We present a generative learning framework for probabilistic sampling based on an extension of the Probabilistic Learning on Manifolds (PLoM) approach, which is designed to generate statistically consistent realizations of a random vector in a finite-dimensional Euclidean space, informed by a limited (yet representative) set of observations. In its original form, PLoM constructs a reduced-order probabilistic model by combining three main components: (a) kernel density estimation to approximate the underlying probability measure, (b) Diffusion Maps to uncover the intrinsic low-dimensional manifold structure, and (c) a reduced-order Ito Stochastic Differential Equation (ISDE) to sample from the learned distribution. A key challenge arises, however, when the number of available data points N is small and the dimensionality of the diffusion-map basis approaches N, resulting in overfitting and loss of generalization. To overcome this limitation, we propose an enabling extension that implements a synthesis of Double Diffusion Maps -- a technique capable of capturing multiscale geometric features of the data -- with Geometric Harmonics (GH), a nonparametric reconstruction method that allows smooth nonlinear interpolation in high-dimensional ambient spaces. This approach enables us to solve a full-order ISDE directly in the latent space, preserving the full dynamical complexity of the system, while leveraging its reduced geometric representation. The effectiveness and robustness of the proposed method are illustrated through two numerical studies: one based on data generated from two-dimensional Hermite polynomial functions and another based on high-fidelity simulations of a detonation wave in a reactive flow.

PLoM (Probabilistic Learning on Manifolds) is a method introduced in 2016 for handling small training datasets by projecting an It\^o equation from a stochastic dissipative Hamiltonian dynamical system, acting as the MCMC generator, for which the KDE-estimated probability measure with the training dataset is the invariant measure. PLoM performs a projection on a reduced-order vector basis related to the training dataset, using the diffusion maps (DMAPS) basis constructed with a time-independent isotropic kernel. In this paper, we propose a new ISDE projection vector basis built from a transient anisotropic kernel, providing an alternative to the DMAPS basis to improve statistical surrogates for stochastic manifolds with heterogeneous data. The construction ensures that for times near the initial time, the DMAPS basis coincides with the transient basis. For larger times, the differences between the two bases are characterized by the angle of their spanned vector subspaces. The optimal instant yielding the optimal transient basis is determined using an estimation of mutual information from Information Theory, which is normalized by the entropy estimation to account for the effects of the number of realizations used in the estimations. Consequently, this new vector basis better represents statistical dependencies in the learned probability measure for any dimension. Three applications with varying levels of statistical complexity and data heterogeneity validate the proposed theory, showing that the transient anisotropic kernel improves the learned probability measure.

Probabilistic approaches to computer vision typically assume a centralized setting, with the algorithm granted access to all observed data points. However, many problems in wide-area surveillance can benefit from distributed modeling, either because of physical or computational constraints. Most distributed models to date use algebraic approaches (such as distributed SVD) and as a result cannot explicitly deal with missing data. In this work we present an approach to estimation and learning of generative probabilistic models in a distributed context where certain sensor data can be missing. In particular, we show how traditional centralized models, such as probabilistic PCA and missing-data PPCA, can be learned when the data is distributed across a network of sensors. We demonstrate the utility of this approach on the problem of distributed affine structure from motion. Our experiments suggest that the accuracy of the learned probabilistic structure and motion models rivals that of traditional centralized factorization methods while being able to handle challenging situations such as missing or noisy observations.

Multivariate sequential data collected in practice often exhibit temporal irregularities, including nonuniform time intervals and component misalignment. However, if uneven spacing and asynchrony are endogenous characteristics of the data rather than a result of insufficient observation, the information content of these irregularities plays a defining role in characterizing the multivariate dependence structure. Existing approaches for probabilistic forecasting either overlook the resulting statistical heterogeneities, are susceptible to imputation biases, or impose parametric assumptions on the data distribution. This paper proposes an end-to-end solution that overcomes these limitations by allowing the observation arrival times to play the central role of model construction, which is at the core of temporal irregularities. To acknowledge temporal irregularities, we first enable unique hidden states for components so that the arrival times can dictate when, how, and which hidden states to update. We then develop a conditional flow representation to non-parametrically represent the data distribution, which is typically non-Gaussian, and supervise this representation by carefully factorizing the log-likelihood objective to select conditional information that facilitates capturing time variation and path dependency. The broad applicability and superiority of the proposed solution are confirmed by comparing it with existing approaches through ablation studies and testing on real-world datasets.

Probabilistic approaches to computer vision typically assume a centralized setting, with the algorithm granted access to all observed data points. However, many problems in wide-area surveillance can benefit from distributed modeling, either because of physical or computational constraints. Most distributed models to date use algebraic approaches (such as distributed SVD) and as a result cannot explicitly deal with missing data. In this work we present an approach to estimation and learning of generative probabilistic models in a distributed context where certain sensor data can be missing. In particular, we show how traditional centralized models, such as probabilistic PCA and missing-data PPCA, can be learned when the data is distributed across a network of sensors.

FlowOps enables optimal experience, predictions and decisions in the operations of factories and supply chains. In human brains, there are three key learning functions related to how we sense, predict and decide. Findings in computational neuroscience [1, 2] suggest that different parts of brain areas play a distinct but connected role in each function. These can be equated with the three Explainable AI (XAI) engines in Noodle.ai's The interplay between deep learning and probabilistic learning are similar to a human brain's thinking fast and slow like in Kahneman's System 1 and System 2. System 1 is a fast, intuitive, heuristic, deterministic, differentiable, and more affective mind, whereas System 2 is a slow, deliberate, logical, probabilistic, integrating, and more cognitive mind. Deep learning (Sentinel) enables fast, scalable, and associative pattern detections from high-dimensional, noisy and temporally correlated data, using differential optimizations on flexible functions with deterministic model parameters.

Probabilistic approaches to computer vision typically assume a centralized setting, with the algorithm granted access to all observed data points. However, many problems in wide-area surveillance can benefit from distributed modeling, either because of physical or computational constraints. Most distributed models to date use algebraic approaches (such as distributed SVD) and as a result cannot explicitly deal with missing data. In this work we present an approach to estimation and learning of generative probabilistic models in a distributed context where certain sensor data can be missing. In particular, we show how traditional centralized models, such as probabilistic PCA and missing-data PPCA, can be learned when the data is distributed across a network of sensors.

This paper tackles the challenge presented by small-data to the task of Bayesian inference. A novel methodology, based on manifold learning and manifold sampling, is proposed for solving this computational statistics problem under the following assumptions: 1) neither the prior model nor the likelihood function are Gaussian and neither can be approximated by a Gaussian measure; 2) the number of functional input (system parameters) and functional output (quantity of interest) can be large; 3) the number of available realizations of the prior model is small, leading to the small-data challenge typically associated with expensive numerical simulations; the number of experimental realizations is also small; 4) the number of the posterior realizations required for decision is much larger than the available initial dataset. The method and its mathematical aspects are detailed. Three applications are presented for validation: The first two involve mathematical constructions aimed to develop intuition around the method and to explore its performance. The third example aims to demonstrate the operational value of the method using a more complex application related to the statistical inverse identification of the non-Gaussian matrix-valued random elasticity field of a damaged biological tissue (osteoporosis in a cortical bone) using ultrasonic waves.

The Koopman operator has emerged as a powerful tool for the analysis of nonlinear dynamical systems as it provides coordinate transformations which can globally linearize the dynamics. Recent deep learning approaches such as Linearly-Recurrent Autoencoder Networks (LRAN) show great promise for discovering the Koopman operator for a general nonlinear dynamical system from a data-driven perspective, but several challenges remain. In this work, we formalize the problem of learning the continuous-time Koopman operator with deep neural nets in a measure-theoretic framework. This induces two forms of models: differential and recurrent form, the choice of which depends on the availability of the governing equation and data. We then enforce a structural parameterization that renders the realization of the Koopman operator provably stable. A new autoencoder architecture is constructed, such that only the residual of the dynamic mode decomposition is learned. Finally, we employ mean-field variational inference (MFVI) on the aforementioned framework in a hierarchical Bayesian setting to quantify uncertainties in the characterization and prediction of the dynamics of observables. The framework is evaluated on a simple polynomial system, the Duffing oscillator, and an unstable cylinder wake flow with noisy measurements.

In a recent paper, the authors proposed a general methodology for probabilistic learning on manifolds. The method was used to generate numerical samples that are statistically consistent with an existing dataset construed as a realization from a non-Gaussian random vector. The manifold structure is learned using diffusion manifolds and the statistical sample generation is accomplished using a projected Ito stochastic differential equation. This probabilistic learning approach has been extended to polynomial chaos representation of databases on manifolds and to probabilistic nonconvex constrained optimization with a fixed budget of function evaluations. The methodology introduces an isotropic-diffusion kernel with hyperparameter {\epsilon}. Currently, {\epsilon} is more or less arbitrarily chosen. In this paper, we propose a selection criterion for identifying an optimal value of {\epsilon}, based on a maximum entropy argument. The result is a comprehensive, closed, probabilistic model for characterizing data sets with hidden constraints. This entropy argument ensures that out of all possible models, this is the one that is the most uncertain beyond any specified constraints, which is selected. Applications are presented for several databases.