We formulate the inverse problem in a Bayesian framework and aim to train a generative model that allows us to simulate (i.e., sample from the likelihood) and do inference (i.e., sample from the posterior). We review the use of triangular normalizing flows for conditional sampling in this context and show how to combine two such triangular maps (an upper and a lower one) in to one invertible mapping that can be used for simulation and inference. We work out several useful properties of this invertible generative model and propose a possible training loss for training the map directly. We illustrate the workings of this new approach to conditional generative modeling numerically on a few stylized examples.

Simulation-based inference (SBI) enables Bayesian analysis when the likelihood is intractable but model simulations are available. Recent advances in statistics and machine learning, including Approximate Bayesian Computation and deep generative models, have expanded the applicability of SBI, yet these methods often face challenges in moderate to high-dimensional parameter spaces. Motivated by the success of gradient-based Monte Carlo methods in Bayesian sampling, we propose a novel SBI method that integrates score matching with Langevin dynamics to explore complex posterior landscapes more efficiently in such settings. Our approach introduces tailored score-matching procedures for SBI, including a localization scheme that reduces simulation costs and an architectural regularization that embeds the statistical structure of log-likelihood scores to improve score-matching accuracy. We provide theoretical analysis of the method and illustrate its practical benefits on benchmark tasks and on more challenging problems in moderate to high dimensions, where it performs favorably compared to existing approaches.

We investigate partially identifiable queries in a class of causal models. We focus on acyclic Structural Causal Models that are quasi-Markovian (that is, each endogenous variable is connected with at most one exogenous confounder). We look into scenarios where endogenous variables are observed (and a distribution over them is known), while exogenous variables are not fully specified. This leads to a representation that is in essence a Bayesian network where the distribution of root variables is not uniquely determined. In such circumstances, it may not be possible to precisely compute a probability value of interest. We thus study the computation of tight probability bounds, a problem that has been solved by multilinear programming in general, and by linear programming when a single confounded component is intervened upon. We present a new algorithm to simplify the construction of such programs by exploiting input probabilities over endogenous variables. For scenarios with a single intervention, we apply column generation to compute a probability bound through a sequence of auxiliary linear integer programs, thus showing that a representation with polynomial cardinality for exogenous variables is possible. Experiments show column generation techniques to be superior to existing methods.

Recent advancements in robot navigation, particularly with end-to-end learning approaches such as reinforcement learning (RL), have demonstrated strong performance. However, successful navigation still depends on two key capabilities: mapping and planning (explicitly or implicitly). Classical approaches rely on explicit mapping pipelines to register egocentric observations into a coherent map. In contrast, end-to-end learning often achieves this implicitly -- through recurrent neural networks (RNNs) that fuse current and historical observations into a latent space for planning. While existing architectures, such as LSTM and GRU, can capture temporal dependencies, our findings reveal a critical limitation: their inability to effectively perform spatial memorization. This capability is essential for integrating sequential observations from varying perspectives to build spatial representations that support planning. To address this, we propose Spatially-Enhanced Recurrent Units (SRUs) -- a simple yet effective modification to existing RNNs -- that enhance spatial memorization. We further introduce an attention-based network architecture integrated with SRUs, enabling long-range mapless navigation using a single forward-facing stereo camera. We also employ regularization techniques to facilitate robust end-to-end recurrent training via RL. Experimental results show 23.5% overall improvement in long-range navigation compared to existing RNNs. With SRU memory, our method outperforms RL baselines -- one relying on explicit mapping and the other on stacked historical observations -- by 29.6% and 105.0%, respectively, across diverse environments requiring long-horizon mapping and memorization. Finally, we address the sim-to-real gap by leveraging large-scale pretraining on synthetic depth data, enabling zero-shot transfer for deployment across diverse and complex real-world environments.

Online 3D Bin Packing Problem (3D-BPP) has widespread applications in industrial automation. Existing methods usually solve the problem with limited resolution of spatial discretization, and/or cannot deal with complex practical constraints well. We propose to enhance the practical applicability of online 3D-BPP via learning on a novel hierarchical representation, packing configuration tree (PCT). PCT is a full-fledged description of the state and action space of bin packing which can support packing policy learning based on deep reinforcement learning (DRL). The size of the packing action space is proportional to the number of leaf nodes, making the DRL model easy to train and well-performing even with continuous solution space. We further discover the potential of PCT as tree-based planners in deliberately solving packing problems of industrial significance, including large-scale packing and different variations of BPP setting. A recursive packing method is proposed to decompose large-scale packing into smaller sub-trees while a spatial ensemble mechanism integrates local solutions into global. For different BPP variations with additional decision variables, such as lookahead, buffering, and offline packing, we propose a unified planning framework enabling out-of-the-box problem solving. Extensive evaluations demonstrate that our method outperforms existing online BPP baselines and is versatile in incorporating various practical constraints. The planning process excels across large-scale problems and diverse problem variations. We develop a real-world packing robot for industrial warehousing, with careful designs accounting for constrained placement and transportation stability. Our packing robot operates reliably and efficiently on unprotected pallets at 10 seconds per box. It achieves averagely 19 boxes per pallet with 57.4% space utilization for relatively large-size boxes.

Despite the tremendous success of graph-based learning systems in handling structural data, it has been widely investigated that they are fragile to adversarial attacks on homophilic graph data, where adversaries maliciously modify the semantic and topology information of the raw graph data to degrade the predictive performances. Motivated by this, a series of robust models are crafted to enhance the adversarial robustness of graph-based learning systems on homophilic graphs. However, the security of graph-based learning systems on heterophilic graphs remains a mystery to us. To bridge this gap, in this paper, we start to explore the vulnerability of graph-based learning systems regardless of the homophily degree, and theoretically prove that the update of the negative classification loss is negatively correlated with the pairwise similarities based on the powered aggregated neighbor features. The theoretical finding inspires us to craft a novel robust graph structural learning strategy that serves as a useful graph mining module in a robust model that incorporates a dual-kNN graph constructions pipeline to supervise the neighbor-similarity-preserved propagation, where the graph convolutional layer adaptively smooths or discriminates the features of node pairs according to their affluent local structures. In this way, the proposed methods can mine the ``better" topology of the raw graph data under diverse graph homophily and achieve more reliable data management on homophilic and heterophilic graphs.

A Markov logic network (MLN) determines a probability distribution on the set of structures, or ``possible worlds'', with an arbitrary finite domain. We study the properties of such distributions as the domain size tends to infinity. Three types of concrete examples of MLNs will be considered, and the properties of random structures with domain sizes tending to infinity will be studied: (1) Arbitrary quantifier-free MLNs over a language with only one relation symbol which has arity 1. In this case we give a pretty complete characterization of the possible limit behaviours of random structures. (2) An MLN that favours graphs with fewer triangles (or more generally, fewer k-cliques). As a corollary of the analysis a ``$δ$-approximate 0-1 law'' for first-order logic is obtained. (3) An MLN that favours graphs with fewer vertices with degree higher than a fixed (but arbitrary) number. The analysis shows that depending on which ``soft constraints'' an MLN uses the limit behaviour of random structures can be quite different, and the weights of the soft constraints may, or may not, have influence on the limit behaviour. It will also be demonstrated, using (1), that quantifier-free MLNs and lifted Bayesian networks (in a broad sense) are asymptotically incomparable, roughly meaning that there is a sequence of distributions on possible worlds with increasing domain sizes that can be defined by one of the formalisms but not even approximated by the other. In a rather general context it is also shown that on large domains the distribution determined by an MLN concentrates almost all its probability mass on a totally different part of the space of possible worlds than the uniform distribution does.

Suppose two Bayesian agents each learn a generative model of the same environment. We will assume the two have converged on the predictive distribution, i.e. distribution over some observables in the environment, but may have different generative models containing different latent variables. Under what conditions can one agent guarantee that their latents are a function of the other agents latents? We give simple conditions under which such translation is guaranteed to be possible: the natural latent conditions. We also show that, absent further constraints, these are the most general conditions under which translatability is guaranteed. Crucially for practical application, our theorems are robust to approximation error in the natural latent conditions.

Parametric partial differential equations (PDEs) are fundamental mathematical tools for modeling complex physical systems, yet their numerical evaluation across parameter spaces remains computationally intensive when using conventional high-fidelity solvers. To address this challenge, we propose a novel physical law-corrected prior Gaussian process (LC-prior GP) surrogate modeling framework that effectively integrates data-driven learning with underlying physical constraints to flexibly handle multi-coupled variables defined on complex geometries. The proposed approach leverages proper orthogonal decomposition (POD) to parameterize high-dimensional PDE solutions via their dominant modes and associated coefficients, thereby enabling efficient Gaussian process (GP) surrogate modeling within a reduced-dimensional coefficient space. A key contribution lies in the incorporation of physical laws together with a limited number of parameter samples to correct the GP posterior mean, thus avoiding reliance on computationally expensive numerical solvers. Furthermore, interpolation functions are constructed to describe the mapping from the full parameter space to the physics-based correction term. This mapping is subsequently backpropagated to constrain the original GP surrogate, yielding a more physically consistent conditional prior. To handle irregular geometries, the radial basis function-finite difference (RBF-FD) method is incorporated during training set computation, with its inherent differentiation matrices providing both computational efficiency and numerical accuracy for physical constraint optimization. The effectiveness of the proposed method is demonstrated through numerical experiments involving a reaction-diffusion model, miscible flooding models, and Navier-Stokes equations with multi-physics coupling defined on irregular domains.

While calibration of probabilistic predictions has been widely studied, this paper rather addresses calibration of likelihood functions. This has been discussed, especially in biometrics, in cases with only two exhaustive and mutually exclusive hypotheses (classes) where likelihood functions can be written as log-likelihood-ratios (LLRs). After defining calibration for LLRs and its connection with the concept of weight-of-evidence, we present the idempotence property and its associated constraint on the distribution of the LLRs. Although these results have been known for decades, they have been limited to the binary case. Here, we extend them to cases with more than two hypotheses by using the Aitchison geometry of the simplex, which allows us to recover, in a vector form, the additive form of the Bayes' rule; extending therefore the LLR and the weight-of-evidence to any number of hypotheses. Especially, we extend the definition of calibration, the idempotence, and the constraint on the distribution of likelihood functions to this multiple hypotheses and multiclass counterpart of the LLR: the isometric-log-ratio transformed likelihood function. This work is mainly conceptual, but we still provide one application to machine learning by presenting a non-linear discriminant analysis where the discriminant components form a calibrated likelihood function over the classes, improving therefore the interpretability and the reliability of the method.