Factor Graphs

Accurate seafloor mapping is essential for marine science, archaeology, and environmental monitoring. However, integrating data from different sensors, such as side-scan sonar and optical cameras, collected across separate survey sessions, remains challenging due to positioning drift and sensor offsets. This paper presents a multi-session, multimodal underwater mapping framework based on factor graph optimization. The method jointly optimizes vehicle trajectories, 3D landmark positions, sensor extrinsics, and per-session global alignment transformations. By combining rigid inter-session corrections with local trajectory deformations, it compensates for both inter-session offsets and intra-session distortions from accumulated navigation errors. The proposed methodology was validated on real-world datasets collected along the Catalan coast. Results show measurable improvements in map consistency over both unoptimized and rigid-alignment baselines across all metrics, including Pixel Accuracy and mean Intersection over Union. The method achieves a 3.4% improvement in pixel accuracy over the unoptimized baseline, corresponding to improved semantic labelling across approximately 14700 $\text{m}^2$ of mapped area. Qualitative results further show consistent co-registration between sonar and optical maps, even in the presence of significant trajectory distortions and inter-session misalignments. These findings demonstrate the potential of the proposed framework to generate coherent multimodal seafloor maps from heterogeneous underwater surveys.

Estimating deformable object states remains a fundamental challenge in robotics and simulation. We propose a novel factor graph-based framework for probabilistic mesh state estimation of deformable objects. The method directly updates a tetrahedral mesh, a rich and physically-grounded representation of an environment, by combining physics priors, noisy sensor measurements, and temporal smoothness constraints within a unified probabilistic formulation. The estimation problem is posed as a nonlinear least-squares optimization and solved using Levenberg-Marquardt. Ex vivo central-airway obstruction experiments and simulations on deforming cube models demonstrate reliable and accurate reconstruction under both rigid motion and deformation, highlighting the potential of this probabilistic approach for principled, measurement-driven mesh state estimation in deformable object reconstruction.

Multi-output Gaussian process regression scales cubically in the number of observations times outputs, and dense kernel-matrix methods need bespoke handling whenever different outputs are observed at different inputs. We express multi-output Gaussian process regression as a Forney-style factor graph in which a nearest-neighbor chain orders a fixed candidate set of $C$ inputs into a one-dimensional sequence. Along this chain, latent Matérn processes evolve through linear-Gaussian transition factors, while the linear model of coregionalization mixes $L$ latent processes into $D$ outputs through a deterministic mixing factor and per-output scalar observation factors. Posterior computation reduces to exact Gaussian message passing on the chain at cost $\mathcal{O}(C(DL^2 + L^3))$ after chain construction, and missing observations omit their local factor without any covariance-matrix restructuring. The formulation therefore scales in the number of data samples and in the rate of missing observations, while remaining best suited to candidate sets in low input dimension.We compare the factor-graph formulation against an exact kernel-matrix baseline, a sparse-variational inducing-point baseline, and a nearest-neighbor baseline on a synthetic input-dimension sweep and on electricity time series forecasting. At low input dimension the factor-graph posterior tracks the exact kernel-matrix posterior closely, and the gap grows gradually as input dimension increases while staying competitive with both approximate baselines. On the electricity time series our factor-graph formulation matches all three baselines in forecast accuracy while scaling linearly in the number of data points, where the exact kernel-matrix method becomes infeasible and the inducing-point baseline remains substantially slower.

Current portfolio construction methods are either agnostic to the effects of idiosyncratic shocks (standard factor models) or to the latent data structure driving systematic returns (recent graph-based approaches). This presents an opportunity to combine the complementary market aspects captured by the factor and graph domains, allowing asset allocations to operate directly on the underlying market structure, rather than on its observed co-movement or its finite-sample artefacts. In this work, we introduce the Mutually-INformed Graph-Locality and Exposures framework (MINGLE), which mutually regularises the factor and graph domains by redefining graph locality through systematic factor exposure profiles, rather than via observed co-movements. This is formalised through a unified Alternating Direction Method of Multipliers (ADMM) framework that jointly learns a latent factor representation and its induced graph topology directly from market returns. The resulting exposure-similarity graph aligns more closely with established economic sectors than conventional correlation-based graphs. Portfolios constructed from this representation are shown to consistently outperform their correlation-based counterparts across a range of volatility regimes and transaction cost levels. For rigour, paired statistical testing confirms that these gains stem from the reconciliation of the graph and factor domains.

We study the convergence properties of the random-sweep Gibbs sampler for Gaussian graphical models with a thin-membrane prior. We demonstrate that the convergence rate of the Gibbs sampler is significantly accelerated in the dual model, which is obtained by applying the Fourier transform to the local factors of the normal factor graph representing the original model. In both domains, we derive the exact convergence rates for homogeneous $k$-regular graphs. We prove that, for all homogeneous models whose graphical representations contain cycles, the convergence rate in the dual domain is universal and independent of the underlying graph topology. Moreover, we show that the effective convergence rate in the dual domain is governed by the algebraic connectivity of the graph, providing an additional acceleration without increasing the computational complexity per sweep. We further establish an explicit algebraic relation between the covariance structures of the primal and dual models, enabling marginal statistics of the primal model to be recovered directly from those of the dual model. Finally, numerical experiments on several graph families confirm our theoretical results and demonstrate substantial improvements in the convergence rates in various settings.

The problem of learning the graphical Markov blanket (MB) of a variable from data has applications in many areas such as structure learning for Bayesian networks and Markov random fields, causal discovery, and feature selection. However, a common assumption most methods make is that the conditional independencies in the distribution imply the same separation in the graphical structure -- also known as the faithfulness assumption. Unfortunately, this assumption can be violated by higher-order dependencies such as XOR and parity-type relations, and -- on finite samples -- by empirical violations that, in extreme cases, even induce spurious dependencies absent from the true distribution. Therefore, in this paper we propose a "k-order" relaxation of the faithfulness assumption that captures parity type relationships between k+2 variables. We then propose a proof of concept algorithm called k-order Markov blanket (kOMB) that uses this relaxation for MB discovery. Finally, we empirically show how kOMB can recover the MB of a variable under both true and empirical violations of faithfulness. Code available at: https://github.com/lklee9/k-order-Markov-blanket

Learning causal graphs from interventional data is a challenging problem with broad applications. In molecular biology, for example, a central goal is to uncover gene regulatory networks from large-scale perturbation data. An ideal algorithm for this task should scale to thousands of nodes, incorporate interventions even when their targets are unknown, quantify uncertainty, and provide identifiability guarantees. However, existing approaches---e.g. approaches using score-based optimization or approximate Bayesian inference---often fail to meet all of these criteria. To address these limitations, we develop Amortized Bayesian Causal Discovery of Extended Factor Graphs (ABCDEFG). Our method guarantees exact acyclicity, scales to graphs with thousands of nodes, and naturally handles interventions even when their targets are unknown. Additionally, ABCDEFG estimates a posterior distribution whose maximum a posteriori estimate provably identifies the true causal graph up to an equivalence class. On simulated datasets, ABCDEFG achieves state-of-the-art accuracy, producing a well-calibrated posterior distribution while outperforming previous score-based and approximate Bayesian methods. Applied to large-scale single-cell perturbation data, ABCDEFG identifies both established and novel gene targets of growth factors.

Multiple Sequence Alignments (MSAs) provide protein language models with explicit evolutionary context, but their large depth makes subsampling unavoidable under limited token budgets. Existing strategies, including random selection, identity-based filtering, and diversity-driven sampling, are effective heuristics, yet provide limited control over the evolutionary signals retained in the subset. In this work, we recast MSA subsampling as an explicit optimization problem, where key evolutionary measures, including query identity and diversity, are treated as controllable objectives. Building on this view, we introduce AP-REASONER, an Affinity-Propagation-based factor-graph approach. With evolution-aware unary factors, exemplar-consistency factors, and two control knobs, AP-REASONER performs factor-graph reasoning through message passing to infer a fixed-budget MSA subset. Experiments on long-range contact prediction and conformational ensemble prediction show that AP-REASONER outperforms baseline subsamplers on structure-sensitive downstream tasks and enables controllable recovery of alternative protein conformations. These results highlight the value of modeling MSA subsampling as a controllable optimization problem, where factor-graph reasoning offers an effective alternative to heuristic selection.

Continuum robots offer key advantages in navigating unstructured environments, but their safe operation requires accurate estimation of the external contact forces acting anywhere along the robot body. Estimating these forces at unknown locations is an ill-conditioned problem, particularly for multiple contacts. We propose a unified shape and force estimation framework formulated on a factor graph. By incorporating a Gaussian mixture force parameterization into a discretized probabilistic Cosserat rod model, we reduce the dimensionality of the unknown external forces and mitigate the ill-conditioning of node-wise force estimation. The framework fuses strain, tendon tension, and pose measurements to simultaneously estimate the robot's shape and external forces while accounting for modeling and sensor uncertainties. Numerical simulations demonstrate that the proposed method outperforms existing methods in terms of force location and magnitude estimation for both single and multi-contact scenarios. We further present a progressive variant that introduces basis functions on demand to estimate contact forces sequentially during a simulated confined-navigation task.

Information lattice learning (ILL) learns interpretable rules of a signal by alternately projecting the signal onto a partition lattice that encodes a hierarchy of abstractions and lifting selected rules back to the signal domain. When the signal is a probability mass function, we show the probabilistic rules learned by ILL admit a natural probabilistic graphical model (PGM) interpretation and develop this interpretation in detail. A partition in ILL induces a deterministic quotient variable, and a rule is the marginal law of that quotient variable. A rule set is therefore a collection of marginal constraints over interpretable abstractions. General lifting is the feasible family of all joint distributions satisfying those constraints, while special lifting chooses a maximum-ignorance reconstruction, implemented in ILL by an L2 uniformity principle closely related to maximum entropy. Under a Shannon-entropy lifting, the same constraints yield a log-linear factor graph whose factors are indexed by learned abstractions. The information lattice itself, however, is not a Bayesian network: its edges encode refinement and coarsening of abstractions, not conditional dependence. Thus ILL is best viewed as structure learning for interpretable constraint-based factor graphs over quotient variables. This view clarifies how ILL relates to graphical models and maximum entropy models, while suggesting new directions for inference, identifiability, and hybrid symbolic-probabilistic learning.

Robust and efficient state estimation is crucial for perception, navigation, and control in robotics. State estimation problems are conveniently modeled using the factor-graph framework as enabled by modern software packages such as GTSAM or g2o. However, the standard solvers included in such frameworks are local and may converge to poor local minima, posing significant safety concerns. Conversely, techniques based on convex relaxations have been shown to provide a means of globally solving or certifying many state estimation problems. However, these relaxations 1) often require substantial effort to formulate, and 2) may incur significantly higher cost compared to efficient local solvers, as they require solving a large semidefinite program (SDP). In this work, we address both shortcomings by 1) creating a new procedure within the GTSAM framework for automatically constructing convex SDP relaxations for any factor graphs with common factor and variable types, and by 2) exploiting the Bayes tree constructions native to GTSAM to decompose the SDP problem, leading to significant speedup in solver time for chordally sparse problems. We demonstrate the favorable scaling of this structure-exploiting global estimator compared to standard local solvers for two case studies: A 3D pose-graph SLAM problem with a ring factor graph and a 2D localization problem with a chain factor graph. The software framework is available at https://github.com/borglab/gtsam.

Stacking probabilistic building blocks into deeper architectures typically breaks closed-form inference. We show that closed-form inference can be preserved. We identify five factor-graph primitives: a bilinear factor, an exponential link, a Gamma prior, a Gaussian likelihood, and an equality node, and prove that any model composed from them admits closed-form variational message passing. The construction works because each primitive preserves a small set of message families: under mean-field factorization, messages on Gaussian variables remain Gaussian and messages on precision variables remain Gamma, while the only non-conjugate interface, the exponential link, remains tractable through the Gaussian moment-generating function and the sufficient statistics of the Gamma family. We demonstrate composition at increasing depth, from static ensembles through input-dependent gating to split-branch routing, and show that stacking routing layers encodes arbitrary decision trees, establishing universal function approximation with closed-form inference. Applied to ensemble time-series forecasting, the framework yields a Bayesian mixture of experts in which gating functions are inferred rather than learned, providing calibrated uncertainty over expert selection across five benchmark datasets.

Exploiting the indistinguishability of objects in a probabilistic graphical model such as a factor graph is key to lifted probabilistic inference algorithms and allows for tractable probabilistic inference problems with respect to domain sizes. A central building block for the exploitation of indistinguishable objects in factor graphs is the identification of commutative factors, i.e., factors whose output values are invariant under permutations of input values assigned to a subset of their arguments. In this paper, we revisit the theoretical foundations underlying the state-of-the-art algorithm to detect commutative factors. Specifically, we show that in its current form, the state-of-the-art algorithm relies on a central theorem that is mistakenly regarded as a sufficient condition to identify commutative factors, while it actually only implies necessary condition. Consequently, the state of the art might, as we show in this paper, deliver incorrect results. To fix the flaws currently present in the state of the art, we prove a slightly modified version of the aforementioned theorem, which serves as a necessary condition to identify commutative factors. Moreover, we present a corrected version of the state-of-the-art algorithm, which keeps its efficiency while ensuring correctness and introduce a complementary algorithm with tighter worst-case bounds.

In Global Navigation Satellite System (GNSS)-degraded environments, pseudolites (PLs) provide additional signal sources to enhance positioning performance, but their integration in optimization-based frameworks remains limited. This paper presents a loosely coupled factor graph optimization (FGO) framework that fuses the GNSS/PL least-squares (LS) solutions with inertial measurement unit (IMU) data. The evaluation considers low GNSS visibility scenarios with four high-elevation GNSS satellites and up to two PL transmitters over an 80~s window. FGO achieves a 22.8% to 41.3% reduction in mean 3D error compared to standard LS methods. Compared to a GNSS-IMU baseline, incorporating PL transmitters further improves positioning accuracy, with performance depending on geometry.

Inferring time-varying graph structures from high-dimensional nodal observations is a fundamental problem arising in neuroscience, finance, climatology, and beyond. Two intrinsic challenges govern this problem: maintaining the \emph{temporal coherence} of the latent graph across successive observation windows, and respecting the \emph{intrinsic Riemannian geometry} of the symmetric positive definite manifold on which precision matrices naturally reside, a curved space whose geodesic structure departs fundamentally from that of the ambient Euclidean space. In this paper we propose dynamic estimation on the Grassmann manifold with a factor model (\textsc{Degfm}), a novel algorithm that jointly addresses both challenges. We model the time-varying precision matrix sequence as a low-rank-plus-diagonal structure governed by a latent elliptical graph factor model, which drastically reduces the effective parameter count and enables reliable estimation in the challenging small-sample regime. Temporal coherence is enforced through a Riemannian geodesic penalty defined on the Grassmann manifold, ensuring that the estimated graph trajectory is smooth with respect to the intrinsic geometry rather than the ambient Euclidean space. To solve the resulting non-convex optimization problem over Grassmann-manifold-valued sequences subject to the LRaD constraint, we derive an efficient Riemannian gradient descent algorithm that respects the manifold structure at every iterate and rigorously establish its convergence to a stationary point. Extensive experiments on both synthetic benchmarks and real-world datasets demonstrate that \textsc{Degfm} consistently outperforms state-of-the-art baselines across all evaluation metrics, confirming the practical effectiveness of the proposed framework.

DynoJEPP is a factor-graph-based framework that jointly formulates and simultaneously optimizes estimation, prediction, and planning in dynamic environments. In conventional factor-graph-based approaches that jointly formulate estimation, prediction, and planning, information from prediction and planning feeds back into state estimation, yielding corrupted estimates, undesired behaviors, and unsafe plans. To address this, DynoJEPP introduces a novel directed factor that enforces directional information flow within the factor graph, preventing prediction and planning from corrupting state estimation. We evaluate the impact of directed factors on inter-module interactions during navigation in both static and dynamic environments. Our results demonstrate that these factors are critical for safe operation, as without them, the robot collides in the majority of experiments. Building on this, we further introduce Cooperative DynoJEPP, which enables the ego robot to incorporate cooperative object behavior into its prediction and trajectory planning.

Continuous-time state estimation is gaining in popularity due to its abilities to provide smooth solutions, handle asynchronous sensors, and interpolate between data points. While there are two main paradigms, parametric (e.g., temporal basis functions, splines) and nonparametric (Gaussian processes), the latter has seen less adoption despite its technical advantages and relative ease of implementation. In this article, we seek to rectify this situation by providing a new simplified explanation of GP continuous-time estimation rooted in the language of factor graphs, which have become the de facto estimation paradigm in much of robotics. To simplify onboarding, we also provide three working examples implemented in the popular GTSAM estimation framework.

Graph representation learning (GRL) has evolved from topology-only graph embeddings to task-specific supervised GNNs, and more recently to reusable representations and graph foundation models (GFMs). However, existing evaluations mainly measure clean transfer, adaptation, and task coverage. It remains unclear whether GRL methods stay reliable when deployment stresses affect graph signals, graph contexts, label support, structural groups, or predictive evidence. We introduce GRL-Safety, a multi-axis safety evaluation benchmark for GRL. GRL-Safety evaluates twelve representative methods, spanning topology-only embedding methods, supervised GNNs, self-supervised graph models, and GFMs, on twenty-five graph datasets under standardized evaluation conditions while preserving method-native adaptation. The evaluation covers five safety axes: corruption robustness, OOD generalization, class imbalance, fairness, and interpretation, with per-axis and sub-condition reporting rather than a single aggregate score. Our analysis yields three cross-axis insights that can inspire future research. First, safety behavior is shaped by the interaction between representation design and the stressed graph factor, rather than by method family alone. Second, foundation-era methods show axis-specific strengths rather than broad safety dominance. Third, several deployment regimes remain difficult even for the best evaluated method, revealing capability gaps that require new robustness, adaptation, or training objectives beyond model selection. The benchmark, evaluation protocols, and code are available at: https://github.com/GXG-CS/GRL-Safety.

Pixel-space diffusion has re-emerged as a promising alternative to latent-space generation because it avoids the representation bottleneck introduced by VAEs. Yet most existing methods still treat image generation as a frequency-homogeneous process, overlooking the distinct roles and learning dynamics of low- and high-frequency components. To address this, we propose FREPix, a FREquency-heterogeneous flow matching framework for Pixel-space image generation. FREPix explicitly decomposes generation into low- and high-frequency components, assigns them separate transport paths, predicts them with a factorized network, and trains them with a frequency-aware objective. In this way, coarse-to-fine generation becomes an explicit design principle rather than an implicit behavior. On ImageNet class-to-image generation, FREPix achieves competitive results among pixel-space generation models, reaching 1.91 FID at $256\times256$ and 2.38 FID at $512\times512$, with particularly strong behavior in the low-NFE regime.

Global navigation satellite system (GNSS) positioning is widely used for urban navigation, but the covariance reported by the GNSS solver is often unreliable in urban canyons. Existing differentiable factor graph optimization (DFGO) methods learn measurement weighting through the solver, but they still use position-only objectives. As a result, the position estimate may improve while the reported covariance remains too small, too large, or incorrectly oriented. We propose CredibleDFGO (CDFGO), a differentiable GNSS factor graph framework that makes covariance credibility an explicit training target. A Weighting Generation Network (WGN) predicts per-satellite reliability weights, and a differentiable Gauss-Newton solver maps these weights to a position estimate and a Hessian-derived posterior covariance. We use proper scoring rules to supervise the East-North predictive distribution end to end. We study negative log-likelihood (NLL), the energy score (ES), and their combination. Results on three UrbanNav test scenes show consistent gains in covariance credibility. Positioning accuracy also improves on the medium-urban and harsh-urban scenes; on the deep-urban scene, both the mean horizontal error and the 95th-percentile error improve. On the harsh-urban Mong Kok (MK) scene, CDFGO-Combined reduces the mean horizontal error from 13.77 m to 11.68 m, reduces NLL from 40.63 to 6.59, and reduces ES from 12.31 to 9.05 relative to DFGO (MAE). Case studies link the MK improvement to better axis-wise consistency, more credible local covariance ellipses, and satellite-level reweighting.

Most neural methods for Vehicle Routing Problems (VRPs) are limited to Euclidean settings or simple graphs. In this work, we instead consider multigraphs, where parallel edges represent distinct travel options with varying trade-offs (e.g., distance vs time). Few methods are designed for such formulations and those that do exist face major scalability issues. We mitigate these scalability issues via a Node-Edge Policy Factorization (NEPF) approach, which splits the routing policy into a node permutation stage and an edge selection stage. To enable the decomposition, we introduce a pre-encoding edge aggregation scheme and a non-autoregressive architecture for the edge stage, as well as a hierarchical reinforcement learning method to train the stages jointly. Our experiments across six VRP variants demonstrate that NEPF matches or outperforms the state-of-the-art in terms of solution quality, while being significantly faster in training and inference.

Single-view RGB object pose estimators have reached a level of precision and efficiency that makes them good candidates for vision-based robot control. However, off-the-shelf methods lack temporal consistency and robustness that are mandatory for a stable feedback control. In this work, we develop a factor graph approach to enforce temporal consistency of the object pose estimates. In particular, the proposed approach: (i) incorporates object motion models, (ii) explicitly estimates the object pose measurement uncertainty, and (iii) integrates the above two components in an online optimization-based estimator. We demonstrate that with appropriate outlier rejection and smoothing using the proposed factor graph approach, we can significantly improve the results on standardized pose estimation benchmarks. We experimentally validate the stability of the proposed approach for a feedback-based robot control task in which the object is tracked by the camera attached to a torque controlled manipulator.

Dense and accurate depth estimation is essential for robotic manipulation, grasping, and navigation, yet currently available depth sensors are prone to errors on transparent, specular, and general non-Lambertian surfaces. To mitigate these errors, large-scale monocular depth estimation approaches provide strong structural priors, but their predictions can be potentially skewed or mis-scaled in metric units, limiting their direct use in robotics. Thus, in this work, we propose a training-free depth grounding framework that anchors monocular depth estimation priors from a depth foundation model in raw sensor depth through factor graph optimization. Our method performs a patch-wise affine alignment, locally grounding monocular predictions in metric real-world depth while preserving fine-grained geometric structure and discontinuities. To facilitate evaluation in challenging real-world conditions, we introduce a benchmark dataset with dense scene-wide ground truth depth in the presence of non-Lambertian objects. Ground truth is obtained via matte reflection spray and multi-camera fusion, overcoming the reliance on object-only CAD-based annotations used in prior datasets. Extensive evaluations across diverse sensors and domains demonstrate consistent improvements in depth performance without any (re-)training. We make our implementation publicly available at https://anchord.cs.uni-freiburg.de.

The Probabilistic Transformer (PT) establishes that the Transformer's self-attention plus its feed-forward block is mathematically equivalent to Mean-Field Variational Inference (MFVI) on a Conditional Random Field (CRF). Under this equivalence the Transformer ceases to be a black-box neural network and becomes a programmable factor graph: graph topology, factor potentials, and the message-passing schedule are all explicit and inspectable primitives that can be engineered. PT was originally developed for natural language and in this report we investigate its potential for time series. We first lift PT into the Spatial-Temporal Probabilistic Transformer (ST-PT) to repair PT's missing channel axis and weak per-step semantics, and adopt ST-PT as a shared cornerstone backbone. We then identify three distinct properties that PT/ST-PT offers as a factor-graph model and derive three Research Questions, one per property, that probe how each property can be exploited in time series: RQ1. The graph topology and potentials are direct programmable primitives. Can this be used to inject symbolic time-series priors into ST-PT through structural graph modifications, especially under data scarcity and noise? RQ2. The CRF's factor matrices are the operator's potentials. Can an external condition program these factor matrices on a per-sample basis, so that conditional generation becomes structural rather than feature-level modulation of a fixed one? RQ3. Each MFVI iteration is a Bayesian posterior update on the factor graph. Can this turn the latent transition of latent-space AutoRegressive (AR) forecasting from an opaque MLP into a principled posterior update, and can a CRF teacher distill its latents into the AR student to counter cumulative error? We give one empirical study per question. Together, these three studies position ST-PT as a programmable framework for time-series modeling.