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1. AUTM Flow: Atomic Unrestricted Time Machine for Monotonic Normalizing Flows

   Cai, D.; Ji, Y.; He, H.; Ye, Q. (January 2022, Uncertainty in artificial intelligence)

   Nonlinear monotone transformations are used extensively in normalizing flows to construct invertible triangular mappings from simple distributions to complex ones. In existing literature, monotonicity is usually enforced by restricting function classes or model parameters and the inverse transformation is often approximated by root-finding algorithms as a closed-form inverse is unavailable. In this paper, we introduce a new integral-based approach termed: Atomic Unrestricted Time Machine (AUTM), equipped with unrestricted integrands and easy-to-compute explicit inverse. AUTM offers a versatile and efficient way to the design of normalizing flows with explicit inverse and unrestricted function classes or parameters. Theoretically, we present a constructive proof that AUTM is universal: all monotonic normalizing flows can be viewed as limits of AUTM flows. We provide a concrete example to show how to approximate any given monotonic normalizing flow using AUTM flows with guaranteed convergence. Our result implies that AUTM can be used to transform an existing flow into a new one equipped with explicit inverse and unrestricted parameters. The performance of the new approach is evaluated on high dimensional density estimation, variational inference and image generation. 

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2. AdaAnn: ADAPTIVE ANNEALING SCHEDULER FOR PROBABILITY DENSITY APPROXIMATION

   https://doi.org/10.1615/Int.J.UncertaintyQuantification.2022043110  

   Cobian, Emma R.; Hauenstein, Jonathan D.; Liu, Fang; Schiavazzi, Daniele E. (January 2023, International Journal for Uncertainty Quantification)

   Approximating probability distributions can be a challenging task, particularly when they are supported over regions of high geometrical complexity or exhibit multiple modes. Annealing can be used to facilitate this task which is often combined with constant a priori selected increments in inverse temperature. However, using constant increments limits the computational efficiency due to the inability to adapt to situations where smooth changes in the annealed density could be handled equally well with larger increments. We introduce AdaAnn, an adaptive annealing scheduler that automatically adjusts the temperature increments based on the expected change in the Kullback-Leibler divergence between two distributions with a sufficiently close annealing temperature. AdaAnn is easy to implement and can be integrated into existing sampling approaches such as normalizing flows for variational inference and Markov chain Monte Carlo. We demonstrate the computational efficiency of the AdaAnn scheduler for variational inference with normalizing flows on a number of examples, including posterior estimation of parameters for dynamical systems and probability density approximation in multimodal and high-dimensional settings. 

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3. Normalizing Flows for Intelligent Manufacturing

   https://doi.org/10.1115/MSEC2023-101281  

   Russell, Matthew; Wang, Peng (June 2023, American Society of Mechanical Engineers)

   Abstract Monitoring machine health and product quality enables predictive maintenance that optimizes repairs to minimize factory downtime. Data-driven intelligent manufacturing often relies on probabilistic techniques with intractable distributions. For example, generative models of data distributions can balance fault classes with synthetic data, and sampling the posterior distribution of hidden model parameters enables prognosis of degradation trends. Normalizing flows can address these problems while avoiding the training instability or long inference times of other generative Deep Learning (DL) models like Generative Adversarial Networks (GAN), Variational Autoencoders (VAE), and diffusion networks. To evaluate normalizing flows for manufacturing, experiments are conducted to synthesize surface defect images from an imbalanced data set and estimate parameters of a tool wear degradation model from limited observations. Results show that normalizing flows are an effective, multi-purpose DL architecture for solving these problems in manufacturing. Future work should explore normalizing flows for more complex degradation models and develop a framework for likelihood-based anomaly detection. Code is available at https://github.com/uky-aism/flows-for-manufacturing. 

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4. OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport

   Onken, D; Wu Fung, S; Li, Xingjian; Ruthotto, L (January 2021, Proceedings of the AAAI Conference on Artificial Intelligence)

   null (Ed.)

   A normalizing flow is an invertible mapping between an arbitrary probability distribution and a standard normal distribution; it can be used for density estimation and statistical inference. Computing the flow follows the change of variables formula and thus requires invertibility of the mapping and an efficient way to compute the determinant of its Jacobian. To satisfy these requirements, normalizing flows typically consist of carefully chosen components. Continuous normalizing flows (CNFs) are mappings obtained by solving a neural ordinary differential equation (ODE). The neural ODE's dynamics can be chosen almost arbitrarily while ensuring invertibility. Moreover, the log-determinant of the flow's Jacobian can be obtained by integrating the trace of the dynamics' Jacobian along the flow. Our proposed OT-Flow approach tackles two critical computational challenges that limit a more widespread use of CNFs. First, OT-Flow leverages optimal transport (OT) theory to regularize the CNF and enforce straight trajectories that are easier to integrate. Second, OT-Flow features exact trace computation with time complexity equal to trace estimators used in existing CNFs. On five high-dimensional density estimation and generative modeling tasks, OT-Flow performs competitively to state-of-the-art CNFs while on average requiring one-fourth of the number of weights with an 8x speedup in training time and 24x speedup in inference. 

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5. Simulator-Based Inference with WALDO: Confidence Regions by Leveraging Prediction Algorithms and Posterior Estimators for Inverse Problems

   Masserano, L.; Dorigo, T.; Izbicki, R.; Kuusela, M.; Lee, A. B. (January 2023, Proceedings of Machine Learning Research)

   Ruiz, F.; Dy, J.; Meent, J.-W. (Ed.)

   Prediction algorithms, such as deep neural networks (DNNs), are used in many domain sciences to directly estimate internal parameters of interest in simulator-based models, especially in settings where the observations include images or complex high-dimensional data. In parallel, modern neural density estimators, such as normalizing flows, are becoming increasingly popular for uncertainty quantification, especially when both parameters and observations are high-dimensional. However, parameter inference is an inverse problem and not a prediction task; thus, an open challenge is to construct conditionally valid and precise confidence regions, with a guaranteed probability of covering the true parameters of the data-generating process, no matter what the (unknown) parameter values are, and without relying on large-sample theory. Many simulator-based inference (SBI) methods are indeed known to produce biased or overly con- fident parameter regions, yielding misleading uncertainty estimates. This paper presents WALDO, a novel method to construct confidence regions with finite-sample conditional validity by leveraging prediction algorithms or posterior estimators that are currently widely adopted in SBI. WALDO reframes the well-known Wald test statistic, and uses a computationally efficient regression-based machinery for classical Neyman inversion of hypothesis tests. We apply our method to a recent high-energy physics problem, where prediction with DNNs has previously led to estimates with prediction bias. We also illustrate how our approach can correct overly confident posterior regions computed with normalizing flows. 

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