Linear structural causal models (SCMs) -- in which each observed variable is generated by a subset of the other observed variables as well as a subset of the exogenous sources -- are pervasive in causal inference and casual discovery. However, for the task of causal discovery, existing work almost exclusively focus on the submodel where each observed variable is associated with a distinct source with non-zero variance. This results in the restriction that no observed variable can deterministically depend on other observed variables or latent confounders. In this paper, we extend the results on structure learning by focusing on a subclass of linear SCMs which do not have this property, i.e., models in which observed variables can be causally affected by any subset of the sources, and are allowed to be a deterministic function of other observed variables or latent confounders. This allows for a more realistic modeling of influence or information propagation in systems. We focus on the task of causal discovery form observational data generated from a member of this subclass. We derive a set of necessary and sufficient conditions for unique identifiability of the causal structure. To the best of our knowledge, this is the first work that gives identifiability results for causal discovery under both latent confounding and deterministic relationships. Further, we propose an algorithm for recovering the underlying causal structure when the aforementioned conditions are satisfied. We validate our theoretical results both on synthetic and real datasets.

We present 3DP3, a framework for inverse graphics that uses inference in a structured generative model of objects, scenes, and images. 3DP3 uses (i) voxel models to represent the 3D shape of objects, (ii) hierarchical scene graphs to decompose scenes into objects and the contacts between them, and (iii) depth image likelihoods based on real-time graphics. Given an observed RGB-D image, 3DP3's inference algorithm infers the underlying latent 3D scene, including the object poses and a parsimonious joint parametrization of these poses, using fast bottom-up pose proposals, novel involutive MCMC updates of the scene graph structure, and, optionally, neural object detectors and pose estimators. We show that 3DP3 enables scene understanding that is aware of 3D shape, occlusion, and contact structure. Our results demonstrate that 3DP3 is more accurate at 6DoF object pose estimation from real images than deep learning baselines and shows better generalization to challenging scenes with novel viewpoints, contact, and partial observability.

Multi-task learning requires accurate identification of the correlations between tasks. In real-world time-series, tasks are rarely perfectly temporally aligned; traditional multi-task models do not account for this and subsequent errors in correlation estimation will result in poor predictive performance and uncertainty quantification. We introduce a method that automatically accounts for temporal misalignment in a unified generative model that improves predictive performance. Our method uses Gaussian processes (GPs) to model the correlations both within and between the tasks. Building on the previous work by Kazlauskaiteet al. [2019], we include a separate monotonic warp of the input data to model temporal misalignment. In contrast to previous work, we formulate a lower bound that accounts for uncertainty in both the estimates of the warping process and the underlying functions. Also, our new take on a monotonic stochastic process, with efficient path-wise sampling for the warp functions, allows us to perform full Bayesian inference in the model rather than MAP estimates. Missing data experiments, on synthetic and real time-series, demonstrate the advantages of accounting for misalignments (vs standard unaligned method) as well as modelling the uncertainty in the warping process(vs baseline MAP alignment approach).

Simulation-based techniques such as variants of stochastic Runge-Kutta are the de facto approach for inference with stochastic differential equations (SDEs) in machine learning. These methods are general-purpose and used with parametric and non-parametric models, and neural SDEs. Stochastic Runge-Kutta relies on the use of sampling schemes that can be inefficient in high dimensions. We address this issue by revisiting the classical SDE literature and derive direct approximations to the (typically intractable) Fokker-Planck-Kolmogorov equation by matching moments. We show how this workflow is fast, scales to high-dimensional latent spaces, and is applicable to scarce-data applications, where a non-parametric SDE with a driving Gaussian process velocity field specifies the model.

From the previous article, we know that Bayesian Neural Network would treat the model weights and outputs as variables. Instead of finding a set of optimal estimates, we are fitting the probability distributions for them. But the problem is "How can we know what their distributions look like?" To answer this, you have to learn what prior, posterior, and Bayes' theorem are. In the following, we will use an example for illustration.

We present a mathematical framework and computational methods to optimally design a finite number of sequential experiments. We formulate this sequential optimal experimental design (sOED) problem as a finite-horizon partially observable Markov decision process (POMDP) in a Bayesian setting and with information-theoretic utilities. It is built to accommodate continuous random variables, general non-Gaussian posteriors, and expensive nonlinear forward models. sOED then seeks an optimal design policy that incorporates elements of both feedback and lookahead, generalizing the suboptimal batch and greedy designs. We solve for the sOED policy numerically via policy gradient (PG) methods from reinforcement learning, and derive and prove the PG expression for sOED. Adopting an actor-critic approach, we parameterize the policy and value functions using deep neural networks and improve them using gradient estimates produced from simulated episodes of designs and observations. The overall PG-sOED method is validated on a linear-Gaussian benchmark, and its advantages over batch and greedy designs are demonstrated through a contaminant source inversion problem in a convection-diffusion field.

We study the problem of constructing coresets for clustering problems with time series data. This problem has gained importance across many fields including biology, medicine, and economics due to the proliferation of sensors facilitating real-time measurement and rapid drop in storage costs. In particular, we consider the setting where the time series data on $N$ entities is generated from a Gaussian mixture model with autocorrelations over $k$ clusters in $\mathbb{R}^d$. Our main contribution is an algorithm to construct coresets for the maximum likelihood objective for this mixture model. Our algorithm is efficient, and under a mild boundedness assumption on the covariance matrices of the underlying Gaussians, the size of the coreset is independent of the number of entities $N$ and the number of observations for each entity, and depends only polynomially on $k$, $d$ and $1/\varepsilon$, where $\varepsilon$ is the error parameter. We empirically assess the performance of our coreset with synthetic data.

Based on rectangle theory of formal concept and set covering theory, the concept reduction preserving binary relations is investigated in this paper. It is known that there are three types of formal concepts: core concepts, relative necessary concepts and unnecessary concepts. First, we present the new judgment results for relative necessary concepts and unnecessary concepts. Second, we derive the bounds for both the maximum number of relative necessary concepts and the maximum number of unnecessary concepts and it is a difficult problem as either in concept reduction preserving binary relations or attribute reduction of decision formal contexts, the computation of formal contexts from formal concepts is a challenging problem. Third, based on rectangle theory of formal concept, a fast algorithm for reducing attributes while preserving the extensions for a set of formal concepts is proposed using the extension bit-array technique, which allows multiple context cells to be processed by a single 32-bit or 64-bit operator. Technically, the new algorithm could store both formal context and extent of a concept as bit-arrays, and we can use bit-operations to process set operations "or" as well as "and". One more merit is that the new algorithm does not need to consider other concepts in the concept lattice, thus the algorithm is explicit to understand and fast. Experiments demonstrate that the new algorithm is effective in the computation of attribute reductions.

We propose a new computationally efficient test for conditional independence based on the $L^{p}$ distance between two kernel-based representatives of well suited distributions. By evaluating the difference of these two representatives at a finite set of locations, we derive a finite dimensional approximation of the $L^{p}$ metric, obtain its asymptotic distribution under the null hypothesis of conditional independence and design a simple statistical test from it. The test obtained is consistent and computationally efficient. We conduct a series of experiments showing that the performance of our new tests outperforms state-of-the-art methods both in term of statistical power and type-I error even in the high dimensional setting.

We describe convolutional deep exponential families (CDEFs) in this paper. CDEFs are built based on deep exponential families, deep probabilistic models that capture the hierarchical dependence between latent variables. CDEFs greatly reduce the number of free parameters by tying the weights of DEFs. Our experiments show that CDEFs are able to uncover time correlations with a small amount of data.