Diffusion models (DMs) demonstrate potent image generation capabilities in various generative modeling tasks. Nevertheless, their primary limitation lies in slow sampling speed, requiring hundreds or thousands of sequential function evaluations through large neural networks to generate high-quality images. Sampling from DMs can be seen as solving corresponding stochastic differential equations (SDEs) or ordinary differential equations (ODEs). In this work, we formulate the sampling process as an extended reverse-time SDE (ER SDE), unifying prior explorations into ODEs and SDEs. Leveraging the semi-linear structure of ER SDE solutions, we offer exact solutions and arbitrarily high-order approximate solutions for VP SDE and VE SDE, respectively. Based on the solution space of the ER SDE, we yield mathematical insights elucidating the superior performance of ODE solvers over SDE solvers in terms of fast sampling. Additionally, we unveil that VP SDE solvers stand on par with their VE SDE counterparts. Finally, we devise fast and training-free samplers, ER-SDE Solvers, elevating the efficiency of stochastic samplers to unprecedented levels. Experimental results demonstrate achieving 3.45 FID in 20 function evaluations and 2.24 FID in 50 function evaluations on the ImageNet 64$\times$64 dataset.
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Flow-based generative models enjoy certain advantages in computing the data generation and the likelihood, and have recently shown competitive empirical performance. Compared to the accumulating theoretical studies on related score-based diffusion models, analysis of flow-based models, which are deterministic in both forward (data-to-noise) and reverse (noise-to-data) directions, remain sparse. In this paper, we provide a theoretical guarantee of generating data distribution by a progressive flow model, the so-called JKO flow model, which implements the Jordan-Kinderleherer-Otto (JKO) scheme in a normalizing flow network. Leveraging the exponential convergence of the proximal gradient descent (GD) in Wasserstein space, we prove the Kullback-Leibler (KL) guarantee of data generation by a JKO flow model to be $O(\varepsilon^2)$ when using $N \lesssim \log (1/\varepsilon)$ many JKO steps ($N$ Residual Blocks in the flow) where $\varepsilon $ is the error in the per-step first-order condition. The assumption on data density is merely a finite second moment, and the theory extends to data distributions without density and when there are inversion errors in the reverse process where we obtain KL-$W_2$ mixed error guarantees. The non-asymptotic convergence rate of the JKO-type $W_2$-proximal GD is proved for a general class of convex objective functionals that includes the KL divergence as a special case, which can be of independent interest.
Generative diffusion models have achieved spectacular performance in many areas of generative modeling. While the fundamental ideas behind these models come from non-equilibrium physics, in this paper we show that many aspects of these models can be understood using the tools of equilibrium statistical mechanics. Using this reformulation, we show that generative diffusion models undergo second-order phase transitions corresponding to symmetry breaking phenomena. We argue that this lead to a form of instability that lies at the heart of their generative capabilities and that can be described by a set of mean field critical exponents. We conclude by analyzing recent work connecting diffusion models and associative memory networks in view of the thermodynamic formulations.
Importance sampling is a popular technique in Bayesian inference: by reweighting samples drawn from a proposal distribution we are able to obtain samples and moment estimates from a Bayesian posterior over some $n$ latent variables. Recent work, however, indicates that importance sampling scales poorly -- in order to accurately approximate the true posterior, the required number of importance samples grows is exponential in the number of latent variables [Chatterjee and Diaconis, 2018]. Massively parallel importance sampling works around this issue by drawing $K$ samples for each of the $n$ latent variables and reasoning about all $K^n$ combinations of latent samples. In principle, we can reason efficiently over $K^n$ combinations of samples by exploiting conditional independencies in the generative model. However, in practice this requires complex algorithms that traverse backwards through the graphical model, and we need separate backward traversals for each computation (posterior expectations, marginals and samples). Our contribution is to exploit the source term trick from physics to entirely avoid the need to hand-write backward traversals. Instead, we demonstrate how to simply and easily compute all the required quantities -- posterior expectations, marginals and samples -- by differentiating through a slightly modified marginal likelihood estimator.
This note introduces an unsupervised learning algorithm to debug errors in finite element (FE) simulation models and details how it was productionised. The algorithm clusters degrees of freedom in the FE model using numerical properties of the adjacency of its stiffness matrix. The algorithm has been deployed as a tool called `Model Stability Analysis' tool within the commercial structural FE suite Oasys GSA (www.oasys-software.com/gsa). It has been used successfully by end-users for debugging real world FE models and we present examples of the tool in action.
This paper presents a new algorithm for generating random inverse-Wishart matrices that directly generates the Cholesky factor of the matrix without computing the factorization. Whenever parameterized in terms of a precision matrix $\Omega=\Sigma^{-1}$, or its Cholesky factor, instead of a covariance matrix $\Sigma$, the new algorithm is more efficient than the current standard algorithm.
We introduce and demonstrate how to effectively train multilingual machine translation models with pixel representations. We experiment with two different data settings with a variety of language and script coverage, demonstrating improved performance compared to subword embeddings. We explore various properties of pixel representations such as parameter sharing within and across scripts to better understand where they lead to positive transfer. We observe that these properties not only enable seamless cross-lingual transfer to unseen scripts, but make pixel representations more data-efficient than alternatives such as vocabulary expansion. We hope this work contributes to more extensible multilingual models for all languages and scripts.
In high-dimensional linear models the problem of constructing adaptive confidence sets for the full parameter is known to be generally impossible. We propose re-weighted loss functions under which constructing fully adaptive confidence sets for the parameter is shown to be possible. We give necessary and sufficient conditions on the weights for adaptive confidence sets to exist, and exhibit a concrete rate-optimal procedure in the feasible regime.
Estimating the structure of directed acyclic graphs (DAGs) from observational data remains a significant challenge in machine learning. Most research in this area concentrates on learning a single DAG for the entire population. This paper considers an alternative setting where the graph structure varies across individuals based on available "contextual" features. We tackle this contextual DAG problem via a neural network that maps the contextual features to a DAG, represented as a weighted adjacency matrix. The neural network is equipped with a novel projection layer that ensures the output matrices are sparse and satisfy a recently developed characterization of acyclicity. We devise a scalable computational framework for learning contextual DAGs and provide a convergence guarantee and an analytical gradient for backpropagating through the projection layer. Our experiments suggest that the new approach can recover the true context-specific graph where existing approaches fail.
Suppose a finite, unweighted, combinatorial graph $G = (V,E)$ is the union of several (degree-)regular graphs which are then additionally connected with a few additional edges. $G$ will then have only a small number of vertices $v \in V$ with the property that one of their neighbors $(v,w) \in E$ has a higher degree $\mbox{deg}(w) > \mbox{deg}(v)$. We prove the converse statement: if a graph has few vertices having a neighbor with higher degree and satisfies a mild regularity condition, then, via adding and removing a few edges, the graph can be turned into a disjoint union of (distance-)regular graphs. The number of edge operations depends on the maximum degree and number of vertices with a higher degree neighbor but is independent of the size of $|V|$.
We introduce a Bayesian conditional autoregressive model for analyzing patient-specific and neighborhood risks of stillbirth and preterm birth within a city. Our fully Bayesian approach automatically learns the amount of spatial heterogeneity and spatial dependence between neighborhoods. Our model provides meaningful inferences and uncertainty quantification for both covariate effects and neighborhood risk probabilities through their posterior distributions. We apply our methodology to data from the city of Philadelphia. Using electronic health records (45,919 deliveries at hospitals within the University of Pennsylvania Health System) and United States Census Bureau data from 363 census tracts in Philadelphia, we find that both patient-level characteristics (e.g. self-identified race/ethnicity) and neighborhood-level characteristics (e.g. violent crime) are highly associated with patients' odds of stillbirth or preterm birth. Our neighborhood risk analysis further reveals that census tracts in West Philadelphia and North Philadelphia are at highest risk of these outcomes. Specifically, neighborhoods with higher rates of women in poverty or on public assistance have greater neighborhood risk for these outcomes, while neighborhoods with higher rates of college-educated women or women in the labor force have lower risk. Our findings could be useful for targeted individual and neighborhood interventions.
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