The ability to selectively remove knowledge from LLMs is highly desirable. However, existing methods often struggle with balancing unlearning efficacy and retain model utility, and lack controllability at inference time to emulate base model behavior as if it had never seen the unlearned data. In this paper, we propose LUNAR, a novel unlearning method grounded in the Linear Representation Hypothesis and operates by redirecting the representations of unlearned data to activation regions that expresses its inability to answer. We show that contrastive features are not a prerequisite for effective activation redirection, and LUNARachieves state-of-the-art unlearning performance and superior controllability. Specifically, LUNARachieves between 2.9 and 11.7 improvement in the combined unlearning efficacy and model utility score (Deviation Score) across various base models and generates coherent, contextually appropriate responses post-unlearning. Moreover, LUNAR effectively reduces parameter updates to a single down-projection matrix, a novel design that significantly enhances efficiency by 20 and robustness. Finally, we demonstrate that LUNARis robust to white-box adversarial attacks and versatile in real-world scenarios, including handling sequential unlearning requests.

The sensitivity of optimal transport (OT) to noise has motivated the study of robust variants. In this paper, we study two such formulations of semi-discrete OT in Rd: (i) the α-optimal partial transport, which minimizes the cost of transporting a mass of α; and (ii) the λ-robust optimal transport, which regularizes the OT problem using the total variation (TV) distance. First, we provide a novel characterization of the optimal solutions in these settings, showing they can be represented as a restricted Laguerre diagram. Second, we exploit this characterization to establish a strong algorithmic connection between the two problems, showing that any solver for one can be adapted to solve the other with comparable precision. Third, we overcome key challenges posed in extending the cost-scaling paradigm to compute these variants of OT and present an algorithm that computes the exact solution up to log(1/ε) bits of precision in nO(d) log(1/ε) time, where nis the support size of the discrete distribution.

Gradient-based data attribution methods, such as influence functions, are critical for understanding the impact of individual training samples without requiring repeated model retraining. However, their scalability is often limited by the high computational and memory costs associated with per-sample gradient computation. In this work, we propose GRASS, a novel gradient compression algorithm and its variants FACTGRASS for linear layers specifically, that explicitly leverage the inherent sparsity of per-sample gradients to achieve sub-linear space and time complexity. Extensive experiments demonstrate the effectiveness of our approach, achieving substantial speedups while preserving data influence fidelity. In particular, FACTGRASS achieves up to 165% faster throughput on billion-scale models compared to the previous state-of-the-art baselines.

Interpreting deep neural networks through concept-based explanations offers a bridge between low-level features and high-level human-understandable semantics. However, existing automatic concept discovery methods often fail to align these extracted concepts with the model's true decision-making process, thereby compromising explanation faithfulness. In this work, we propose FACE (Faithful Automatic Concept Extraction), a novel framework that augments Non-negative Matrix Factorization (NMF) with a Kullback-Leibler (KL) divergence regularization term to ensure alignment between the model's original and concept-based predictions. Unlike prior methods that operate solely on encoder activations, FACE incorporates classifier supervision during concept learning, enforcing predictive consistency and enabling faithful explanations. We provide theoretical guarantees showing that minimizing the KL divergence bounds the deviation in predictive distributions, thereby promoting faithful local linearity in the learned concept space. Systematic evaluations on ImageNet, COCO, and CelebA datasets demonstrate that FACE outperforms existing methods across faithfulness and sparsity metrics.

The empirical emergence of neural collapse--a surprising symmetry in the feature representations of the training data in the penultimate layer of deep neural networks--has spurred a line of theoretical research aimed at its understanding. However, existing work focuses on data-agnostic models or, when data structure is taken into account, it remains limited to multi-layer perceptrons. Our paper fills both these gaps by analyzing modern architectures in a data-aware regime: we prove that global optima of deep regularized transformers and residual networks (ResNets) with LayerNorm trained with cross entropy or mean squared error loss are approximately collapsed, and the approximation gets tighter as the depth grows. More generally, we formally reduce any end-to-end large-depth ResNet or transformer training into an equivalent unconstrained features model, thus justifying its wide use in the literature even beyond data-agnostic settings. Our theoretical results are supported by experiments on computer vision and language datasets showing that, as the depth grows, neural collapse indeed becomes more prominent.

Despite recent progress in identifying latent causal structures using causal representation learning (CRL), what makes learned representations useful for causal downstream tasks and how to evaluate them are still not well understood. In this paper, we reinterpret CRL using a measurement model framework, where the learned representations are viewed as proxy measurements of the latent causal variables. Our approach clarifies the conditions under which learned representations support downstream causal reasoning and provides a principled basis for quantitatively assessing the quality of representations using a new Test-based Measurement EXclusivity (T-MEX) score. We validate T-MEX across diverse causal inference scenarios, including numerical simulations and real-world ecological video analysis, demonstrating that the proposed framework and corresponding score effectively assess the identification of learned representations and their usefulness for causal downstream tasks.

Wasserstein gradient flow (WGF) is a common method to perform optimization over the space of probability measures. While WGF is guaranteed to converge to a first-order stationary point, for nonconvex functionals the converged solution does not necessarily satisfy the second-order optimality condition; i.e., it could converge to a saddle point. In this work, we propose a new algorithm for probability measure optimization, perturbed Wasserstein gradient flow (PWGF), that achieves second-order optimality for general nonconvex objectives. PWGF enhances WGF by injecting noisy perturbations near saddle points via a Gaussian process-based scheme. By pushing the measure forward along a random vector field generated from a Gaussian process, PWGF helps the solution escape saddle points efficiently by perturbing the solution towards the smallest eigenvalue direction of the Wasserstein Hessian. We theoretically derive the computational complexity for PWGF to achieve a second-order stationary point. Furthermore, we prove that PWGF converges to a global optimum in polynomial time for strictly benign objectives.

We study the problem of estimating the average treatment effect (ATE) in adaptive experiments where treatment can only be encouraged--rather than directly assigned--via a binary instrumental variable. Building on semiparametric efficiency theory, we derive the efficiency bound for ATE estimation under arbitrary, history-dependent instrument-assignment policies, and show it is minimized by a variance-aware allocation rule that balances outcome noise and compliance variability. Leveraging this insight, we introduce AMRIV--an Adaptive, Multiply-Robust estimator for Instrumental-Variable settings with variance-optimal assignment. AMRIV pairs (i) an online policy that adaptively approximates the optimal allocation with (ii) a sequential, influence-function-based estimator that attains the semiparametric efficiency bound while retaining multiply-robust consistency. We establish asymptotic normality, explicit convergence rates, and anytime-valid asymptotic confidence sequences that enable sequential inference. Finally, we demonstrate the practical effectiveness of our approach through empirical studies, showing that adaptive instrument assignment, when combined with the AMRIV estimator, yields improved efficiency and robustness compared to existing baselines.

Predicting changes in binding free energy ( G) is essential for understanding protein-protein interactions, which are critical in drug design and protein engineering. However, existing methods often rely on pre-trained knowledge and heuristic features, limiting their ability to accurately model complex mutation effects, particularly higher-order and many-body interactions. To address these challenges, we propose H3-DDG, a Hypergraph-driven Hierarchical network to capture Higherorder many-body interactions across multiple scales.

Dynamic pricing algorithms typically assume continuous price variables, which may not reflect real-world scenarios where prices are often discrete. This paper demonstrates that leveraging discrete price information within a semi-parametric model can substantially improve performance, depending on the size of the support set of the price variable relative to the time horizon. Specifically, we propose a novel semi-parametric contextual dynamic pricing algorithm, namely BayesCoxCP, based on a Bayesian approach to the Cox proportional hazards model. Our theoretical analysis establishes high-probability regret bounds that adapt to the sparsity level γ, proving that our algorithm achieves a regret upper bound of eO(T(1+γ)/2 + dT) for γ < 1/3 and eO(T2/3 + dT) for γ 1/3, where γ represents the sparsity of the price grid relative to the time horizon T. Through numerical experiments, we demonstrate that our proposed algorithm significantly outperforms an existing method, particularly in scenarios with sparse discrete price points.