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arXiv:2510.08409v2 Announce Type: replace Abstract: We identify and analyze a surprising phenomenon of Latent Diffusion Models (LDMs) where the final steps of the diffusion can degrade sample quality. In contrast to conventional arguments that justify early stopping for numerical stability, this phenomenon is intrinsic to the dimensionality reduction in LDMs. We provide a principled explanation by analyzing the interaction between latent dimension and stopping time. Under a Gaussian framework with linear autoencoders, we characterize the conditions under which […]

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arXiv:2510.07088v4 Announce Type: replace Abstract: Fourier analysis on the Boolean hypercube is fundamentally defined as the orthogonal decomposition of the space of pseudo-Boolean functions with respect to the uniform probability measure. In this work, we propose an ANOVA-based generalization of the Fourier decomposition on the Boolean hypercube endowed with any arbitrary probability measure. We provide an emph{explicit} decomposition basis which generalizes the Walsh-Hadamard (or parity functions) basis under any emph{arbitrary} probability measure on the Boolean hypercube. We formulate […]

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arXiv:2510.00504v2 Announce Type: replace Abstract: When training large-scale models, the performance typically scales with the number of parameters and the dataset size according to a slow power law. A fundamental theoretical and practical question is whether comparable performance can be achieved with significantly smaller models and substantially less data. In this work, we provide a positive and constructive answer. We prove that a generic permutation-invariant function of $d$ objects can be asymptotically compressed into a function of $operatorname{polylog} […]

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arXiv:2509.26560v2 Announce Type: replace Abstract: The global dimensionality of a neural representation manifold provides rich insight into the computational process underlying both artificial and biological neural networks. However, all existing measures of global dimensionality are sensitive to the number of samples, i.e., the number of rows and columns of the sample matrix. We show that, in particular, the participation ratio of eigenvalues, a popular measure of global dimensionality, is highly biased with small sample sizes, and propose a […]

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arXiv:2509.22459v2 Announce Type: replace Abstract: While achieving exceptional generative quality, modern diffusion, flow, and other matching models suffer from slow inference, as they require many steps of iterative generation. Recent distillation methods address this by training efficient one-step generators under the guidance of a pre-trained teacher model. However, these methods are often constrained to only one specific framework, e.g., only to diffusion or only to flow models. Furthermore, these methods are naturally data-free, and to benefit from the […]

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arXiv:2507.07469v3 Announce Type: replace Abstract: We introduce Galerkin-ARIMA and Galerkin-SARIMA, a projection-based extension of classical ARIMA/SARIMA that replaces rigid linear lag operators with low-dimensional Galerkin basis expansions while preserving the familiar AR-MA decomposition. Experiments on synthetic series and on quarterly GDP and daily S&P 500 returns show that Galerkin-SARIMA matches or improves forecast accuracy relative to classical ARIMA/SARIMA. Estimation is closed-form via a two-stage least-squares procedure, and the closed-form two-stage estimator enables efficient rolling-window re-estimation while preserving the […]

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arXiv:2501.04272v2 Announce Type: replace Abstract: We investigate the problem of weight uncertainty originally proposed by [Blundell et al. (2015). Weight uncertainty in neural networks. In International conference on machine learning, 1613-1622, PMLR.] in the context of neural networks designed for regression tasks, and we extend their framework by incorporating variance uncertainty into the model. Our analysis demonstrates that explicitly modeling uncertainty in the variance parameter can significantly enhance the predictive performance of Bayesian neural networks. By considering a […]

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arXiv:2501.04134v3 Announce Type: replace Abstract: We study the mixing time of the projected Langevin algorithm (LA) and the privacy curve of noisy Stochastic Gradient Descent (SGD), beyond nonexpansive iterations. Specifically, we derive new mixing time bounds for the projected LA which are, in some important cases, dimension-free and poly-logarithmic on the accuracy, closely matching the existing results in the smooth convex case. Additionally, we establish new upper bounds for the privacy curve of the subsampled noisy SGD algorithm. […]

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arXiv:2412.16031v2 Announce Type: replace Abstract: This paper introduces a novel approach to learning sparsity-promoting regularizers for solving linear inverse problems. We develop a bilevel optimization framework to select an optimal synthesis operator, denoted as $B$, which regularizes the inverse problem while promoting sparsity in the solution. The method leverages statistical properties of the underlying data and incorporates prior knowledge through the choice of $B$. We establish the well-posedness of the optimization problem, provide theoretical guarantees for the learning […]

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arXiv:2411.19305v2 Announce Type: replace Abstract: Data assimilation techniques are crucial for accurately tracking complex dynamical systems by integrating observational data with numerical forecasts. Recently, score-based data assimilation methods emerged as powerful tools for high-dimensional and nonlinear data assimilation. However, these methods still incur substantial computational costs due to the need for expensive forward simulations. In this work, we propose LD-EnSF, a novel score-based data assimilation method that eliminates the need for full-space simulations by evolving dynamics directly in […]

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