation, and SPIF to $χ^2$-regularized self-play. Since these divergences exhibit different strengths depending on the distributional gap between model and target, no single choice appears to provide favorable learning dynamics across training stages. We propose IRIS (Interpolative Rényi Iterative Self-play), a Rényi-based self-play fine-tuning framework with a continuously adjustable objective. IRIS decomposes into two independent tilted risk terms over annotated and synthetic data, with exponential importance weights controlled by the order parameter $α$. We show that several self-play objectives can be interpreted as limiting or representative regimes at particular values of $α$, providing a unified theoretical perspective on these methods. An adaptive order schedule further adjusts $α$ to the distributional gap, shifting from sharper importance weighting early in training to smoother refinement near convergence. Theoretically, we establish the fixed-point property of IRIS and analyze how $α$ controls gradient concentration. Experiments on Zephyr-7B and Qwen2.5-3B across ten benchmarks show that IRIS improves upon baselines, reaching 44.57% average score with gains across iterations. In our setting, IRIS with only 26$k$ annotated samples surpasses standard supervised fine-tuning trained on the full 200$k$ dataset.

We consider the problem of sampling from a probability distribution $π$. It is well known that this can be written as an optimisation problem over the space of probability distributions in which we aim to minimise the Kullback--Leibler divergence from $π$. We consider the effect of replacing $π$ with a sequence of moving targets $(π_t)_{t\ge0}$ defined via geometric tempering on the Wasserstein and Fisher--Rao gradient flows. We show that convergence occurs exponentially in continuous time, providing novel bounds in both cases. We also consider popular time discretisations and explore their convergence properties. We show that in the Fisher--Rao case, replacing the target distribution with a geometric mixture of initial and target distribution never leads to a convergence speed up both in continuous time and in discrete time. Finally, we explore the gradient flow structure of tempered dynamics and derive novel adaptive tempering schedules.

Mode collapse remains a fundamental challenge in training generative adversarial networks (GANs). While existing works have primarily focused on inter-mode collapse, such as mode dropping, intra-mode collapse-where many latent variables map to the same or highly similar outputs-has received significantly less attention. In this work, we propose a pairing regularizer jointly optimized with the generator to mitigate the many-to-one collapse by enforcing local consistency between latent variables and generated samples. We show that the effect of pairing regularization depends on the dominant failure mode of training. In collapse-prone regimes with limited exploration, pairing encourages structured local exploration, leading to improved coverage and higher recall. In contrast, under stabilized training with sufficient exploration, pairing refines the generator's induced data density by discouraging redundant mappings, thereby improving precision without sacrificing recall. Extensive experiments on both toy distributions and real-image benchmarks demonstrate that the proposed regularizer effectively complements existing stabilization techniques by directly addressing intra-mode collapse.

Damage identification is a core task in structural health monitoring. In practice, however, its reliability is often compromised by confounding non-damage effects, such as variations in excitation and environmental conditions, which can induce changes comparable to or larger than those caused by structural damage. To address this challenge, this study proposes a self-supervised label-free disentangled representation learning framework for robust vibration-based structural damage identification. The proposed framework employs an autoencoder with two latent representations to learn directly from raw vibration acceleration signals. A self-supervised invariance regularization, implemented via Variance-Invariance-Covariance Regularization (VICReg), is imposed on one latent representation using baseline data where structural damage is assumed constant but operational and environmental conditions vary. In addition, a frequency-domain constraint is introduced to enforce agreement between the power spectral density reconstructed from the latent representation and that computed from the corresponding input time series. Together, these mechanisms promote disentanglement, enabling the learned representation to be sensitive to damage-related characteristics while remaining invariant to nuisance variability. The framework is trained in a fully end-to-end and label-free manner, requiring no prior information on damage, excitation, or environmental conditions, making it well-suited for real-world applications. Its effectiveness is validated on two distinct real-world vibration datasets, including a bridge and a gearbox. The results demonstrate robustness to operational variability, strong generalization capability, and good performance in both damage detection and quantification.

Recovering corrupted images is one of the most challenging problems in image processing. Among various restoration tasks, blind image deblurring has been extensively studied due to its practical importance and inherent difficulty. In this problem, both the point spread function (PSF) and the underlying latent sharp image must be estimated simultaneously. This problem cannot be solved directly due to its ill-posed nature. One powerful tool for solving such problems is total variation (TV) regularization. The $\ell_0$-norm regularization within the TV framework has been widely adopted to promote sparsity in image gradients or transform domains, leading to improved preservation of edges and fine structures. However, the use of the $\ell_0$-norm results in a highly nonconvex and computationally intractable optimization problem, which limits its practical applicability. To overcome these difficulties, we employ the minimax concave penalty (MCP), which promotes enhanced sparsity and provides a closer approximation to the $\ell_0$-norm. In addition, a reweighted $\ell_1$-norm regularization is incorporated to further reduce estimation bias and improve the preservation of fine image details and textures. After introducing the proposed model, a numerical algorithm is developed to solve the resulting optimization problem. The effectiveness of the proposed approach is then demonstrated through experimental evaluations on several test images.

Unification of automatic speech recognition (ASR) systems reduces development and maintenance costs, but training a single model to perform well in both offline and low-latency streaming settings remains challenging. We present a Unified ASR framework for Transducer (RNNT) training that supports both offline and streaming decoding within a single model, using chunk-limited attention with right context and dynamic chunked convolutions. To further close the gap between offline and streaming performance, we introduce an efficient Triton implementation of mode-consistency regularization for RNNT (MCR-RNNT), which encourages agreement across training modes. Experiments show that the proposed approach improves streaming accuracy at low latency while preserving offline performance and scaling to larger model sizes and training datasets. The proposed Unified ASR framework and the English model checkpoint are open-sourced.

Semi-supervised learning with manifold regularization is a classical framework for jointly learning from both labeled and unlabeled data, where the key requirement is that the support of the unknown marginal distribution has the geometric structure of a Riemannian manifold. Typically, the Laplace-Beltrami operator-based manifold regularization can be approximated empirically by the Laplacian regularization associated with the entire training data and its corresponding graph Laplacian matrix. However, the graph Laplacian matrix depends heavily on the prespecified similarity metric and may lead to inappropriate penalties when dealing with redundant or noisy input variables. To address the above issues, this paper proposes a new Semi-Supervised Meta Additive Model (S$^2$MAM) based on a bilevel optimization scheme that automatically identifies informative variables, updates the similarity matrix, and simultaneously achieves interpretable predictions. Theoretical guarantees are provided for S$^2$MAM, including the computing convergence and the statistical generalization bound. Experimental assessments across 4 synthetic and 12 real-world datasets, with varying levels and categories of corruption, validate the robustness and interpretability of the proposed approach.

Scaling critic capacity is a promising direction for improving off-policy reinforcement learning (RL). However, recent work shows that larger critics are prone to overfitting and instability in replay-based bootstrapped training. In this paper, we propose using Low-Rank Adaptation (LoRA) as a structural regularizer for critic learning. Our approach freezes randomly initialized base matrices and optimizes only the corresponding low-rank adapters, thereby constraining critic updates to a low-dimensional subspace. We evaluate our method across different off-policy RL algorithms, including SAC and FastTD3 based on different network architectures. Empirically, LoRA efficiently reduces critic loss during training and improves overall policy performance, achieving the best or competitive results on most tasks. Extensive experiments demonstrate that our low-rank updates provide a simple and effective form of structural regularization for critic learning in off-policy RL.

LLMs frequently generate fictitious yet convincing citations, often expressing high confidence even when the underlying reference is wrong. We study this failure across 9 models and 108{,}000 generated references, and find that author names fail far more often than other fields across all models and settings. Citation style has no measurable effect, while reasoning-oriented distillation degrades recall. Probes trained on one field transfer at near-chance levels to the others, suggesting that hallucination signals do not generalize across fields. Building on this finding, we apply elastic-net regularization with stability selection to neuron-level CETT values of Qwen2.5-32B-Instruct and identify a sparse set of field-specific hallucination neurons (FH-neurons). Causal intervention further confirms their role: amplifying these neurons increases hallucination, while suppressing them improves performance across fields, with larger gains in some fields. These results suggest a lightweight approach to detecting and mitigating citation hallucination using internal model signals alone.

In continual learning, the primary challenge is to learn new information without forgetting old knowledge. A common solution addresses this trade-off through regularization, penalizing changes to parameters critical for previous tasks. In most cases, this regularization term is directly added to the training loss and optimized with standard gradient descent, which blends learning and retention signals into a single update and does not explicitly separate essential parameters from redundant ones. As task sequences grow, this coupling can over-constrain the model, limiting forward transfer and leading to inefficient use of capacity. We propose a different approach that separates task learning from stability enforcement via operator splitting. The learning step focuses on minimizing the current task loss, while a proximal stability step applies a sparse regularizer to prune unnecessary parameters and preserve task-relevant ones. This turns the stability-plasticity into a negotiated update between two complementary operators, rather than a conflicting gradient. We provide theoretical justification for the splitting method on the continual-learning objective, and demonstrate that our proposed solver achieves state-of-the-art results on standard benchmarks, improving both stability and adaptability without the need for replay buffers, Bayesian sampling, or meta-learning components.

This paper studies how empirical dialogue-flow statistics can be incorporated into Next Dialogue Act Prediction (NDAP). A KL regularization term is proposed that aligns predicted act distributions with corpus-derived transition patterns. Evaluated on a 60-class German counselling taxonomy using 5-fold cross-validation, this improves macro-F1 by 9--42% relative depending on encoder and substantially improves dialogue-flow alignment. Cross-dataset validation on HOPE suggests that improvements transfer across languages and counselling domains. In systematic ablations across pretrained encoders and architectures, the findings indicate that transition regularization provides consistent gains and disproportionately benefits weaker baseline models. The results suggest that lightweight discourse-flow priors complement pretrained encoders, especially in fine-grained, data-sparse dialogue tasks.

Continual learning (CL) is concerned with learning multiple tasks sequentially without forgetting previously learned tasks. Despite substantial empirical advances over recent years, the theoretical development of CL remains in its infancy. At the heart of developing CL theory lies the challenge that the data distribution varies across tasks, and we argue that properly addressing this challenge requires understanding this variation--dependency among tasks. To explicitly model task dependency, we consider nonlinear regression tasks and propose the assumption that these tasks are dependent in such a way that the data of the current task is a nonlinear transformation of previous data. With this model and under natural assumptions, we prove statistical recovery guarantees (more specifically, bounds on estimation errors) for several CL paradigms in practical use, including experience replay with data-independent regularization and data-independent weights that balance the losses of tasks, replay with data-dependent weights, and continual learning with data-dependent regularization (e.g., knowledge distillation). To the best of our knowledge, our bounds are informative in cases where prior work gives vacuous bounds.

Recent text embedding models are often adapted to specialized domains via contrastive pre-finetuning (PFT) on a naive collection of scattered, heterogeneous tasks. However, this approach often introduces task-induced bias alongside domain knowledge, leading to uncontrolled representation shifts that distort the pretrained embedding geometry and cause substantial performance degradation. To address this issue, we propose REZE, a representation regularization framework that explicitly controls representation shift during embedding pre-finetuning. REZE operates on the relations of anchor-positive pairs and decomposes them in an eigenspace. It then measures task-wise dispersion along each eigencomponent to identify task-variant directions and applies adaptive soft-shrinkage to suppress task-induced noise while preserving task-invariant semantic structure, without inference-time overhead. Experiments across multiple embedding backbones and specialized benchmarks show that REZE outperforms standard pre-finetuning and isotropy-oriented post-hoc regularization in most settings, remaining stable where existing PFT variants collapse. Embedding space analyses further confirm that REZE induces controlled shifts aligned with the original embedding manifold, underscoring representation shift control as a key principle for robust embedding pre-finetuning under heterogeneous supervision.

Fine-tuning well-aligned large language models (LLMs) on new domains often degrades their safety alignment, even when using benign datasets. Existing safety alignment techniques primarily focus on pretraining, leaving fine-tuned models vulnerable to behavioral shifts. In this work, we introduce safety token regularization (STR), a lightweight method designed to preserve safety properties during fine-tuning. Our approach identifies salient tokens from rejection templates of well-aligned models and constrains their associated logits during training, preventing the loss of critical safety behaviors. Unlike reinforcement learning or preference optimization methods, STR requires minimal additional computation and seamlessly integrates with parameter-efficient fine-tuning techniques such as LoRA. Comprehensive experiments demonstrate that our approach achieves safety performance on par with state-of-the-art methods, while preserving task-specific utility and requiring minimal implementation overhead. Furthermore, we show that safety token regularization enhances training stability and overall performance beyond safety considerations alone. This work offers a practical and readily deployable strategy for continual safety alignment in fine-tuned LLMs.

Iterative alignment methods based on purely greedy updates are remarkably effective in practice, yet existing theoretical guarantees of $O(\log T)$ KL-regularized regret can seem pessimistic relative to their empirical performance. In this paper, we argue that this mismatch arises from the regret criterion itself: KL-regularized regret conflates the statistical cost of learning with the exploratory randomization induced by the softened training policy. To separate these effects, we study the traditional temperature-zero regret criterion, which evaluates only the top-ranked response at inference time. Under this decision-centric notion of performance, we prove that standard greedy online alignment methods, including online RLHF and online DPO, achieve constant $(O(1))$ cumulative regret. By isolating the cost of identifying the best response from the stochasticity induced by regularization, our results provide a sharper theoretical explanation for the practical superb efficiency of greedy alignment.

Task arithmetic provides an efficient, training-free way to edit pre-trained models, yet lacks a fundamental theoretical explanation for its success. The existing concept of ``weight disentanglement" describes the ideal outcome of non-interfering task composition but does not reveal its underlying cause. Crucially, what intrinsic properties of the pre-trained model ($θ_0$) or the task vectors ($τ_t$) enable this disentanglement remains underexplored. In this paper, we introduce Task-Feature Specialization (TFS), a model's ability to allocate distinct internal features to different tasks, as the fundamental principle. We first prove that TFS is a sufficient condition for weight disentanglement. More importantly, we find that TFS also gives rise to an observable geometric consequence: weight vector orthogonality. This positions TFS as the common cause for both the desired functional outcome (disentanglement) and a measurable geometric property (orthogonality). This relationship provides the key insight for our method: since the abstract TFS property is intractable to enforce directly, we can instead promote weight disentanglement by shaping its concrete geometric consequence, orthogonality. Therefore, we propose OrthoReg, a simple and effective regularization method that actively enforces an internal orthogonal structure on weight updates ($ΔW$) that constitute $τ_t$ during fine-tuning. And we theoretically prove that OrthoReg promotes disentanglement. Extensive experiments demonstrate that OrthoReg consistently and significantly enhances the performance of various task arithmetic methods. Code is available at \href{https://github.com/RL-MIND/OrthoReg}{https://github.com/RL-MIND/OrthoReg}.

The paper considers the computation of L1 regularization paths in a state space setting, which includes L1 regularized Kalman smoothing, linear SVM, LASSO, and more. The paper proposes two new algorithms, which are duals of each other; the first algorithm applies to L1 regularization of independent variables while the second applies to L1 regularization of dependent variables. The heart of the proposed algorithms is parametric Gaussian message passing (i.e., Kalman-type forward-backward recursions) in the pertinent factor graphs. The proposed methods are broadly applicable, they (usually) require only matrix multiplications, and their complexity can be competitive with prior methods in some cases.

Existing feature-interpretation pipelines typically operate on uniformly sampled units or exhaustive feature sets, incurring massive costs on units irrelevant to target behaviors. To address this, we introduce the first CLT-native end-to-end pruning framework, PIE, which pioneers the paradigm of pruning first and interpreting later. PIE connects Pruning, automatic Interpretation, and interpretation Evaluation, establishing a comprehensive benchmarking environment to systematically measure behavioral fidelity and downstream interpretability under pruning. Within this framework, we adapt strong relevance baselines and propose Feature Attribution Patching (FAP), a patch-grounded attribution method that scores CLT features by aggregating gradient-weighted write contributions. Furthermore, we introduce FAP-Synergy, a systematic synergy-aware reranking procedure. We evaluate pruning using KL-divergence behavior retention and assess interpretation quality with FADE-style metrics across IOI and Doc-String datasets. Across budget constraints of K in {50, 100, 200, 400, 800}, our rigorous benchmarking reveals distinct operational regimes: while base FAP and adapted baselines perform robustly at relaxed budgets, FAP-Synergy excels in highly constrained, strict-budget regimes. Crucially, we demonstrate a practical "Effective Budget" advantage: on the IOI task for both Llama-3.2-1B and Gemma-2-2B, FAP-Synergy at K=50 functionally matches the behavioral fidelity of baseline circuits at K=75. Because downstream evaluation costs scale linearly per feature, Synergy effectively grants the pipeline 25 "free" features, achieving K=75 fidelity while reducing interpretation costs by 33%.

Standard supervised learning optimizes for predictive accuracy but remains agnostic to the internal geometry of learned features, often yielding representations that are entangled and brittle. We propose Class-Conditional Activation Regularization (CCAR) to explicitly engineer the feature space, imposing a block-diagonal structure via a soft inductive bias. By shaping the latent representation to confine class energy to orthogonal subspaces, we create an intrinsic geometric scaffold that naturally filters noise and adversarial perturbations. We provide theoretical analysis linking this structural constraint to the maximization of the Fisher Discriminant Ratio, establishing a formal connection between geometric disentanglement and algorithmic stability. Empirically, this approach demonstrates that robustness is an emergent property of a well-engineered feature space, significantly outperforming baselines on label noise and input corruption benchmarks.

We propose HILBERT (HIerarchical Long-sequence Balanced Embedding with Reciprocal contrastive Training), a cross-attentive multimodal framework for learning document-level audio-text representations from long, segmented sequences in low-resource data settings. HILBERT leverages frozen pre-trained speech and language encoders to extract segment-level features, which are aggregated via cross-modal attention and self-attentive pooling to form modality-specific document representations and a joint cross-attentive embedding. To align modalities while preserving modality-specific structure under severe audio-text dimensional imbalance, we introduce a reciprocal dual contrastive objective that simultaneously aligns audio-to-joint and text-to-joint representations, rather than directly contrasting audio and text alone. Two auxiliary regularizers further stabilize long-sequence fusion: a Centered Kernel Alignment (CKA) loss that preserves structural consistency between each modality and the joint embedding, and a mutual information balancing loss that prevents dominance of a single modality by equalizing information flow from audio and text into the joint space. For downstream prediction, HILBERT employs a Mixture-of-Experts (MoE) classifier over concatenated audio, text, and joint representations to accommodate heterogeneous label regimes. Extensive evaluation across multiple audio-text backbone combinations demonstrates that HILBERT learns semantically meaningful long-sequence representations and achieves superior performance on highly imbalanced multi-class settings.

Predictive multiplicity and chaotic dynamics represent two fundamental challenges in machine learning that have evolved independently despite their conceptual connections. We bridge this gap by introducing horizon-constrained Rashomon sets, a theoretical framework that characterizes how model multiplicity evolves with prediction horizon in chaotic systems. Unlike static prediction tasks where the Rashomon set remains fixed, chaos induces exponential divergence among initially similar models, fundamentally transforming the nature of predictive equivalence. We prove that the effective Rashomon set contracts exponentially with lead time at a rate determined by the maximum Lyapunov exponent and introduce Lyapunov-weighted metrics that provide tighter bounds on predictive disagreement. Leveraging these insights, we develop decision-aligned selection algorithms that choose among near-optimal models based on downstream utility rather than forecast accuracy alone. Extensive experiments on synthetic chaotic systems (Lorenz-96, Kuramoto-Sivashinsky) and real-world applications (wind power, traffic, weather) demonstrate that our framework improves decision quality by 18-34% while maintaining competitive predictive performance. This work establishes the first rigorous connection between chaos theory and predictive multiplicity, providing principled guidance for deploying machine learning in safety-critical chaotic domains.

Proximal operators are fundamental across many applications in signal processing and machine learning, including solving ill-posed inverse problems. Recent work has introduced Learned Proximal Networks (LPNs), providing parametric functions that compute exact proximals for data-driven and potentially non-convex regularizers. However, in many settings it is important to include additional structure to these regularizers--and their corresponding proximals--such as shift and scale equivariance. In this work, we show how to obtain learned functions parametrized by neural networks that provably compute exact proximal operators while being equivariant to shifts and scaling, which we dub Affine-Equivariant Learned Proximal Networks (AE-LPNs). We demonstrate our results on synthetic, constructive examples, and then on real data via denoising in out-of-distribution settings. Our equivariant learned proximals enhance robustness to noise distributions and affine shifts far beyond training distributions, improving the practical utility of learned proximal operators

KV caching is a ubiquitous optimization in autoregressive transformer inference, long presumed to be numerically equivalent to cache-free computation. This assumption fails under standard FP16 precision: cache-ON and cache-OFF execution paths employ different floating-point accumulation orderings which, due to FP16 non-associativity, produce a deterministic divergence in decoded token sequences. Across three open-weight models (LLaMA-2-7B, Mistral-7B-v0.3, Gemma-2-2B) evaluated on GSM8K, we observe a 100% token divergence rate across all sampling strategies, including greedy decoding, which rules out sampling randomness as a cause, and also with cache-ON yielding higher accuracy in 8 of 9 conditions, where the accuracy difference serves as an indicator that the divergence direction is systematic rather than random. Controlled FP32 falsification reduces divergence by eight orders of magnitude, eliminates token flips, and drops the flip rate to exactly 0.0%, confirming FP16 non-associativity as the sole causal driver. Layer-wise drift profiling reveals architecturally predictable propagation patterns: models using Grouped-Query Attention exhibit sharp divergence at the first layer, while Gemma's larger head dimension and sliding window attention produce uniform accumulation across all layers. Finally, activation patching of the entire residual stream fails to recover the cache-free trajectory, localizing the causal variable to the stateful KV cache. These findings establish that FP16 KV cache inference is fundamentally non-equivalent to recomputation and provide a mechanistic framework for understanding numerical instability in modern LLM inference systems.

Recent work has shown that diffusion models trained with the denoising score matching (DSM) objective often violate the Fokker--Planck (FP) equation that governs the evolution of the true data density. Directly penalizing these deviations in the objective function reduces their magnitude but introduces a significant computational overhead. It is also observed that enforcing strict adherence to the FP equation does not necessarily lead to improvements in the quality of the generated samples, as often the best results are obtained with weaker FP regularization. In this paper, we investigate whether simpler penalty terms can provide similar benefits. We empirically analyze several lightweight regularizers, study their effect on FP residuals and generation quality, and show that the benefits of FP regularization are available at substantially lower computational cost. Our code is available at https://github.com/OnnoNiemann/fp_diffusion_analysis.

Conventional Graph Contrastive Learning (GCL) on Text-Attributed Graphs (TAGs) relies on blind stochastic augmentations, inadvertently entangling task-relevant signals with noise. We propose SDM-SCR, a robust framework anchored in Approximate Orthogonal Decomposition. First, the Semantic Decoupling Module (SDM) leverages the instruction-following capability of Large Language Models (LLMs) to actively parse raw attributes into asymmetric, task-oriented signal and noise views. This shifts the paradigm from random perturbation to semantic-aware disentanglement. Subsequently, Semantic Consistency Regularization (SCR) exploits the spectral observation that semantic signals are topologically smooth while residual noise is high-frequency. SCR functions as a selective spectral filter, enforcing consistency only on the signal subspace to eliminate LLM hallucinations without over-smoothing. This ``Disentangle-then-Refine'' mechanism ensures rigorous signal purification. Extensive experiments demonstrate that SDM-SCR achieves SOTA performance in accuracy and efficiency.

Data-driven Model Predictive Control (MPC) has lately been the core research subject in the field of control theory. The combination of an optimal control framework with deep learning paradigms opens up the possibility to accurately track control tasks without the need for complex analytical models. However, the system dynamics are often nuanced and the neural model lacks the potential to understand physical properties such as inertia and conservation of energy. In this work, we propose a novel energy-based regularization loss function which is applied to the training of a neural model that learns the residual dynamics of an omnidirectional aerial robot. Our energy-based regularization encourages the neural network to cause control corrections that stabilize the energy of the system. The residual dynamics are integrated into the MPC framework and improve the positional mean absolute error (MAE) over three real-world experiments by 23% compared to an analytical MPC. We also compare our method to a standard neural MPC implementation without regularization and primarily achieve a significantly increased flight stability implicitly due to the energy regularization and up to 15% lower MAE. Our code is available under: https://github.com/johanneskbl/jsk_aerial_robot/tree/develop/neural_MPC.

Zeroth-order (ZO) methods are widely used when gradients are unavailable or prohibitively expensive, including black-box learning and memory-efficient fine-tuning of large models, yet their optimization dynamics in deep learning remain underexplored. In this work, we provide an explicit step size condition that exactly captures the (mean-square) linear stability of a family of ZO methods based on the standard two-point estimator. Our characterization reveals a sharp contrast with first-order (FO) methods: whereas FO stability is governed solely by the largest Hessian eigenvalue, mean-square stability of ZO methods depends on the entire Hessian spectrum. Since computing the full Hessian spectrum is infeasible in practical neural network training, we further derive tractable stability bounds that depend only on the largest eigenvalue and the Hessian trace. Empirically, we find that full-batch ZO methods operate at the edge of stability: ZO-GD, ZO-GDM, and ZO-Adam consistently stabilize near the predicted stability boundary across a range of deep learning training problems. Our results highlight an implicit regularization effect specific to ZO methods, where large step sizes primarily regularize the Hessian trace, whereas in FO methods they regularize the top eigenvalue.

Clinical language models (LMs) are increasingly applied to support clinical risk prediction from free-text notes, yet their uncertainty estimates often remain poorly calibrated and clinically unreliable. In this work, we propose Clinical Uncertainty Risk Alignment (CURA), a framework that aligns clinical LM-based risk estimates and uncertainty with both individual error likelihoods and cohort-level ambiguities. CURA first fine-tunes domain-specific clinical LMs to obtain task-adapted patient embeddings, and then performs uncertainty fine-tuning of a multi-head classifier using a bi-level uncertainty objective. Specifically, an individual-level calibration term aligns predictive uncertainty with each patient's likelihood of error, while a cohort-aware regularizer pulls risk estimates toward event rates in their local neighborhoods in the embedding space and places extra weight on ambiguous cohorts near the decision boundary. We further show that this cohort-aware term can be interpreted as a cross-entropy loss with neighborhood-informed soft labels, providing a label-smoothing view of our method. Extensive experiments on MIMIC-IV clinical risk prediction tasks across various clinical LMs show that CURA consistently improves calibration metrics without substantially compromising discrimination. Further analysis illustrates that CURA reduces overconfident false reassurance and yields more trustworthy uncertainty estimates for downstream clinical decision support.

Existing methods for quantifying predictive uncertainty in neural networks are either computationally intractable for large language models or require access to training data that is typically unavailable. We derive a lightweight alternative through two approximations: a first-order Taylor expansion that expresses uncertainty in terms of the gradient of the prediction and the parameter covariance, and an isotropy assumption on the parameter covariance. Together, these yield epistemic uncertainty as the squared gradient norm and aleatoric uncertainty as the Bernoulli variance of the point prediction, from a single forward-backward pass through an unmodified pretrained model. We justify the isotropy assumption by showing that covariance estimates built from non-training data introduce structured distortions that isotropic covariance avoids, and that theoretical results on the spectral properties of large networks support the approximation at scale. Validation against reference Markov Chain Monte Carlo estimates on synthetic problems shows strong correspondence that improves with model size. We then use the estimates to investigate when each uncertainty type carries useful signal for predicting answer correctness in question answering with large language models, revealing a benchmark-dependent divergence: the combined estimate achieves the highest mean AUROC on TruthfulQA, where questions involve genuine conflict between plausible answers, but falls to near chance on TriviaQA's factual recall, suggesting that parameter-level uncertainty captures a fundamentally different signal than self-assessment methods.

This work uncovers variational principles behind symmetrizing the Bregman divergences induced by generic mirror maps over the cone of positive definite matrices. We show that computing the canonical means for this symmetrization can be posed as minimizing the desired symmetrized divergences over a set of mean functionals defined axiomatically to satisfy certain properties. For the forward symmetrization, we prove that the arithmetic mean over the primal space is canonical for any mirror map over the positive definite cone. For the reverse symmetrization, we show that the canonical mean is the arithmetic mean over the dual space, pulled back to the primal space. Applying this result to three common mirror maps used in practice, we show that the canonical means for reverse symmetrization, in those cases, turn out to be the arithmetic, log-Euclidean and harmonic means. Our results improve understanding of existing symmetrization practices in the literature, and can be seen as a navigational chart to help decide which mean to use when.

In low-resource multilingual speech-to-text translation, uniform architectural sharing across languages frequently introduces representation conflicts that impede convergence. This work proposes a principled methodology to automatically determine layer-specific sharing patterns by mining training gradient information. Our approach employs three distinct analysis strategies: distance-based language clustering, self/cross-task divergence metrics for capacity allocation, and joint factorization coupled with canonical correlation analysis for subspace alignment. Extensive evaluation across four language pairs (using the SeamlessM4T-Medium architecture) demonstrates persistent improvements in translation quality metrics.

Classical sparse recovery treats all nonzero entries equally, though numerical noise often creates long tails of negligible coefficients. This paper develops an entropy-based notion of effective sparsity to measure the coefficients carrying significant mass. The central quantity, the effective number of nonzeros (ENZ), is obtained by exponentiating the Shannon entropy of the normalized magnitude distribution. We show that ENZ decomposes exactly into the support cardinality multiplied by a distributional efficiency factor, thereby making precise its relation to the $\ell_0$ count and explaining how it discounts uninformative coefficients. Furthermore, the Shannon ENZ is embedded into a parallel Rényi family that recovers several scale-invariant sparsity measures, including the $\ell_1/\ell_2$ ratio, as special cases. We then prove a stability result under a restricted isometry condition, establishing an explicit bound that depends on the tail energy, measurement perturbation, and restricted isometry constant. For computation, a separable unnormalized entropy surrogate is introduced to avoid global coupling. Numerical experiments on sparse signal recovery and gradient-domain image denoising demonstrate that the resulting regularizer is robust, computationally efficient, and competitive with standard sparsity penalties.

Establishing accurate point-to-point correspondences between non-rigid 3D shapes remains a critical challenge, particularly under non-isometric deformations and topological noise. Existing functional map pipelines suffer from ambiguities that geometric descriptors alone cannot resolve, and spatial inconsistencies inherent in the projection of truncated spectral bases to dense pointwise correspondences. In this paper, we introduce SGMatch, a learning-based framework that couples 3D-lifted semantic cues with trajectory-level feature transport regularization. Specifically, we design a Semantic-Guided Local Cross-Attention module that integrates semantic features from vision foundation models into geometric descriptors while preserving local structural continuity. Furthermore, we adapt conditional flow matching as a time-conditioned feature transport regularizer that promotes spatially coherent point-wise recovery. Experimental results on multiple benchmarks demonstrate that SGMatch achieves competitive performance across near-isometric settings and consistent improvements under non-isometric deformations and topological noise.

Multi--task learning seeks to improve the generalization error by leveraging the common information shared by multiple related tasks. One challenge in multi--task learning is identifying formulations capable of uncovering the common information shared between different but related tasks. This paper provides a precise asymptotic analysis of a popular multi--task formulation associated with misspecified perceptron learning models. The main contribution of this paper is to precisely determine the reasons behind the benefits gained from combining multiple related tasks. Specifically, we show that combining multiple tasks is asymptotically equivalent to a traditional formulation with additional regularization terms that help improve the generalization performance. Another contribution is to empirically study the impact of combining tasks on the generalization error. In particular, we empirically show that the combination of multiple tasks postpones the double descent phenomenon and can mitigate it asymptotically.

Reinforcement Learning from Human Feedback (RLHF) or Verifiable Rewards (RLVR) are two key steps in the post-training of modern Language Models (LMs). A common problem is reward hacking, where the policy may exploit inaccuracies of the reward and learn an unintended behavior. Most previous works address this by limiting the policy update with a Kullback-Leibler (KL) penalty towards a reference model. We propose a different framing: Train the LM in a way that biases policy updates towards regions in which the reward is more accurate. First, we derive a theoretical connection between the accuracy of a reward model and the flatness of an optimum at convergence. Gradient regularization (GR) can then be used to bias training to flatter regions and thereby maintain reward model accuracy. We confirm these results by showing that the gradient norm and reward accuracy are empirically correlated in RLHF. We then empirically show that Reference Resets of the KL penalty find flatter regions with a higher reward accuracy. We further improve on this by proposing to use explicit GR with an efficient finite-difference estimate. Empirically, GR performs better than a KL penalty across a diverse set of RL experiments with LMs. GR achieves a higher GPT-judged win-rate in RLHF, avoids overly focusing on the format in rule-based math rewards, and prevents hacking the judge in LLM-as-a-Judge math tasks.

For a broad family of discriminative models that includes autoregressive language models, identifiability results imply that if two models induce the same conditional distributions, then their internal representations are equal up to an invertible linear transformation. We ask whether an analogous conclusion holds approximately when the distributions are close instead of equal. Building on the observation of Nielsen et al. (2025) that closeness in KL divergence need not imply high linear representational similarity, we study a distributional distance based on logit differences and show that closeness in this distance does yield linear similarity guarantees. Specifically, we define a representational dissimilarity measure based on the models' identifiability class and prove that it is bounded by the logit distance. We further show that, when model probabilities are bounded away from zero, KL divergence upper-bounds logit distance; yet the resulting bound fails to provide nontrivial control in practice. As a consequence, KL-based distillation can match a teacher's predictions while failing to preserve linear representational properties, such as linear-probe recoverability of human-interpretable concepts. In distillation experiments on synthetic and image datasets, logit-distance distillation yields students with higher linear representational similarity and better preservation of the teacher's linearly recoverable concepts.

Diffusion models over discrete spaces have recently shown striking empirical success, yet their theoretical foundations remain incomplete. In this paper, we study the sampling efficiency of score-based discrete diffusion models under a continuous-time Markov chain (CTMC) formulation, with a focus on $τ$-leaping-based samplers. We establish sharp convergence guarantees for attaining $\varepsilon$ accuracy in Kullback-Leibler (KL) divergence for both uniform and masking noising processes. For uniform discrete diffusion, we show that the $τ$-leaping algorithm achieves an iteration complexity of order $\tilde O(d/\varepsilon)$, with $d$ the ambient dimension of the target distribution, eliminating linear dependence on the vocabulary size $S$ and improving existing bounds by a factor of $d$; moreover, we establish a matching algorithmic lower bound showing that linear dependence on the ambient dimension is unavoidable in general. For masking discrete diffusion, we introduce a modified $τ$-leaping sampler whose convergence rate is governed by an intrinsic information-theoretic quantity, termed the effective total correlation, which is bounded by $d \log S$ but can be sublinear or even constant for structured data. As a consequence, the sampler provably adapts to low-dimensional structure without prior knowledge or algorithmic modification, yielding sublinear convergence rates for various practical examples (such as hidden Markov models, image data, and random graphs). Our analysis requires no boundedness or smoothness assumptions on the score estimator beyond control of the score entropy loss.

The widespread availability of fine-tuned LoRA modules for open pre-trained models has led to an interest in methods that can adaptively merge LoRAs to improve performance. These methods typically include some way of selecting LoRAs from a pool and tune merging coefficients based on a task-specific dataset. While adaptive merging methods have demonstrated improvements in some settings, no past work has attempted to recycle LoRAs found "in the wild" on model repositories like the Hugging Face Hub. To address this gap, we consider recycling from a pool of nearly 1,000 user-contributed LoRAs trained from the Llama 3.1 8B-Instruct language model. Our empirical study includes a range of adaptive and non-adaptive merging methods in addition to a new method designed via a wide search over the methodological design space. We demonstrate that adaptive merging methods can improve performance over the base model but provide limited benefit over training a new LoRA on the same data used to set merging coefficients. We additionally find not only that the specific choice of LoRAs to merge has little importance, but that using LoRAs with randomly initialized parameter values yields similar performance. This raises the possibility that adaptive merging from recycled LoRAs primarily works via some kind of regularization effect, rather than by enabling positive cross-task transfer. To better understand why past work has proven successful, we confirm that positive transfer is indeed possible when there are highly relevant LoRAs in the pool. We release the model checkpoints and code online.

This paper addresses the issue of inversion in cases where (1) the observation system is modeled by a linear transformation and additive error, (2) the problem is ill-posed and regularization relies on a Bayesian strategy, (3)~the prior is modeled by a diffusion process adjusted on an available large set of examples. In this context, it is known that the issue of posterior sampling is a thorny one and the paper introduces a Gibbs algorithm. It appears that this avenue has not been explored, and we show that it is particularly effective and remarkably simple. In addition, it provides clear elements regarding convergence guarantees in a specific case and arguments supporting such guarantees in practical cases. The results are clearly confirmed by numerical simulations based on a toy example.

We derive two attention operators from generalized statistical entropies. Kaniadakis entropy yields an exact full-support normalization whose weights and low-score sensitivities decay algebraically, rather than exponentially as in Softmax or by exact truncation as in entmax. Classical Abe entropy yields an implicit reciprocal-symmetric operator. With $q=e^ε$, the involution $q\leftrightarrow q^{-1}$ removes every odd correction about Softmax; we obtain the normalized second- and fourth-order terms, including the deformation of the normalization multiplier. These stationary laws follow from a Fisher-metric Lagrangian on the probability simplex, whose Shannon sector recovers scaled dot-product Softmax. We also give a tangent-gradient test for deciding whether changing the entropy changes the attention profile or only its scale. Rényi and two-parameter Sharma--Mittal entropies retain the Tsallis--entmax inverse-gradient shape, but their global moments make the effective temperature input dependent when the external temperature is fixed. Distinguishing profile-shape equivalence from fixed-parameter operator equivalence separates new normalization shapes from adaptive rescalings and organizes the operators by support, tail behavior, and realization complexity.

GRPO-style reinforcement learning (RL)-based LLM fine-tuning algorithms have recently gained popularity. Relying on heuristic trust-region approximations, however, they can lead to brittle optimization behavior, as global importance-ratio clipping and group-wise normalization fail to regulate samples whose importance ratios fall outside the clipping range. We propose Query-Adaptive Trust-Region policy Optimization (QUATRO), which directly enforces trust-region constraints through a principled optimization. This yields a clear and interpretable objective that enables explicit control over policy updates and stable, entropy-controlled optimization, with a stabilizer terms arising intrinsically from the exact trust-region formulation. Empirically verified on diverse mathematical reasoning benchmarks, QUATRO shows stable training under increased policy staleness and aggressive learning rates, maintaining well-controlled entropy throughout training.

Score-based diffusion models are a recently developed framework for posterior sampling in Bayesian inverse problems, enabling high-quality reconstructions in inverse problems by leveraging expressive prior distributions learned from empirical data. Despite their strong empirical performance and growing interest within the machine learning community, their behaviour in realistic, severely ill-posed inverse problems with experimental measurement data remains under-explored. Diffuse optical tomography (DOT) is an inverse boundary value problem that uses boundary measurements of near-infrared light to recover spatially varying absorption and scattering parameters in biological tissue. The problem is highly ill-posed and particularly sensitive to both measurement noise and modelling errors. We introduce a regularization strategy by constructing a mixed score consisting of a learned component and a model-based component. We show that the resulting mixed score approximates the score of a corresponding mixture distribution locally and in the small diffusion-time regime, providing a theoretical justification for the approach. We compare four approaches for difference imaging in DOT: a classical model-based method, an approximate score-based diffusion method (DPS), an exact posterior sampling method (UCoS) and a novel, regularized version of UCoS. We show that both the model-based approach and approximate diffusion-based sampling degrade significantly in the presence of limited-view geometry and real experimental data, whereas UCoS yields more accurate reconstructions.

This paper studies optimization for a family of problems termed $\textbf{compositional entropic risk minimization}$, in which each data's loss is formulated as a Log-Expectation-Exponential (Log-E-Exp) function. The Log-E-Exp formulation serves as an abstraction of the Log-Sum-Exponential (LogSumExp) function when the explicit summation inside the logarithm is taken over a gigantic number of items and is therefore expensive to evaluate. While entropic risk objectives of this form arise in many machine learning problems, existing optimization algorithms suffer from several fundamental limitations including non-convergence, numerical instability, and slow convergence rates. To address these limitations, we propose a geometry-aware stochastic algorithm, termed $\textbf{SCENT}$, for the dual formulation of entropic risk minimization cast as a min--min optimization problem. The key to our design is a $\textbf{stochastic proximal mirror descent (SPMD)}$ update for the dual variable, equipped with a Bregman divergence induced by a negative exponential function that faithfully captures the geometry of the objective. Our main contributions are threefold: (i) we establish an $O(1/\sqrt{T})$ convergence rate of the proposed SCENT algorithm for convex problems; (ii) we theoretically characterize the advantages of SPMD over standard SGD update for optimizing the dual variable; and (iii) we demonstrate the empirical effectiveness of SCENT on extreme classification, partial AUC maximization, contrastive learning and distributionally robust optimization, where it consistently outperforms existing baselines. Code is available at https://github.com/Optimization-AI/SCENT.

Molecular function is largely determined by structure. Accurately aligning molecular structure with natural language is therefore essential for enabling large language models (LLMs) to reason about downstream chemical tasks. However, the substantial cost of human annotation makes it infeasible to construct large-scale, high-quality datasets of structure-grounded descriptions. In this work, we propose a fully automated annotation framework for generating precise molecular descriptions that preserve complete structural details at scale. Our approach builds upon and extends a rule-based chemical nomenclature parser to interpret IUPAC names and construct enriched, structural XML metadata that explicitly encodes molecular structure. This metadata is then used to guide LLMs in producing accurate natural-language descriptions. Using this framework, we curate a large-scale dataset of approximately $163$k molecule--description pairs. A rigorous validation protocol combining LLM-based and expert human evaluation on a subset of $2,000$ molecules demonstrates a high description precision of $98.6$%. The proposed annotation framework is readily beneficial to broader chemical tasks that rely on structural descriptions, with the resulting dataset providing a reliable foundation for molecule--language alignment. The source code and dataset are hosted at https://github.com/TheLuoFengLab/MolLangData and https://huggingface.co/datasets/ChemFM/MolLangData, respectively.

The problem of corrupted data, missing features, or missing modalities continues to plague the modern machine learning landscape. To address this issue, a class of regularization methods that enforce consistency between imputed and fully observed data has emerged as a promising approach for improving model generalization, particularly in partially observed settings. We refer to this class of methods as Measure Consistency Regularization (MCR). Despite its empirical success in various applications, such as image inpainting, data imputation and semi-supervised learning, a fundamental understanding of the theoretical underpinnings of MCR remains limited. This paper bridges this gap by offering theoretical insights into when MCR yields a more favorable finite-sample estimation-error upper bound, viewed through the lens of neural network distance. Under ideal interpolation and compatibility conditions, we show that the MCR estimation-error upper bound is no larger than vanilla supervised training and becomes strictly smaller when the mixed-sample alternative is favorable. We then extend the analysis to the non-ideal regime, where optimization and compatibility residuals can potentially offset this finite-sample advantage. Guided by these insights, we propose a novel practical diagnostic that leverages the duality gap and a calibrated drift estimator to infer the potential benefit of MCR training. We present detailed empirical evidence to support our theoretical claims and to show the effectiveness and accuracy of our practical diagnostic. We further provide simulations on real-world datasets to show the versatility of MCR under different model architectures designed for different data sources.

Deep learning has substantially advanced medical image segmentation, yet achieving robust generalization across diverse imaging modalities and anatomical structures remains a major challenge. A key contributor to this limitation lies in how existing architectures, ranging from CNNs to Transformers and their hybrids, primarily encode spatial information while overlooking frequency-domain representations that capture rich structural and textural cues. Although recent studies have begun exploring spectral information at the feature level, supervision-level integration of frequency cues-crucial for fine-grained object localization-remains largely untapped. To this end, we propose Phi-SegNet, a CNN-based architecture that incorporates phase-aware information at both architectural and optimization levels. The network integrates Bi-Feature Mask Former (BFMF) modules that blend neighboring encoder features to reduce semantic gaps, and Reverse Fourier Attention (RFA) blocks that refine decoder outputs using phase-regularized features. A dedicated phase-aware loss aligns these features with structural priors, forming a closed feedback loop that emphasizes boundary precision. Evaluated on five public datasets spanning X-ray, US, histopathology, MRI, and colonoscopy, Phi-SegNet consistently achieved state-of-the-art performance, with an average relative improvement of 1.54+/-1.26% in IoU and 0.98+/-0.71% in F1-score over the next best-performing model. In cross-dataset generalization scenarios involving unseen datasets from the known domain, it also exhibits robust and superior performance, highlighting its adaptability and modality-agnostic design. These findings demonstrate the potential of leveraging spectral priors in both feature representation and supervision, paving the way for generalized segmentation frameworks that excel in fine-grained object localization. Our code is available on GitHub.

Training deep computer vision models requires manual oversight or hyperparameter tuning of the learning rate (LR) schedule. While existing adaptive optimizers schedule the LR automatically, they suffer from computational and memory overhead, incompatibility with regularization, and suboptimal LR choices. In this work, we introduce the ZENITH (Zero-overhead Evolution using Norm-Informed Training History) optimizer, which adapts the LR using the temporal evolution of the gradient norm. Image classification experiments spanning 6 CNN architectures and 6 benchmarks demonstrate that ZENITH achieves higher test accuracy in lower wall-clock time than baselines. It also yielded superior mAP in object detection, keypoint detection, and instance segmentation on MS COCO using the R-CNN family of models. Furthermore, its compatibility with regularization enables even better generalization.

This paper introduces an analytical approach to quantifying and optimizing the distributional discrepancy in generative diffusion models. For a multivariate Gaussian source, we explicitly derive the closed-form evolution trajectory and the resulting Kullback-Leibler (KL) divergence between the distributions of the source data and the reversely sampled data. Asymptotic analysis via the Euler-Maclaurin expansion characterizes the convergence behavior of this KL divergence, extracting its dominant term as an explicit functional of the noise schedule. Minimizing this dominant term via the calculus of variations yields a noise schedule described by a tangent law, inherently determined by the source covariance spectrum. We further prove that the Gaussian source exhibits an extremal property for the KL divergence among general source distributions with a given covariance. We also utilize the analytical KL divergence as a principled metric to identify efficient time discretization strategies for pretrained diffusion models, and demonstrate via experiments over diverse datasets that the identified strategies consistently outperform established baselines, particularly under constrained function evaluation budgets.

Pretraining of large language models is not only expensive but also prone to certain training instabilities. A specific instability that often occurs at the end of training is output logit divergence. The most widely used mitigation strategies, z-loss and logit soft-capping, merely address the symptoms rather than the underlying cause of the problem. In this paper, we analyze the instability from the perspective of the output embeddings' geometry and identify anisotropic embeddings as its source. Based on this, we propose output embedding centering (OEC) as a new mitigation strategy, and demonstrate that it suppresses output logit divergence. OEC can be implemented in two different ways: as a deterministic operation called $μ$-centering, or a regularization method called $μ$-loss. Our experiments show that both variants outperform z-loss in terms of training stability, while being on par with logit soft-capping. This holds true both in the presence and the absence of weight tying. As a secondary result, we find that $μ$-loss is significantly less sensitive to regularization hyperparameter tuning than z-loss.

In this work, we introduce $\textit{basic inequalities}$ for first-order iterative optimization algorithms, forming a simple yet versatile framework which connects implicit and explicit regularization. Building on related comparison inequalities for optimization iterates that already exist in the literature, we extend and unify these arguments to produce a general framework, which can be used as a tool for statistical analysis. In more detail, let $f$ denote the objective function to be optimized. Given a first-order iterative algorithm initialized at $θ_0$, with current iterate $θ_T$, the basic inequality upper bounds $f(θ_T) - f(z)$ for any reference point $z$ in terms of the accumulated step sizes, and the distances between $θ_0$, $θ_T$, and $z$. These distances are measured in a geometry inherent to the optimization algorithm, which then translates into a notion of regularization being applied across the path of iterates. In addition to refining existing results on gradient descent, we provide new results for mirror descent and other first-order methods. We then show how to use these basic inequalities to derive elementary yet useful bounds on the prediction risk of early-stopped gradient descent and exponentiated gradient descent iterates in generalized linear models. We also supplement these findings with numerical experiments.