Foundations of Top-$k$ Decoding For Language Models

Top-$k$ decoding is a widely used method for sampling from LLMs: at each token, only the largest $k$ next-token-probabilities are kept, and the next token is sampled after re-normalizing them to sum to unity. Top-$k$ and other sampling methods are motivated by the intuition that true next-token distributions are sparse, and the noisy LLM probabilities need to be truncated. However, to our knowledge, a precise theoretical motivation for the use of top-$k$ decoding is missing. In this work, we develop a theoretical framework that both explains and generalizes top-$k$ decoding. We view decoding at a fixed token as the recovery of a sparse probability distribution. We consider \emph{Bregman decoders} obtained by minimizing a separable Bregman divergence (for both the \emph{primal} and \emph{dual} cases) with a sparsity-inducing $\ell_0$ regularization. Despite the combinatorial nature of the objective, we show how to optimize it efficiently for a large class of divergences. We show that the optimal decoding strategies are greedy, and further that the loss function is discretely convex in $k$, so that binary search provably and efficiently finds the optimal $k$. We show that top-$k$ decoding arises as a special case for the KL divergence, and identify new decoding strategies that have distinct behaviors (e.g., non-linearly up-weighting larger probabilities after re-normalization).

Time Tracker: Mixture-of-Experts-Enhanced Foundation Time Series Forecasting Model with Decoupled Training Pipelines

In the past few years, time series foundation models have achieved superior predicting accuracy. However, real-world time series often exhibit significant diversity in their temporal patterns across different time spans and domains, making it challenging for a single model architecture to fit all complex scenarios. In addition, time series data may have multiple variables exhibiting complex correlations between each other. Recent mainstream works have focused on modeling times series in a channel-independent manner in both pretraining and finetuning stages, overlooking the valuable inter-series dependencies. To this end, we propose \textbf{Time Tracker} for better predictions on multivariate time series data. Firstly, we leverage sparse mixture of experts (MoE) within Transformers to handle the modeling of diverse time series patterns, thereby alleviating the learning difficulties of a single model while improving its generalization. Besides, we propose Any-variate Attention, enabling a unified model structure to seamlessly handle both univariate and multivariate time series, thereby supporting channel-independent modeling during pretraining and channel-mixed modeling for finetuning. Furthermore, we design a graph learning module that constructs relations among sequences from frequency-domain features, providing more precise guidance to capture inter-series dependencies in channel-mixed modeling. Based on these advancements, Time Tracker achieves state-of-the-art performance in predicting accuracy, model generalization and adaptability.

An Analytical Characterization of Sloppiness in Neural Networks: Insights from Linear Models

Recent experiments have shown that training trajectories of multiple deep neural networks with different architectures, optimization algorithms, hyper-parameter settings, and regularization methods evolve on a remarkably low-dimensional "hyper-ribbon-like" manifold in the space of probability distributions. Inspired by the similarities in the training trajectories of deep networks and linear networks, we analytically characterize this phenomenon for the latter. We show, using tools in dynamical systems theory, that the geometry of this low-dimensional manifold is controlled by (i) the decay rate of the eigenvalues of the input correlation matrix of the training data, (ii) the relative scale of the ground-truth output to the weights at the beginning of training, and (iii) the number of steps of gradient descent. By analytically computing and bounding the contributions of these quantities, we characterize phase boundaries of the region where hyper-ribbons are to be expected. We also extend our analysis to kernel machines and linear models that are trained with stochastic gradient descent.

Physics-Informed Inference Time Scaling via Simulation-Calibrated Scientific Machine Learning

High-dimensional partial differential equations (PDEs) pose significant computational challenges across fields ranging from quantum chemistry to economics and finance. Although scientific machine learning (SciML) techniques offer approximate solutions, they often suffer from bias and neglect crucial physical insights. Inspired by inference-time scaling strategies in language models, we propose Simulation-Calibrated Scientific Machine Learning (SCaSML), a physics-informed framework that dynamically refines and debiases the SCiML predictions during inference by enforcing the physical laws. SCaSML leverages derived new physical laws that quantifies systematic errors and employs Monte Carlo solvers based on the Feynman-Kac and Elworthy-Bismut-Li formulas to dynamically correct the prediction. Both numerical and theoretical analysis confirms enhanced convergence rates via compute-optimal inference methods. Our numerical experiments demonstrate that SCaSML reduces errors by 20-50% compared to the base surrogate model, establishing it as the first algorithm to refine approximated solutions to high-dimensional PDE during inference. Code of SCaSML is available at https://github.com/Francis-Fan-create/SCaSML.

A Survey on Mechanistic Interpretability for Multi-Modal Foundation Models

The rise of foundation models has transformed machine learning research, prompting efforts to uncover their inner workings and develop more efficient and reliable applications for better control. While significant progress has been made in interpreting Large Language Models (LLMs), multimodal foundation models (MMFMs) - such as contrastive vision-language models, generative vision-language models, and text-to-image models - pose unique interpretability challenges beyond unimodal frameworks. Despite initial studies, a substantial gap remains between the interpretability of LLMs and MMFMs. This survey explores two key aspects: (1) the adaptation of LLM interpretability methods to multimodal models and (2) understanding the mechanistic differences between unimodal language models and crossmodal systems. By systematically reviewing current MMFM analysis techniques, we propose a structured taxonomy of interpretability methods, compare insights across unimodal and multimodal architectures, and highlight critical research gaps.

Trustworthy Evaluation of Generative AI Models

Generative AI (GenAI) models have recently achieved remarkable empirical performance in various applications, however, their evaluations yet lack uncertainty quantification. In this paper, we propose a method to compare two generative models based on an unbiased estimator of their relative performance gap. Statistically, our estimator achieves parametric convergence rate and asymptotic normality, which enables valid inference. Computationally, our method is efficient and can be accelerated by parallel computing and leveraging pre-storing intermediate results. On simulated datasets with known ground truth, we show our approach effectively controls type I error and achieves power comparable with commonly used metrics. Furthermore, we demonstrate the performance of our method in evaluating diffusion models on real image datasets with statistical confidence.

TeZO: Empowering the Low-Rankness on the Temporal Dimension in the Zeroth-Order Optimization for Fine-tuning LLMs

Zeroth-order optimization (ZO) has demonstrated remarkable promise in efficient fine-tuning tasks for Large Language Models (LLMs). In particular, recent advances incorporate the low-rankness of gradients, introducing low-rank ZO estimators to further reduce GPU memory consumption. However, most existing works focus solely on the low-rankness of each individual gradient, overlooking a broader property shared by all gradients throughout the training, i.e., all gradients approximately reside within a similar subspace. In this paper, we consider two factors together and propose a novel low-rank ZO estimator, TeZO, which captures the low-rankness across both the model and temporal dimension. Specifically, we represent ZO perturbations along the temporal dimension as a 3D tensor and employ Canonical Polyadic Decomposition (CPD) to extract each low-rank 2D matrix, significantly reducing the training cost. TeZO can also be easily extended to the Adam variant while consuming less memory than MeZO-SGD, and requiring about only 35% memory of MeZO-Adam. Both comprehensive theoretical analysis and extensive experimental research have validated its efficiency, achieving SOTA-comparable results with lower overhead of time and memory.

MolGraph-xLSTM: A graph-based dual-level xLSTM framework with multi-head mixture-of-experts for enhanced molecular representation and interpretability

Predicting molecular properties is essential for drug discovery, and computational methods can greatly enhance this process. Molecular graphs have become a focus for representation learning, with Graph Neural Networks (GNNs) widely used. However, GNNs often struggle with capturing long-range dependencies. To address this, we propose MolGraph-xLSTM, a novel graph-based xLSTM model that enhances feature extraction and effectively models molecule long-range interactions. Our approach processes molecular graphs at two scales: atom-level and motif-level. For atom-level graphs, a GNN-based xLSTM framework with jumping knowledge extracts local features and aggregates multilayer information to capture both local and global patterns effectively. Motif-level graphs provide complementary structural information for a broader molecular view. Embeddings from both scales are refined via a multi-head mixture of experts (MHMoE), further enhancing expressiveness and performance. We validate MolGraph-xLSTM on 10 molecular property prediction datasets, covering both classification and regression tasks. Our model demonstrates consistent performance across all datasets, with improvements of up to 7.03% on the BBBP dataset for classification and 7.54% on the ESOL dataset for regression compared to baselines. On average, MolGraph-xLSTM achieves an AUROC improvement of 3.18\% for classification tasks and an RMSE reduction of 3.83\% across regression datasets compared to the baseline methods. These results confirm the effectiveness of our model, offering a promising solution for molecular representation learning for drug discovery.

Unsupervised Domain Adaptation with Dynamic Clustering and Contrastive Refinement for Gait Recognition

Gait recognition is an emerging identification technology that distinguishes individuals at long distances by analyzing individual walking patterns. Traditional techniques rely heavily on large-scale labeled datasets, which incurs high costs and significant labeling challenges. Recently, researchers have explored unsupervised gait recognition with clustering-based unsupervised domain adaptation methods and achieved notable success. However, these methods directly use pseudo-label generated by clustering and neglect pseudolabel noise caused by domain differences, which affects the effect of the model training process. To mitigate these issues, we proposed a novel model called GaitDCCR, which aims to reduce the influence of noisy pseudo labels on clustering and model training. Our approach can be divided into two main stages: clustering and training stage. In the clustering stage, we propose Dynamic Cluster Parameters (DCP) and Dynamic Weight Centroids (DWC) to improve the efficiency of clustering and obtain reliable cluster centroids. In the training stage, we employ the classical teacher-student structure and propose Confidence-based Pseudo-label Refinement (CPR) and Contrastive Teacher Module (CTM) to encourage noisy samples to converge towards clusters containing their true identities. Extensive experiments on public gait datasets have demonstrated that our simple and effective method significantly enhances the performance of unsupervised gait recognition, laying the foundation for its application in the real-world.The code is available at https://github.com/YanSun-github/GaitDCCR

FGATT: A Robust Framework for Wireless Data Imputation Using Fuzzy Graph Attention Networks and Transformer Encoders

Missing data is a pervasive challenge in wireless networks and many other domains, often compromising the performance of machine learning and deep learning models. To address this, we propose a novel framework, FGATT, that combines the Fuzzy Graph Attention Network (FGAT) with the Transformer encoder to perform robust and accurate data imputation. FGAT leverages fuzzy rough sets and graph attention mechanisms to capture spatial dependencies dynamically, even in scenarios where predefined spatial information is unavailable. The Transformer encoder is employed to model temporal dependencies, utilizing its self-attention mechanism to focus on significant time-series patterns. A self-adaptive graph construction method is introduced to enable dynamic connectivity learning, ensuring the framework's applicability to a wide range of wireless datasets. Extensive experiments demonstrate that our approach outperforms state-of-the-art methods in imputation accuracy and robustness, particularly in scenarios with substantial missing data. The proposed model is well-suited for applications in wireless sensor networks and IoT environments, where data integrity is critical.