Deep generative models such as flow matching and diffusion models have shown potential for learning complex dynamical systems, but typically act as black boxes that neglect underlying physical structure, while physics-based models governed by partial differential equations are often incomplete due to missing source terms, or uncertain parametrisations. We present Climate Physics Dynamic Matching (ClimPhyDM), a variational simulation-free dynamics informed framework for weather forecasting that combines an advection-type physics prior with data-driven components in a variational framework. % to capture the stochasticity and multi-modality of unresolved atmospheric dynamics. On the ERA5 benchmark at hourly (42-hour) and monthly (5-month) resolutions, ClimPhyDM outperforms ClimODE, and GB-DM, keeping the lower error at extended horizon, indicating improved temporal stability and resistance to error accumulation, while its simulation-free paradigm also enables training on a single modest 12 GB consumer GPU.
Outliers can substantially distort Gaussian process regression (GPR) due to its conventional Gaussian observation likelihood, leading to inaccurate model learning and prediction. To address this limitation, this article introduces a generative GPR model that captures observation-specific contamination and adaptively mitigates the influence of outliers. Subsequently, a variational generalized expectation-maximization procedure is used to learn the latent variables and GPR model parameters. Experiments on synthetic and real datasets under different contamination settings demonstrate that the proposed method remains competitive with-and in several cases outperforms-robust GPR baselines in prediction accuracy. Moreover, the proposed method shares the cubic computational scaling of the compared GPR methods.
Mykola Lukashchuk, Kyrylo Yemets, Alex Ledbetter +1cs.LG cs.AI
We show that the natural-gradient stationary condition of variational inference has an edge-local form on a Forney-style factor graph. We start from the Bethe free energy and constrain a selected edge marginal to an exponential family. At a stationary point, the natural parameter of that edge equals the sum of two projected messages, one from each incident factor. Each projected message is the natural-gradient projection of the exact belief-propagation log-message at the current receiving marginal, or equivalently, the gradient of its expectation in the so-called mean coordinates. We call the resulting scheme natural-gradient message passing (NGMP). The rule is local; each edge may carry its own exponential family, and the message a factor sends depends on the marginal that receives it. Compared with variational message passing, NGMP keeps the part of the exact message that the receiving family can represent instead of averaging the factor under the neighboring beliefs. The two coincide when the uncertainty on the edges entering a non-conjugate factor vanishes, and NGMP is more accurate when that uncertainty persists, for example, along a partially observed latent chain or when parameters are filtered through successive data batches. Experiments on Poisson smoothing, heteroskedastic regression, and hourly ETTh forecasting confirm this and show that the gain appears mainly in uncertainty calibration.
Incomplete multi-view clustering (IMVC) aims to uncover shared cluster structures from data with partially observed views. Although recent imputation-free methods based on variational inference demonstrate robustness to missing views, they commonly rely on a conditional independence assumption across views in the posterior aggregation stage, which fails to capture the inherently structured and potentially correlated nature of multi-view data. In this paper, we propose a variational framework that explicitly goes beyond this assumption by introducing a learnable cross-view correlation structure. Specifically, we explicitly model and learn correlations between views by utilizing the covariance structure of posterior estimation errors during aggregation. To facilitate robust and efficient learning, the correlation matrix is parameterized through a normalized Cholesky decomposition, ensuring positive definiteness and enabling the entire model to be trained jointly through a unified variational objective. Extensive experiments on multiple IMVC benchmarks demonstrate that our method consistently outperforms state-of-the-art approaches across diverse missing-view settings while introducing only a negligible number of learnable parameters. These results highlight the effectiveness of adaptive correlation modeling in variational IMVC, demonstrating the need to go beyond the independence assumption in IMVC. The code is available at https://github.com/zmxu196/ACOVA.
We study the implementable finite-batch particle algorithm for mean-field variational inference as a fully discrete stochastic approximation of the projected Wasserstein dynamics. The target potential is globally smooth but need not be strongly convex. The departure from contractivity is quantified by the curvature defect \[ \mathfrak d_α(x,y) = \bigl[α\|x-y\|^2- \langle\nabla V(x)-\nabla V(y),x-y\rangle\bigr]_+, \] which is the additive loss in the one-step Euler contraction estimate. We prove a non-asymptotic Wasserstein stability bound that separates initialization, product-empirical approximation, finite-batch drift error, time discretization, and the defects accumulated along the coupled trajectories. Under the uniform bound $\mathfrak d_α\leqβ$, the particle iterates remain within $O(\sqrt{β/α})$ of any MFVI minimizer, up to explicit errors in the particle number, batch size, and step size. The proof uses a stationary comparison array whose population law is an MFVI minimizer but whose particle-level law is a random product empirical measure, and it controls the resulting projected-drift discrepancy explicitly. We also give coordinatewise defect estimates and structural conditions for dimension-independent projected-drift sensitivity, construct an arbitrary-dimensional smooth nonconvex benchmark with a closed-form MFVI minimizer, and explain why polynomially growing drifts require a modification of the untamed explicit scheme.
Bayesian model selection couples a discrete model indicator with a model-specific continuous parameter space. We introduce structured dimension-matched variational transdimensional inference (SM-VTI) for finite enumerable model spaces. A rooted construction graph expresses a model as a sequence of local stop/child decisions. Each typed edge compiles a declared scientific parent-child edit into an exact native-coordinate dimension-matching lifting; an edge-conditioned flow then learns the residual continuous transport. The resulting local policy and conditional flow define one direct joint variational distribution, without embedding every model in a saturated maximum-dimensional surrogate. We derive its exact path density and optimize the joint reverse-KL objective. On a controlled 15-model target, SM-VTI-Joint recovers terminal masses, local actions, and nonlinear conditional geometry. On a 128-model misspecified robust variable-selection problem, a 10-data-set nearly parameter-matched affine comparison with AVTI shows stronger early model-mass recovery and competitive final joint accuracy under the same target-evaluation budget.
Bayesian causal discovery seeks to determine the posterior distribution of causal theories, which are interpreted as directed acyclic graphs (DAGs) that explain the observed data. The resulting posterior allows systematic reasoning regarding epistemic uncertainty within these theories. Nonetheless, finding such graphs is difficult due to identifiability problems and limited observational data. Furthermore, precisely approximating posterior over graphs is challenging given vast range of potential DAGs. Recent Bayesian approaches have addressed some of these challenges, yet they remain limited as they fail to encode dependencies between edges, and lack principled ways to incorporate domain knowledge as inductive biases during the search process. To overcome these limitations, we propose SVI-DAG, a structured variational inference approach to Bayesian causal discovery using observational data and prior beliefs that uses normalizing flows to model dependencies between edges, supporting expressive and multimodal posterior learning over DAGs. To mitigate mode seeking behaviour in evidence lower bound optimization and promote mode coverage, we use stein variational gradient descent to update the node potentials using a kernel in acyclicity space. We evaluate SVI-DAG against 5 state-of-the-art Bayesian DAG learning methods and demonstrate superior performance in uncertainty quantification while remaining competitive in terms of structural accuracy.
Carson J. Cook, Ahmed J. Zerouali, Anthony Schmidt +3cs.LG cs.CY
We introduce the Unified Neural Variational Measurement of Proficiency (UNVaMP) architecture, a knowledge tracing method that integrates observed student-item interactions with internal memory to produce evolving latent representations of student knowledge. These representations support accurate predictions of future responses while enabling explicit control over the smoothness of estimated learning trajectories. UNVaMP can be configured as either a purely neural model or a hybrid model that predicts responses through an interpretable measurement function over the latent space. We show that a pure neural configuration (UNVaMP-MLP) achieves the strongest predictive performance among compared models on three out of four datasets. Meanwhile, a hybrid configuration (UNVaMP-MIRT, using a 1PL MIRT measurement function) lags only slightly behind UNVaMP-MLP, indicating that the predictive cost of interpretability is modest. Beyond predictive accuracy, UNVaMP provides the following: a principled mechanism for controlling volatility when estimating student latent variables, quantification of uncertainty over student knowledge state estimates, and flexible input specification that supports heterogeneous student-item interaction features. In addition, the hybrid UNVaMP-MIRT configuration generates interpretable moment-in-time student knowledge state estimates. Using an experimental dataset, we show that auxiliary inputs induce structured changes in the predictive behavior of UNVaMP-MIRT, consistent with sensitivity to underlying structure beyond response correctness. Furthermore, through a simulation study, we show that UNVaMP yields well-behaved knowledge state estimates under controlled measurement conditions. In total, these results indicate that UNVaMP is both useful for real-world education systems and capable of recovering underlying structure from student-item interactions.
Classical neural networks frequently produce overconfident predictions on ambiguous or out-of-distribution (OOD) data, a liability that grows with each AI system deployed in safety-critical real-world scenarios. Bayesian neural networks (BNNs) provide a principled framework for uncertainty-aware prediction by replacing deterministic parameters with probability distributions, but repeated sampling increases latency, memory traffic, and energy consumption. Photonic probabilistic computing offers a promising alternative by exploiting intrinsic optical stochasticity for fast and parallel sampling. However, photonic BNNs are not ideal samplers: analog constraints on quantization, programming error, dynamic range, and representable mean and variance restrict the variational families that can be implemented in hardware. In this work, we study which hardware-imposed constraints limit scalable photonic BNN inference, how these constraints can be represented, and which ranges can be tolerated by photonic BNNs beyond small proof-of-concept networks. We formulate photonic BNN inference as constrained stochastic variational inference and perform a systematic ablation study over stochasticity location, stochasticity modality, quantization, programming error, and mean/variance bounds. From these results, we derive concrete co-design guidelines that distinguish hardware constraints that can be compensated by training from those requiring hardware or architecture intervention. We validate these guidelines under coupled, hardware-realistic constraints on Dirty-MNIST, CIFAR-10, and CINIC-10, using Fashion-MNIST and SVHN as OOD benchmarks, showing that hardware-aware training recovers predictive performance and uncertainty quality whenever the required variational family remains representable, whereas violations of representational limits require targeted hardware modifications.
Pei-Hsuan Hsia, Lars H. Heyen, Arvid Weyrauch +4cs.LG cs.AI
In Bayesian neural networks (BNNs), variational inference is a widely adopted framework for modeling uncertainty in a distributional way, with the evidence lower bound (ELBO) serving as the standard objective function. Several distributions contribute to the ELBO loss, such as the prior, approximated posterior, and likelihood distribution. Typically, these distributions are all approximated by a Gaussian distribution, since it is easy to compute, allows for reparameterized gradients, and provides a closed-form loss for training. However, several works have highlighted that this assumption may not generally hold, posing the risk of model misspecification. Alternative distributions have been proposed for the prior specifically, while the effect of distribution choice on the likelihood distribution remains unexplored. In this work, our aim is to close this gap by investigating whether alternative assumptions for the likelihood distribution can outperform the commonly used Gaussian. We compare several likelihood distribution assumptions, such as skewed or heavy-tailed, across regression tasks on both artificial and real-world datasets using standard multilayer perceptrons (MLPs). Our findings demonstrate that Student's t yields better predictive performance than a Gaussian likelihood distribution, independent of the data distribution and MLP architecture (depth and width). In some cases, Student's t can also lead to shorter training times, while still being easy to implement.
Nour Jamoussi, Ikram Dridi, Giuseppe Serra +1cs.LG
Differential privacy provides formal privacy guarantees for training neural networks on sensitive data, while Bayesian deep learning offers a principled framework for uncertainty-aware prediction. Combining these two objectives remains challenging, as privacy noise can interact with the stochasticity introduced by Bayesian posterior sampling. In this work, we investigate differentially private variational Bayesian learning through the Improved Variational Online Newton (IVON) optimizer. We introduce DP-IVON-Gradsq, a private variant of IVON. The proposed method constructs its curvature estimate from the privatized gradient using a noise-corrected squared-gradient estimator, reducing the direct interaction between posterior-sampling noise and privacy noise while preserving the Adam-like computational efficiency of IVON. We evaluate DP-IVON-Gradsq on CIFAR-10 against the standard private optimizers DP-SGD and DP-Adam over a range of privacy budgets. The results show that DP-IVON-Gradsq is competitive under weak-to-moderate privacy constraints, i.e., large-to-moderate values of $\varepsilon$, while degrading under strong privacy. Code is available at https://github.com/NourJamoussi/DP-IVON-Gradsq.git.
Laura M. Montaldo, Ricardo A. Borsoi, Sebastian Miron +1stat.ML cs.LG eess.SP
Modeling shared and subject-specific structure in multisubject spatiotemporal data remains challenging, particularly in neuroimaging, where both spatial and temporal patterns exhibit rich variability across subjects. Existing matrix and tensor decompositions provide interpretable factorizations, but rely on fixed multilinear structures or coupling schemes that may limit their flexibility in capturing complex variability. In this work, we introduce a spatiotemporal variational tensor decomposition (ST-VTD) framework that combines a tensor factorization generative model with structured priors to jointly represent spatial maps and temporal dynamics. Spatial factors are regularized to promote a low-rank structure inspired by the LL1 decomposition, while temporal factors are modeled using a learned Long short-term memory (LSTM)-based prior, enabling flexible and adaptive dynamics. Posterior inference is performed using an amortized variational formulation by unrolling iterations of an optimization algorithm, leading to an interpretable and parameter-efficient architecture. The proposed inference framework employs a warm-start strategy based on group independent component analysis, which we found to improve optimization performance. Experiments on a realistic synthetic functional MRI (fMRI) dataset demonstrate that the proposed approach significantly improves latent factor recovery compared with representative classical and probabilistic decomposition benchmarks.
Sophisticated Inference is a variant of active inference often associated with recursive belief modeling and tree search. We argue that its central computational role is simpler: within a planning horizon, it makes active inference closed-loop by allowing future actions to depend on future states and observations. This closed-loop structure can be represented in the epistemic-prior variational free energy framework. Epistemic priors supply the active-inference objective, while a joint posterior over future states and actions supplies the state-contingent control structure. We evaluate this decomposition in the Reactivity Maze, a stochastic benchmark designed to separate epistemic incentive from inner-horizon closed-loop control. The comparison includes three variational objectives with the same state-action posterior family, an action-state factorized active inference objective, Sophisticated Inference, and standard Expected Free Energy planning. The results show that neither ingredient is sufficient on its own. Methods without an epistemic component do not seek information, while methods that prevent future actions from depending on future states cannot turn information into reliable goal-reaching. By contrast, both Sophisticated Inference and full-joint epistemic-prior active inference solve the environment by combining epistemic drive with closed-loop inference. These results show that the advantage associated with Sophisticated Inference need not be specific to tree search itself. It arises from the closed-loop form of active inference, and this form can be represented in epistemic-prior variational inference when the posterior keeps future actions dependent on future states.
A language model $p_θ(y \mid x)$ trained on reasoning tasks learns to solve problems via multiple distinct strategies, yet these strategies are implicit and entangled within the model's response distribution. We study the problem of decomposing the response distribution of a given pretrained language model into a structured, strategy-conditioned representation. Specifically, we learn a latent-variable factorization $p_θ(y \mid x) \leadsto (r_φ(z \mid x), g_φ(y \mid x,z))$, where a router $r$ maps each input to a distribution over latent strategies $z$ and a generator $g$ produces the response conditioned on that strategy. A key challenge is that the generator, initialized from the base model, already represents $p_θ(y \mid x)$ without using $z$. Standard variational inference therefore gives the model no incentive to route information through $z$ and can yield a severe form of posterior collapse. To address this, we propose a variational objective that measures fractional information gain relative to the base model's response loss and concentrates reconstruction pressure on tokens with high base model surprisal, encouraging $z$ to encode strategy-relevant response variation. We introduce a benchmark of multi-strategy algorithmic tasks and show that this objective recovers latent codes aligned with distinct reference strategies while preserving the base model's response distribution.
Gene regulatory networks (GRNs) link transcription factor (TF) proteins to their target genes, yet reconstructing these networks from genome-wide data remains challenging under practical and methodological constraints. Many methods couple modeling assumptions to a specific inference procedure and rely on heuristic model selection, while evaluation is constrained by incomplete reference networks and point-estimate outputs that lack uncertainty. GRN reconstruction also depends on prior knowledge to constrain TF-gene interactions, yet available priors are often assay-dependent and difficult to transfer across species and less-characterized systems. In this thesis, we develop two complementary frameworks that address these limitations. In the first, PMF-GRN casts GRN inference as a probabilistic graphical model optimized by variational inference, enabling principled model selection and uncertainty-aware edge estimates. In the second, GLM-Prior addresses the prior bottleneck by fine-tuning the pretrained Nucleotide Transformer to predict TF-target gene interactions directly from nucleotide sequence, while generalizing across yeast, mouse, and human settings. Together, PMF-GRN and GLM-Prior motivate a dual-stage view of GRN reconstruction in which sequence-derived priors provide a transferable starting scaffold and probabilistic inference refines regulatory estimates with quantified uncertainty under incomplete evaluation resources.
The bird's eye view (BEV) has emerged as a pivotal approach for environmental perception in autonomous driving, providing a unified spatial representation for vehicles. Nevertheless, despite BEV's significance in addressing the challenges inherent to autonomous driving, effectively fusing data from multiple camera sensors and operating in complex external driving environments remains a considerable challenge. To mitigate this issue, we recast the BEV segmentation problem within a variational inference framework. In this paper, we propose a novel transformer-based variational flow transformation network for BEV segmentation, denoted as TVB. Our architecture implicitly learns the mapping from multiple camera views to a unified canonical BEV map during training by exploiting posterior BEV supervision. TVB employs a conditional variational auto encoder (CVAE) as its backbone and produces multiple BEV map candidates. To augment the realism of the generated BEV maps, we integrate normalizing flows into the map generation process, enabling the construction of more complex and expressive probability distributions. Furthermore, we design a BEV-attention fusion (BAF) module that harnesses attention mechanisms to adaptively integrate the multiple candidate BEV maps. Experimental results, evaluated on both the nuScenes and OPV2Vdatasets, demonstrate that our proposed method achieves superior performance in multi-camera view BEV segmentation and lane environment perception.
Runze Gan, Qing Li, Simon J. Godsill +2cs.LG cs.CV eess.SP
Multi-object detection and tracking from noisy point clouds remain challenging in many data-scarce radar applications. Current Bayesian trackers based on Poisson measurement models offer a training-free solution but struggle to achieve accuracy and efficiency under severe clutter, large object populations, and full-resolution Doppler point clouds. We address this with PiVoT, a fast, clutter-resilient multi-object tracker for both positional and Doppler measurements. PiVoT performs end-to-end detection and tracking of a large and time-varying number of objects without external clustering or detectors, through joint inference of object states, shapes, existence probabilities, data association, and measurement rates. Its efficiency is driven by several variational inference innovations, such as theoretically justified birth pruning, quadratic-to-linear complexity reductions for exact updates, and a computationally efficient Doppler Poisson model. Experiments show that PiVoT substantially outperforms existing Bayesian trackers in challenging scenes, while also demonstrating exceptional scalability to a thousand objects, robustness to clutter visually inseparable from objects, and real-time operation on full-scale modern automotive radar datasets, where it attains performance comparable to a deep-learning detection benchmark as a training-free joint detector and tracker.
Radiative Gaussian splatting has made sparse-view CT reconstruction fast, but existing methods output point estimates with no notion of where the reconstruction can be trusted. We exploit a property of transmissive X-ray imaging that RGB splatting cannot claim -- projection and voxelization are strictly linear in the per-Gaussian densities -- to equip radiative Gaussians with a variational density posterior whose predictive variance propagates in closed form, exactly, in a single forward pass, in both volume space ($σ^2(x)=\sum_i g_i(x)^2 s_i^2$) and projection space ($\mathrm{Var}[I_p]=\sum_i w_{i,p}^2 s_i^2$). We present the first systematic calibration study for Gaussian-splatting CT (Spearman / AUSE / ECE with temperature scaling), showing that the resulting per-voxel uncertainty ranks true reconstruction error on 14 of 15 scenes of the official benchmark across three view budgets -- 9 of 15 additionally meeting our magnitude-calibration target after a single temperature -- while the perturbation-ensemble heuristic of concurrent work, transplanted to voxel space under the same protocol on our development scenes, does not (rank correlation as low as $-0.08$). We then dissect why uncalibrated acquisition scores can nevertheless select acceptable views, identifying three regimes -- flat (isotropic, balanced), pathological (degenerate coverage), and anisotropic -- and showing, in controlled single-scene testbeds, that principled uncertainty earns a measurable premium only in the last, motivating a coverage-gated, maturity-scheduled acquisition policy; the same calibrated posterior further points toward a dose-adaptive stopping rule, whose experimental validation we leave to future work.
Learning neural set functions is pivotal to a wide range of important applications, including compound selection in AI-driven drug discovery and product recommendation. Recent work has introduced optimal subset oracles to implicitly learn set functions under practical weakly supervised settings, where model parameters are optimized through mean-field variational inference. However, these frameworks rely on Monte Carlo sampling to estimate gradients of the evidence lower bound when updating the variational distribution. Repeated sampling across iterations incurs substantial computational overhead, while the resulting stochasticity can destabilize the optimization trajectory. In this work, we reinterpret the evidence lower bound as a continuous relaxation of the set function and learn a surrogate objective that replaces sampling-based ELBO gradient estimation during variational optimization. The learned surrogate provides stable and efficient gradients throughout the continuous domain, thereby reducing computational overhead and accelerating inference. Furthermore, we establish an approximation guarantee for the proposed framework under submodular maximization and characterize its connection to variational free energy. Experiments on a variety of real-world tasks demonstrate consistent improvements over existing baselines.
Jie Qiao, Ruichu Cai, Zijian Li +6cs.LG cs.AI stat.ML
Causal discovery, the process of recovering underlying causal structures from observational data, is a fundamental pursuit across scientific disciplines. Over the past decades, numerous algorithms have been developed to tackle this challenge through workflows tailored to the specific causal mechanisms underlying each type of dataset, demonstrating effectiveness across a wide range of applications. However, as the volume and heterogeneity of real-world data continue to grow, this dataset-specific approach inevitably leads to a fragmented, test-driven paradigm that struggles to scale to the demands of modern scientific discovery. To address this, we formulate the Causal Discovery Foundation Model (CDFM) as a unified, general-purpose framework for zero-shot structural inference. To ensure reliable generalization across unknown domains, we first investigate the theoretical boundaries of causal identifiability, revealing the indispensable role of causal prior mechanisms in this process. Building on these insights, we formulate a principled variational framework that treats unknown causal mechanisms as latent variables and mathematically decomposes the intractable marginal likelihood into distinct, tractable learning modules. The variational decomposition provides a conceptual design principle for the architecture design of CDFM, while comprehensive causal knowledge guides the large-scale synthesis of our pretraining data. By pretraining on a massive, highly diverse space of synthetic structural causal models, CDFM successfully internalizes complex statistical asymmetries. Extensive experiments demonstrate that CDFM consistently outperforms traditional algorithms, driving a paradigm shift toward a general-purpose causal discovery foundation model.
3D Gaussian Splatting represents scenes as finite mixtures of anisotropic Gaussians whose number of components $K$ is set by heuristic density control or user caps. Variational Bayes Gaussian Splatting (VBGS) recast splat fitting as conjugate variational inference, but $K$ remains fixed. We replace the finite symmetric Dirichlet over mixture weights with a truncated stick-breaking Dirichlet-process prior -- and, as a theory-backed alternative, a sparse overfitted finite Dirichlet -- so that the number of occupied components adapts to the data while every update remains a closed-form coordinate-ascent step; a natural-gradient stochastic variant makes the per-step cost independent of the number of points. We give an exact monotonicity guarantee, a rigorous truncation-error bound correcting an anti-conservative large-$α$ approximation in common use, and an honest account of what the fitted number of components estimates. Empirically: (i) the effective complexity $\hat{K}$ adapts to scene complexity and recovers the true $K$ within $\pm 1$ on well-separated synthetic data with regime-appropriate concentration; (ii) a deconfounded comparison shows the DP prior's contribution is complexity selection, not per-component efficiency -- converged DP fits exceed single-pass fixed-$K$ VBGS by +2.7 dB at matched budgets yet tie an equally converged fixed-$K$ baseline, and on 3D scenes DP-Splat matches or exceeds VBGS's held-out color prediction with 5.9-7.6x fewer components; (iii) the posterior-predictive color variance is well calibrated on model-matched synthetic data; and (iv) the ordering suggested by exact-posterior asymptotics reverses under mean-field coordinate ascent: the DP prior resists over-splitting while the sparse finite mixture saturates its truncation, a gap between variational practice and posterior asymptotics documented across three orders of magnitude in $N$.
Hong Hai Nguyen, Sy Phan Van, Soo-Hyung Kim +1cs.CV
Facial affect in the wild is naturally multi-task: valence-arousal, discrete expressions, and facial action units describe the same face. Yet real corpora annotate these tasks only partially and unevenly, so most systems mask the missing labels or impute pseudo-labels and forgo the cross-task signal. We instead cast partially-labeled multi-task learning as marginalization over a shared affect latent: one variational bottleneck mediates all three task decoders, so a frame annotated for one task shapes the representation the others use, and the masked objective reappears as the reconstruction term of an evidence lower bound. On s-Aff-Wild2, where only 37% of frames carry all three labels, the classes are severely imbalanced, and pretraining on the source data is disallowed, we isolate where this coupling acts. On a single backbone it lifts expression macro-F1 from 0.403 for a dedicated specialist to 0.446, which the masked-loss model does not reach; a second, near-peer backbone with decorrelated errors then breaks an action-unit ceiling that external action-unit data could not, while valence-arousal stays within noise. Every gain is disciplined by a matched-control negative; together these controls indicate that the rare-class failure is representational, not a matter of loss shaping. As each task's source is chosen on the evaluation split, we report the assembled result, a combined multi-task score of 1.679 on validation, as an in-sample endpoint and rest our conclusions on the controlled comparisons; a small, regime-dependent transfer of the expression advantage to AffectNet and RAF-DB is presented as exploratory rather than conclusive.
Federico L. Perlino, Oliver Hamelijnck, Adam M. Johansen +1stat.ML cs.LG math.ST stat.CO stat.ME
Many real-world processes can be represented as compositions of functions along a directed acyclic graph (DAG). In causal modelling, these correspond to the underlying mechanisms; in engineering, to multiple fidelity levels; and in gene-regulatory networks, to transcription factors. These functions are partially observed across the DAG, with noisy and heterogeneously sampled measurements, posing significant challenges for reconstruction, uncertainty propagation, and inference. To tackle these challenges, we place priors over functions and naturally arrive at Deep Gaussian Processes over DAGs. We theoretically study their prior-collapse behaviour, and the effect of graph topology and intermediate observations on the preservation of information. We obtain almost-sure lower bounds on the asymptotic frequency of depths at which the distinction between inputs is preserved, identify broad kernel classes for which these hold, and prove an observation by \cite{dunlop2018} on the role of input connections. We offer a structured variational approximation that retains graph dependencies, preserves compositional uncertainty, and captures the explaining-away behaviour of colliders. Finally, we empirically validate our theoretical results and our methodology, and model a latent-collider DAG, a protein signalling network, and a multi-fidelity heavy-ion collision emulation task, attaining state-of-the-art performance while recovering low-fidelity contributions and yielding interpretability of the simulator hierarchy.
Human decision-making is highly flexible -- some actions are taken immediately; others require longer deliberation. Language models have exhibited a similar capacity for adaptive "reasoning." However, transferring this capability to continuous control policies has been challenging, as directly reasoning in language space may lack the granularity for spatial understanding and precise motions. In this work, we show that reasoning for control policies can emerge by organizing information in an autoregressive latent space reminiscent of a memory palace, where retrieval is iterative and adaptive. Our method, Latent Memory Palace (LMP), formulates reasoning as variational inference with an autoregressive latent distribution. We derive a latent-space reinforcement learning technique to tractably optimize its variational lower bound. The resulting policy, LMP-$π$, achieves strong empirical performance in simulation and real-world domains while exhibiting interpretable, adaptive allocation of test-time compute. We further show that the same framework yields a variable-length action tokenizer, LMP-$\texttt{tok}$, which significantly improves the performance of downstream autoregressive policies. Together, these results present a new perspective on latent reasoning for control through the lens of variational inference.
Quoc Phong Nguyen, Paul Albert, Long Vuong +2cs.IR cs.LG
Two-tower retrievers compress each user into a single embedding, limiting their ability to serve diverse interests. Multi-interest models give each user several heads scored by a maximum inner product, but their hard-routing training under-utilizes heads (routing collapse) and gives no per-user estimate of how much each interest matters for serving. We present \textbf{BACH} (\emph{Bayesian Admixture of Contrastive Heads}), which casts multi-interest two-tower retrieval as a per-user mixture over the heads, fit by variational inference. The soft mixture trains every head (mitigating collapse), produces a per-user weighting of the interests that is reused at serving, and admits a shared global-codebook variant with precomputable retrieval. On three large-scale benchmarks, MovieLens-20M, Taobao, and Netflix, BACH improves top-of-ranking retrieval over hard-routing multi-interest and single-vector baselines at every head count; we further find that scoring every candidate by its best head, consistent with serving, outperforms the usual target-routed training, and that BACH improves further still.
Benjamin Wiriyapong, Oktay Karakus, Can Eyupoglu +1cs.LG stat.ML
Normalising flows provide a powerful variational family for approximate inference, yet individual architectures often fail to generalise across heterogeneous posterior geometries. We revisit mixture-based flow formulations and introduce \emph{AMF\mbox{-}VI\mbox{-}sEMA}, a two-stage framework featuring a \emph{stable global weighting} mechanism based on a \emph{Simplex Exponential Moving Average} (sEMA) update. In Stage~1, a heterogeneous set of experts (\textsc{RealNVP}, \textsc{MAF}, \textsc{RBIG}) are trained independently to specialise in distinct structural regimes. In Stage~2, expert parameters are frozen and global mixture weights are learned through a temperature-controlled softmax of average log-likelihoods, followed by a smooth EMA update on the probability simplex. This design produces a tractable, data-agnostic gating mechanism (without per-sample gating or gradient backpropagation through weights) that adaptively reallocates capacity while avoiding component collapse. We evaluate the framework on ten posterior benchmarks: six canonical 2D synthetic families (Banana, X-Shaped, Bimodal, Multimodal, Two-moons, Rings) and four real/low-dimensional Bayesian targets (BLR, BPR, Weibull, Real-GMM2), with stronger baselines (\textsc{NICE}, \textsc{ResFlow}, and EM-Mixing). Comprehensive evaluation covers NLL, KL divergence, Wasserstein-2 distance, and MMD, together with diagnostics of mixture dynamics, hyperparameter sensitivity, and cross-seed robustness. Empirically, \emph{AMF\mbox{-}VI\mbox{-}sEMA} achieves consistent NLL improvements over its predecessor \emph{AMF\mbox{-}VI} and avoids the catastrophic transport failures of single-flow baselines, while maintaining stable weight trajectories ($N_{\mathrm{eff}}{>}1.4$ on all datasets) with minimal computational overhead.
Traditional variational Kalman filtering with unknown noise statistics suffers from inconsistent process covariance estimation and slow convergence speed, limiting its practical utility. To address these issues, we introduce a surrogate variable representing the process-noise-free state, which enables explicit modeling and inference of process noise statistics. In addition, we reformulate the conventional coordinate ascent variation inference (CAVI) as a marginalized maximum a posteriori problem, followed by a single-step hyperparameter fitting. This reformulation obviates the need for multiple inner iterations inherent to CAVI and decouples the design of the covariance tracking filters. Consequently, this architecture permits the deployment of higher-order filters for covariance tracking and enables sliding-window hyperparameter estimation. Notably, when this window encompasses all historical data, the covariance tracking estimator intrinsically operates as a zero-phase filter. Numerical simulations validate the theoretical framework, demonstrating the enhanced convergence speed and superior estimation accuracy compared with existing methods.
Variational Autoencoders (VAEs) commonly assume a standard isotropic Gaussian prior over the latent space, an assumption that often fails to capture the true distribution of latent representations for complex datasets. This mismatch can limit reconstruction accuracy, reduce sample quality, and constrain the expressive power of the learned latent space. We propose the eXact-Prior Variational Autoencoder (X-VAE), a framework that replaces the conventional standard normal prior with a Gaussian prior derived from the latent representations of a pretrained autoencoder (AE). Specifically, the empirical mean and standard deviation of the AE latent codes are used to parameterize a data-adaptive prior that more closely reflects the underlying structure of the training data. During generation, X-VAE introduces a latent scaling factor that enables explicit control over the variance of the sampled latent vectors, providing a simple mechanism for balancing sample diversity and fidelity. This flexibility makes the proposed approach particularly well suited for applications such as industrial and engineering design, where generated solutions must satisfy strict structural or functional constraints while still permitting meaningful design exploration. We present the mathematical formulation of well-suited X-VAE, derive the corresponding KL divergence objective for the proposed prior, and evaluate the method on standard benchmark datasets. Experimental results demonstrate that X-VAE preserves reconstruction quality while producing latent representations that better align with the empirical data distribution, leading to improved controllability and more realistic generated samples.
The use of ordinary and stochastic differential equations has led to substantial progress in generative machine learning with applications to, for example, image, video and biomolecule generation. This paper provides a self-contained and informal introduction to the differential equations, the probabilistic framework for using them in generative modeling and the Fokker--Planck equation that governs the temporal evolution of the marginal distribution of the stochastic variables of the differential equations. The variational lower bound on the log-likelihood (the evidence lower bound, ELBO) is derived and used as a general starting point for a discussion of diffusion models, score matching, and flow matching. All of these approaches may be viewed as specific parameterizations of the most general variational approach. A one-dimensional density modeling problem is used as a simple example to compare different parameterizations.
Symmetries are important for many deep learning tasks, ranging from applications in the sciences to medical imaging. However, there is an ongoing debate about whether to impose symmetry constraints on the neural network architecture (yielding equivariant neural networks) or learn them from augmented training data. Although equivariant networks are well-studied theoretically, much less is known about data augmentation, since analyzing augmentation requires control over the training dynamics. Inspired by recent results that show that augmented infinite deep ensembles are exactly equivariant, we study data augmentation for Bayesian neural networks (BNNs) trained with variational inference. We focus on variational distributions in the exponential family and derive conditions under which exact equivariance is reached. We furthermore obtain bounds on the equivariance error and introduce three novel symmetrization techniques which boost the effect of data augmentation in this setting. We conduct extensive numerical experiments which show that one of our symmetrization methods (orbit expansion) outperforms the baseline in both equivariance and overall performance. Our code is available at github.com/dmw1998/augment-BNNs