In this work, we propose a new approach to R-D estimation from the perspective of optimal transport.

We propose two new algorithms, SVRS and AccSVRS, motivated by previous works.

Feature attributions are post-training analysis methods that assess how various input features of a machine learning model contribute to an output prediction. Their interpretation is straightforward when features act independently, but becomes less direct when the predictive model involves interactions such as multiplicative relationships or joint feature contributions. In this work, we propose a general theory of higher-order feature attribution, which we develop on the foundation of Integrated Gradients (IG). This work extends existing frameworks in the literature on explainable AI. When using IG as the method of feature attribution, we discover natural connections to statistics and topological signal processing. We provide several theoretical results that establish the theory, and we validate our theory on a few examples.

Existing methods for evaluating graph generative models primarily rely on Maximum Mean Discrepancy (MMD) metrics based on graph descriptors. While these metrics can rank generative models, they do not provide an absolute measure of performance. Their values are also highly sensitive to extrinsic parameters, namely kernel and descriptor parametrization, making them incomparable across different graph descriptors. We introduce PolyGraph Discrepancy (PGD), a new evaluation framework that addresses these limitations. It approximates the Jensen-Shannon distance of graph distributions by fitting binary classifiers to distinguish between real and generated graphs, featurized by these descriptors. The data log-likelihood of these classifiers approximates a variational lower bound on the JS distance between the two distributions. Resulting metrics are constrained to the unit interval [0,1] and are comparable across different graph descriptors. We further derive a theoretically grounded summary metric that combines these individual metrics to provide a maximally tight lower bound on the distance for the given descriptors. Thorough experiments demonstrate that PGD provides a more robust and insightful evaluation compared to MMD metrics. The PolyGraph framework for benchmarking graph generative models is made publicly available at https://github.com/BorgwardtLab/polygraph-benchmark.

Mixtures of linear dynamical systems (MoLDS) provide a path to model time-series data that exhibit diverse temporal dynamics across trajectories. However, its application remains challenging in complex and noisy settings, limiting its effectiveness for neural data analysis. Tensor-based moment methods can provide global identifiability guarantees for MoLDS, but their performance degrades under noise and complexity. Commonly used expectation-maximization (EM) methods offer flexibility in fitting latent models but are highly sensitive to initialization and prone to poor local minima. Here, we propose a tensor-based method that provides identifiability guarantees for learning MoLDS, which is followed by EM updates to combine the strengths of both approaches. The novelty in our approach lies in the construction of moment tensors using the input-output data to recover globally consistent estimates of mixture weights and system parameters. These estimates can then be refined through a Kalman EM algorithm, with closed-form updates for all LDS parameters. We validate our framework on synthetic benchmarks and real-world datasets. On synthetic data, the proposed Tensor-EM method achieves more reliable recovery and improved robustness compared to either pure tensor or randomly initialized EM methods. We then analyze neural recordings from the primate somatosensory cortex while a non-human primate performs reaches in different directions. Our method successfully models and clusters different conditions as separate subsystems, consistent with supervised single-LDS fits for each condition. Finally, we apply this approach to another neural dataset where monkeys perform a sequential reaching task. These results demonstrate that MoLDS provides an effective framework for modeling complex neural data, and that Tensor-EM is a reliable approach to MoLDS learning for these applications.

Guiding pretrained flow-based generative models for conditional generation or to produce samples with desired target properties enables solving diverse tasks without retraining on paired data. We present ESS-Flow, a gradient-free method that leverages the typically Gaussian prior of the source distribution in flow-based models to perform Bayesian inference directly in the source space using Elliptical Slice Sampling. ESS-Flow only requires forward passes through the generative model and observation process, no gradient or Jacobian computations, and is applicable even when gradients are unreliable or unavailable, such as with simulation-based observations or quantization in the generation or observation process. We demonstrate its effectiveness on designing materials with desired target properties and predicting protein structures from sparse inter-residue distance measurements. In generative modeling, we are given data samples and aim to construct a sampler that approximates the data distribution. Diffusion models (Ho et al., 2020; Song et al., 2021) and continuous normalizing flows (Lipman et al., 2023; Liu et al., 2023; Albergo et al., 2023) achieve this by transporting samples from a simple source distribution to the data distribution.

Gradient-based methods are the primary approach for training ne ural networks. In recent years, research in learning theory has shown that neural networks can efficiently lea rn various data classes using empirical risk minimization (ERM) methods. In many real-world settings, data a rrive sequentially in a non-stationary manner, requiring the learner to maintain performance on past tas ks while acquiring new capabilities. In such cases, a learning model must be continually learnable, meaning it should retain previously acquired knowledge when trained on new tasks. On the other hand, various le arning systems, including deep learning architectures, can be prone to catastrophic forgetting, that is, updating a model on new data causes a dramatic drop in performance on previously learned tasks [ McCloskey and Cohen, 1989, Goodfellow et al., 2013 ]. The goal of continual (lifelong) learning is to develop models and methods that, even without retraining on old data, experience minimal forgetting when incorporating new inform ation. Despite deep learning's ubiquity, characterizing the power and limitat ions of neural networks is still an ongoing research direction. While several recent works aim to unde rstand the power of gradient descent (GD) for training neural networks with stylized data distributions, these works are still limited to single-task scenarios (for some examples see [ Du et al., 2019, Bartlett et al., 2021, Abbe et al., 2022 ]).

Low rank inference on matrices is widely conducted by optimizing a cost function augmented with a penalty proportional to the nuclear norm $\Vert \cdot \Vert_*$. However, despite the assortment of computational methods for such problems, there is a surprising lack of understanding of the underlying probability distributions being referred to. In this article, we study the distribution with density $f(X)\propto e^{-λ\Vert X\Vert_*}$, finding many of its fundamental attributes to be analytically tractable via differential geometry. We use these facts to design an improved MCMC algorithm for low rank Bayesian inference as well as to learn the penalty parameter $λ$, obviating the need for hyperparameter tuning when this is difficult or impossible. Finally, we deploy these to improve the accuracy and efficiency of low rank Bayesian matrix denoising and completion algorithms in numerical experiments.

Assessing the capabilities and risks of frontier AI systems is a critical area of research, and recent work has shown that repeated sampling from models can dramatically increase both. For instance, repeated sampling has been shown to increase their capabilities, such as solving difficult math and coding problems, but it has also been shown to increase their potential for harm, such as being jailbroken. Such results raise a crucial question for both capability and safety forecasting: how can one accurately predict a model's behavior when scaled to a massive number of attempts, given a vastly smaller sampling budget? This question is directly relevant to model providers, who serve hundreds of millions of users daily, and to governmental regulators, who seek to prevent harms. To answer this questions, we make three contributions. First, we find that standard methods for fitting these laws suffer from statistical shortcomings that hinder predictive accuracy, especially in data-limited scenarios. Second, we remedy these shortcomings by introducing a robust estimation framework, which uses a beta-binomial distribution to generate more accurate predictions from limited data. Third, we propose a dynamic sampling strategy that allocates a greater budget to harder problems. Combined, these innovations enable more reliable prediction of rare risks and capabilities at a fraction of the computational cost.

Part I of this work [Gao25] establishes online scaled gradient methods (OSGM), a framework that utilizes online convex optimization to adapt stepsizes in gradient methods. This paper focuses on the practical aspects of OSGM. We leverage the OSGM framework to design new adaptive first-order methods and provide insights into their empirical behavior. The resulting method, OSGM-Best, matches the performance of quasi-Newton variants while requiring less memory and cheaper iterations. We also extend OSGM to nonconvex optimization and outline directions that connect OSGM to existing branches of optimization theory and practice.