Modern approximations to Gaussian processes are suitable for "tall data", with a cost that scales well in the number of observations, but under-performs on ``wide data'', scaling poorly in the number of input features. That is, as the number of input features grows, good predictive performance requires the number of summarising variables, and their associated cost, to grow rapidly. We introduce a kernel that allows the number of summarising variables to grow exponentially with the number of input features, but requires only linear cost in both number of observations and input features. This scaling is achieved through our introduction of the B\'ezier buttress, which allows approximate inference without computing matrix inverses or determinants. We show that our kernel has close similarities to some of the most used kernels in Gaussian process regression, and empirically demonstrate the kernel's ability to scale to both tall and wide datasets.

Whereas diverse variations of diffusion models exist, extending the linear diffusion into a nonlinear diffusion process is investigated by very few works. The nonlinearity effect has been hardly understood, but intuitively, there would be promising diffusion patterns to efficiently train the generative distribution towards the data distribution. This paper introduces a data-adaptive nonlinear diffusion process for score-based diffusion models. The proposed Implicit Nonlinear Diffusion Model (INDM) learns by combining a normalizing flow and a diffusion process. Specifically, INDM implicitly constructs a nonlinear diffusion on the \textit{data space} by leveraging a linear diffusion on the \textit{latent space} through a flow network. This flow network is key to forming a nonlinear diffusion, as the nonlinearity depends on the flow network. This flexible nonlinearity improves the learning curve of INDM to nearly Maximum Likelihood Estimation (MLE) against the non-MLE curve of DDPM++, which turns out to be an inflexible version of INDM with the flow fixed as an identity mapping. Also, the discretization of INDM shows the sampling robustness. In experiments, INDM achieves the state-of-the-art FID of 1.75 on CelebA. We release our code at https://github.com/byeonghu-na/INDM.

The application of machine learning (ML) techniques in wireless communication domain has seen a tremendous growth over the years especially in the wireless sensing domain. However, the questions surrounding the ML model's inference reliability, and uncertainty associated with its predictions are never answered or communicated properly. This itself raises a lot of questions on the transparency of these ML systems. Developing ML systems with probabilistic modeling can solve this problem easily, where one can quantify uncertainty whether it is arising from the data (irreducible error or aleotoric uncertainty) or from the model itself (reducible or epistemic uncertainty). This paper describes the idea behind these types of uncertainty quantification in detail and uses a real example of WiFi channel state information (CSI) based sensing for motion/no-motion cases to demonstrate the uncertainty modeling. This work will serve as a template to model uncertainty in predictions not only for WiFi sensing but for most wireless sensing applications ranging from WiFi to millimeter wave radar based sensing that utilizes AI/ML models.

Computer simulations have proven a valuable tool for understanding complex phenomena across the sciences. However, the utility of simulators for modelling and forecasting purposes is often restricted by low data quality, as well as practical limits to model fidelity. In order to circumvent these difficulties, we argue that modellers must treat simulators as idealistic representations of the true data generating process, and consequently should thoughtfully consider the risk of model misspecification. In this work we revisit neural posterior estimation (NPE), a class of algorithms that enable black-box parameter inference in simulation models, and consider the implication of a simulation-to-reality gap. While recent works have demonstrated reliable performance of these methods, the analyses have been performed using synthetic data generated by the simulator model itself, and have therefore only addressed the well-specified case. In this paper, we find that the presence of misspecification, in contrast, leads to unreliable inference when NPE is used naively. As a remedy we argue that principled scientific inquiry with simulators should incorporate a model criticism component, to facilitate interpretable identification of misspecification and a robust inference component, to fit 'wrong but useful' models. We propose robust neural posterior estimation (RNPE), an extension of NPE to simultaneously achieve both these aims, through explicitly modelling the discrepancies between simulations and the observed data. We assess the approach on a range of artificially misspecified examples, and find RNPE performs well across the tasks, whereas naively using NPE leads to misleading and erratic posteriors.

Probabilistic Graphical Models (PGMs) are generative models of complex systems. They rely on conditional independence assumptions between variables to learn sparse representations which can be visualized in a form of a graph. Such models are used for domain exploration and structure discovery in poorly understood domains. This work introduces a novel technique to perform sparse graph recovery by optimizing deep unrolled networks. Assuming that the input data $X\in\mathbb{R}^{M\times D}$ comes from an underlying multivariate Gaussian distribution, we apply a deep model on $X$ that outputs the precision matrix $\hat{\Theta}$, which can also be interpreted as the adjacency matrix. Our model, uGLAD, builds upon and extends the state-of-the-art model GLAD to the unsupervised setting. The key benefits of our model are (1) uGLAD automatically optimizes sparsity-related regularization parameters leading to better performance than existing algorithms. (2) We introduce multi-task learning based `consensus' strategy for robust handling of missing data in an unsupervised setting. We evaluate model results on synthetic Gaussian data, non-Gaussian data generated from Gene Regulatory Networks, and present a case study in anaerobic digestion.

The Kalman filter (KF) is a widely-used algorithm for tracking the latent state of a dynamical system from noisy observations. For systems that are well-described by linear Gaussian state space models, the KF minimizes the mean-squared error (MSE). However, in practice, observations are corrupted by outliers, severely impairing the KFs performance. In this work, an outlier-insensitive KF is proposed, where robustness is achieved by modeling each potential outlier as a normally distributed random variable with unknown variance (NUV). The NUVs variances are estimated online, using both expectation-maximization (EM) and alternating maximization (AM). The former was previously proposed for the task of smoothing with outliers and was adapted here to filtering, while both EM and AM obtained the same performance and outperformed the other algorithms, the AM approach is less complex and thus requires 40 percentage less run-time. Our empirical study demonstrates that the MSE of our proposed outlier-insensitive KF outperforms previously proposed algorithms, and that for data clean of outliers, it reverts to the classic KF, i.e., MSE optimality is preserved

Retraining deep neural networks when new data arrives is typically computationally expensive. Moreover, certain applications do not allow such costly retraining due to time or computational constraints. Fast Bayesian updates are a possible solution to this issue. Therefore, we propose a Bayesian update based on Monte-Carlo samples and a last-layer Laplace approximation for different Bayesian neural network types, i.e., Dropout, Ensemble, and Spectral Normalized Neural Gaussian Process (SNGP). In a large-scale evaluation study, we show that our updates combined with SNGP represent a fast and competitive alternative to costly retraining. As a use case, we combine the Bayesian updates for SNGP with different sequential query strategies to exemplarily demonstrate their improved selection performance in active learning. Extending a dataset with new samples to train a deep learning model typically poses two problems. Updating a trained model may cause catastrophic forgetting while retraining may require high computational effort. Although the generalization performance typically justifies exhaustive retraining procedures, in some applications, retraining is not possible due to, for example, (1) the high number of retraining procedures in applications where data arrives sequentially and immediate updates are beneficial, e.g., in active learning (Settles, 2009) or when working on data streams (Sahoo et al., 2018), (2) the lack of computational power, e.g., for execution on embedded hardware (Taylor et al., 2018), (3) privacy reasons, e.g., when new data cannot be sent to distributed computing units (Taylor et al., 2018). Therefore, we suggest using Bayesian neural networks (BNNs, Fortuin, 2022) in the above examples as they not only provide additional uncertainty estimates or out-of-distribution detection capabilities but also allow updating the predictions with additional data without retraining the network (Kirsch et al., 2022). In this article, we develop a fast Bayesian update algorithm for BNNs. Figure 1 (left) shows the idea of sampling an ensemble of probabilistic hypotheses, each representing a possible true solution for the learning task (white samples).

We present a rotated hyperbolic wrapped normal distribution (RoWN), a simple yet effective alteration of a hyperbolic wrapped normal distribution (HWN). The HWN expands the domain of probabilistic modeling from Euclidean to hyperbolic space, where a tree can be embedded with arbitrary low distortion in theory. In this work, we analyze the geometric properties of the diagonal HWN, a standard choice of distribution in probabilistic modeling. The analysis shows that the distribution is inappropriate to represent the data points at the same hierarchy level through their angular distance with the same norm in the Poincar\'e disk model. We then empirically verify the presence of limitations of HWN, and show how RoWN, the proposed distribution, can alleviate the limitations on various hierarchical datasets, including noisy synthetic binary tree, WordNet, and Atari 2600 Breakout. The code is available at https://github.com/ml-postech/RoWN.

Sampling methods, as important inference and learning techniques, are typically designed for unconstrained domains. However, constraints are ubiquitous in machine learning problems, such as those on safety, fairness, robustness, and many other properties that must be satisfied to apply sampling results in real-life applications. Enforcing these constraints often leads to implicitly-defined manifolds, making efficient sampling with constraints very challenging. In this paper, we propose a new variational framework with a designed orthogonal-space gradient flow (O-Gradient) for sampling on a manifold $\mathcal{G}_0$ defined by general equality constraints. O-Gradient decomposes the gradient into two parts: one decreases the distance to $\mathcal{G}_0$ and the other decreases the KL divergence in the orthogonal space. While most existing manifold sampling methods require initialization on $\mathcal{G}_0$, O-Gradient does not require such prior knowledge. We prove that O-Gradient converges to the target constrained distribution with rate $\widetilde{O}(1/\text{the number of iterations})$ under mild conditions. Our proof relies on a new Stein characterization of conditional measure which could be of independent interest. We implement O-Gradient through both Langevin dynamics and Stein variational gradient descent and demonstrate its effectiveness in various experiments, including Bayesian deep neural networks.

A Bayesian pseudocoreset is a small synthetic dataset for which the posterior over parameters approximates that of the original dataset. While promising, the scalability of Bayesian pseudocoresets is not yet validated in realistic problems such as image classification with deep neural networks. On the other hand, dataset distillation methods similarly construct a small dataset such that the optimization using the synthetic dataset converges to a solution with performance competitive with optimization using full data. Although dataset distillation has been empirically verified in large-scale settings, the framework is restricted to point estimates, and their adaptation to Bayesian inference has not been explored. This paper casts two representative dataset distillation algorithms as approximations to methods for constructing pseudocoresets by minimizing specific divergence measures: reverse KL divergence and Wasserstein distance. Furthermore, we provide a unifying view of such divergence measures in Bayesian pseudocoreset construction. Finally, we propose a novel Bayesian pseudocoreset algorithm based on minimizing forward KL divergence. Our empirical results demonstrate that the pseudocoresets constructed from these methods reflect the true posterior even in high-dimensional Bayesian inference problems.