Papers

Physical AI agents, such as robots and other embodied systems operating under tight and fluctuating resource constraints, remain far less capable than biological agents in open-ended real-world environments. This paper argues that Active Inference (AIF), grounded in the Free Energy Principle, offers a principled foundation for closing that gap. We develop this argument from first principles, following a chain from probability theory through Bayesian machine learning and variational inference to active inference and reactive message passing. From the FEP perspective, systems that maintain their structural and functional integrity over time can, under suitable assumptions, be described as minimizing variational free energy (VFE), and AIF operationalizes this by unifying perception, learning, planning, and control within a single computational objective. We show that VFE minimization is naturally realized by reactive message passing on factor graphs, where inference emerges from local, parallel computations. This realization is well matched to the constraints of physical operation, including hard deadlines, asynchronous data, fluctuating power budgets, and changing environments. Because reactive message passing is event-driven, interruptible, and locally adaptable, performance degrades gracefully under reduced resources while model structure can adjust online. We further show that, under suitable coupling and coarse-graining conditions, coupled AIF agents can be described as higher-level AIF agents, yielding a homogeneous architecture based on the same message-passing primitive across scales. Our contribution is not empirical benchmarking, but a clear theoretical and architectural case for the engineering community.

Uncertainty quantification for image data is dominated by complex deep learning methods, yet the field lacks an interpretable, mathematically grounded baseline. We propose Bayesian scattering to fill this gap, serving as a first-step baseline akin to the role of Bayesian linear regression for tabular data. Our method couples the wavelet scattering transform-a deep, non-learned feature extractor-with a simple probabilistic head. Because scattering features are derived from geometric principles rather than learned, they avoid overfitting the training distribution. This helps provide sensible uncertainty estimates even under significant distribution shifts. We validate this on diverse tasks, including medical imaging under institution shift, wealth mapping under country-to-country shift, and Bayesian optimization of molecular properties. Our results suggest that Bayesian scattering is a solid baseline for complex uncertainty quantification methods.

Data-driven discovery of governing equations is well developed for deterministic systems but much less so for stochastic ones. We present a sparse-symbolic algorithm that recovers both the drift $b(x)$ and the diffusion $a(x)$ of a stochastic differential equation from a single discretely sampled trajectory. It replaces the time-indexed test functions of weak-form SINDy with state-dependent spatial kernels. Fixed-width time-window weighting induces a persistent endogeneity bias: for the Ornstein-Uhlenbeck process it erases the drift coefficient entirely, whatever the window shape. Spatial kernels avoid this through a martingale strong-law argument, linked to the estimating-function theory of Bibby and Sorensen. We prove consistency of the drift and diffusion estimators for the Euler-Maruyama coefficients, a $T^{-1/2}$ central limit theorem for the drift, a finite-step correction that links drift and diffusion estimation, and a correction for additive Gaussian observation noise. The theorems assume ergodicity of the Euler-Maruyama chain, which holds for the OU and multiplicative systems but not for the cubic double well, so the double-well results are empirical. The method is validated on three benchmark SDEs (Ornstein-Uhlenbeck, a double well, and multiplicative diffusion), on eight further systems identified from single trajectories with elastic-net-stabilized sparse regression, and in a support-recovery experiment. Exact support recovery from one trajectory needs a long record (about $T \gtrsim 200$ on the double well) and is not guaranteed for shorter records or weakly identified systems.

Data unfolding – the removal of noise or artifacts from measurements – is a fundamental task across the experimental sciences. Of particular interest are applications in physics, where the dominant approach is Richardson-Lucy (RL) deconvolution. The classical RL approach aims to find denoised data that, once passed through the noise model, is as close as possible to the measured data in terms of Kullback-Leibler (KL) divergence. This requires that the support of the measured data overlaps with the output of the noise model, a hypothesis typically enforced by binning, which introduces numerical error. As a counterpoint, the present work studies an alternative formulation using a Wasserstein loss. We establish sharp conditions for existence and uniqueness of optimizers, answering open questions of Li, et al., regarding necessary conditions for uniqueness in the case of transport map noise models. We then develop a provably convergent generalized Sinkhorn algorithm to compute approximate optimizers. Our algorithm requires only empirical observations of the noise model and measured data and scales with the size of the data, rather than the ambient dimension. Numerical experiments on one- and two-dimensional problems inspired by jet mass unfolding in particle physics demonstrate that the optimal transport approach offers robust, accurate performance compared to classical RL deconvolution, particularly when binning artifacts are significant.

Data assimilation algorithms estimate the state of a dynamical system from partial observations, where the successful performance of these algorithms hinges on costly parameter tuning and on employing an accurate model for the dynamics. This paper introduces a framework for jointly learning the state, dynamics, and parameters of filtering algorithms in data assimilation through a process we refer to as auto-differentiable filtering. The framework leverages a theoretically motivated loss function that enables learning from partial, noisy observations via gradient-based optimization using auto-differentiation. We further demonstrate how several well-known data assimilation methods can be learned or tuned within this framework. To underscore the versatility of auto-differentiable filtering, we perform experiments on dynamical systems spanning multiple scientific domains, such as the Clohessy-Wiltshire equations from aerospace engineering, the Lorenz-96 system from atmospheric science, and the generalized Lotka-Volterra equations from systems biology. Finally, we provide guidelines for practitioners to customize our framework according to their observation model, accuracy requirements, and computational budget.

Evaluating treatment effects is critical in clinical trials but sometimes involves lengthy, invasive, or costly follow-up procedures. In these cases, surrogate markers, which provide intermediate measures of the long-term treatment effect, allow clinicians to obtain results faster and more efficiently than would have otherwise been possible. Prior to adoption, it is vital that the utility of surrogate markers (i.e., their ability to capture the treatment effect on the primary outcome) is statistically validated. Many frameworks for evaluating surrogate markers have been proposed, but they do not account for missing data. Instead, they rely on complete cases (the subset of patients without missing data), which can be inefficient and biased. To improve on this, we propose methods to accommodate missing data in nonparametric and parametric surrogate evaluation via inverse probability weighting (IPW) and semiparametric maximum likelihood estimation (SMLE). Through simulation studies, we demonstrate that the proposed methods remain unbiased under a broader range of missing data mechanisms than complete case analysis and can help retain the statistical precision of the full trial. We illustrate their practical utility through an application to a diabetes clinical trial. Moreover, our missing data corrections have complementary strengths with respect to computational ease, robustness, and statistical efficiency. All methods are implemented in the MissSurrogate R package.

Large-scale longitudinal molecular profiling is now firmly established in biomedical research, prompted by the need to uncover coordinated biomarker trajectories reflecting the dynamics of underlying biological mechanisms and characterise patient heterogeneity in disease progression. While a range of statistical tools exist for either longitudinal modelling or high-dimensional analysis, there is no unified framework tailored to address these questions jointly. Motivated by a longitudinal COVID-19 study conducted in Cambridge hospitals, we propose a Bayesian functional factor model to address this gap. The framework combines latent factor modelling with functional principal component analysis to represent shared temporal programmes across subsets of variables while capturing individual variation through low-dimensional functional scores. We specify sparsity-inducing priors that yield interpretable factor structure and allow the effective number of factors to be inferred via overspecification. An annealed variational algorithm ensures efficient joint posterior inference at scale. The approach achieves accurate recovery of temporal structure in simulations with up to 20 000 variables. Applied to a panel of 57 circulating biomarkers and to whole-blood expression of 13 177 genes from the COVID-19 cohort, the model recovers coordinated inflammatory, interferon and immune-cell signatures, with subject-level scores that follow disease severity. The methodology is implemented in the R package bayesSYNC.

This paper studies finite-sample set-membership identification for discrete-time bilinear systems under bounded symmetric log-concave disturbances. Our analysis considers trajectory-dependent regressors and allows marginally stable dynamics with polynomial mean-square state growth. We prove that the diameter of the feasible parameter set shrinks with sample complexity $\widetilde{\mathcal O}(1/\epsilon)$ where $\epsilon$ is the estimation error. Simulation supports the theory and illustrates the advantage of the proposed estimator for uncertainty quantification.

We propose a nonparametric test of spatial independence for data observed on irregular, non-lattice point clouds $\mathcal{V}_{n}\subset\mathbb{R}^{2}$. For each location $v\in\mathcal{V}_{n}$, we encode the local spatial configuration through the ordinal pattern of the $m$ nearest-neighbour observations, obtaining a symbolic representation that is invariant under strictly monotone transformations and robust to outliers. Under the null hypothesis of spatial independence, the local ordinal patterns are i.i.d.\ and uniformly distributed over the symmetric group $\mathcal{S}_{m}$, regardless of the unknown marginal distribution $F$. We exploit this characterisation to construct a test statistic $L_{n}$ based on the additive log-ratio (ALR) transformation of the empirical ordinal-pattern frequencies. Invoking a central limit theorem for graph-dependent processes under a graph-based $\alpha$-mixing condition, we establish that $L_{n}$ converges in distribution to a $\chi^{2}_{m!-1}$ random variable, yielding an asymptotically pivotal procedure with no nuisance parameters. An extensive Monte Carlo study confirms that the $\chi^{2}_{m!-1}$ approximation is accurate already at moderate sample sizes, that the test controls size at the nominal level, and that power increases monotonically with the strength of spatial dependence. Notably, the test detects dependence in both linear and nonlinearly transformed spatial autoregressive models, illustrating the robustness that is characteristic of ordinal-pattern methods. Our framework extends the spatial ordinal-pattern testing paradigm from regular lattices to general spatial supports, opening the door to ordinal-pattern inference in the many applied settings where observations are irregularly located.

Calibration weighting is a fundamental tool in survey sampling for incorporating auxiliary population information into design-based estimators. Classical formulations measure distance between calibrated and design weights on the multiplicative ratio scale. We develop a unified framework based on Bregman divergence defined directly on the weight vector. The framework reveals a primal–dual symmetry in which both the weight-space and multiplier-space optimization problems are themselves Bregman projections, and the calibrated weights satisfy a generalized Pythagorean decomposition with respect to the constraint manifold. The resulting estimator is asymptotically equivalent to a debiased prediction estimator whose regression coefficient depends explicitly on the Bregman generator, in contrast to the generalized regression estimator equivalent of classical calibration. Exploiting this dependence, we identify a contrast-entropy generator that achieves design-optimality under Poisson sampling. Two extensions are developed: cross-fitted estimation under non-probability sampling, yielding doubly robust inference under standard product-rate conditions; and a regularized extension whose Lagrangian dual produces a Hölder-conjugate penalty for soft balance under high-dimensional auxiliary variables. Simulations and an analysis of National Oceanic and Atmospheric Administration (NOAA)'s Large Pelagics Intercept Survey illustrate the framework.

In this paper we investigate the asymptotic statistical theory of irreducible quantum Markov chains, focusing on identifiability properties and asymptotic convergence of associated quantum statistical models. We show that the space of identifiable parameters for the stationary output is a stratified space called an orbifold, which is obtained as the quotient of the manifold of irreducible dynamics by a compact group of state preserving symmetries. We analyse the orbifold's geometric properties, the connection between periodicity and strata, and provide orbifold charts as the starting point for the local asymptotic theory. The quantum Fisher information rate of the system and output state is expressed in terms of a canonical inner product on the identifiable tangent space. We then show that the joint system and output model satisfies quantum local asymptotic normality while the stationary output model converges to a product between a quantum Gaussian shift model and a mixture of quantum Gaussian shift models, reflecting the underlying periodicity. These strong convergence results provide the basis for constructing asymptotically optimal estimators of dynamical parameters. We provide an in-depth analysis of the model with smallest dimensions, consisting of two-dimensional system and environment units.

Regression with compositional responses is challenging due to the nonlinear geometry of the simplex and the limitations of Euclidean methods. We propose a regression framework for manifold-valued data based on mappings to statistically tractable intermediate spaces. For compositional data, responses are embedded in the positive orthant of the sphere and analysed using Principal Nested Spheres (PNS), yielding a cylindrical intermediate space with a circular leading score and Euclidean higher-order scores. Regression is performed in this intermediate space and fitted values are mapped back to the simplex. A simulation study demonstrates good performance of PNS-based regression. An application to environmental chemical exposure data illustrates the interpretability and practical utility of the method.

For two time series $\{ (Y_t, Z_t^Y) \}_{t}$ and $\{(X_t, Z_t^X)\}_{t}$, the directional dependence of $\{ X_t \}_{t}$ on $\{ Y_t \}_{t}$ while removing the impact of $Z_t^X$ on $X_t$ and the impact of $Z_t^Y$ on $ Y_t$ can be measured by cross-quantilograms. When the two time series are obeserved over two periods of time, it can be of interest to learn whether the cross-quantilograms remain the same for the two periods of time. We propose a test for this purpose, and the cross-quantilograms are estimated using the estimators proposed by Han (2016). The $p$-value of the proposed test is obtained based on a bootstrap approach.

Existing high-dimensional online learning methods often face the challenge that their error bounds, or per-batch sample sizes, diverge as the number of data batches increases. To address this issue, we propose an asynchronous decomposition framework that leverages summary statistics to construct a surrogate score function for current-batch learning. This framework is implemented via a dynamic-regularized iterative hard thresholding algorithm, providing a computationally and memory-efficient solution for sparse online optimization. We provide a unified theoretical analysis that accounts for both the streaming computational error and statistical accuracy, establishing that our estimator maintains non-divergent error bounds and $\ell_0$ sparsity across all batches. Furthermore, the proposed estimator adaptively achieves additional gains as batches accumulate, attaining the oracle accuracy as if the entire historical dataset were accessible and the true support were known. These theoretical properties are further illustrated through an example of the generalized linear model.

The problem of constrained online convex optimization is considered, where at each round, once a learner commits to an action $x_t \in \mathcal{X} \subset \mathbb{R}^d$, a convex loss function $f_t$ and a convex constraint function $g_t$ that drives the constraint $g_t(x)\le 0$ are revealed. The objective is to simultaneously minimize the static regret and cumulative constraint violation (CCV) compared to the benchmark that knows the loss functions and constraint functions $f_t$ and $g_t$ for all $t$ ahead of time, and chooses a static optimal action that is feasible with respect to all $g_t(x)\le 0$. In recent prior work Sinha and Vaze [2024], algorithms with simultaneous regret of $O(\sqrt{T})$ and CCV of $O(\sqrt{T})$ or (CCV of $O(1)$ in specific cases Vaze and Sinha [2025], e.g. when $d=1$) have been proposed. It is widely believed that CCV is $\Omega(\sqrt{T})$ for all algorithms that ensure that regret is $O(\sqrt{T})$ with the worst case input for any $d\ge 2$. In this paper, we refute this and show that the algorithm of Vaze and Sinha [2025] simultaneously achieves regret of $O(\sqrt{T})$ regret and CCV of $O(T^{1/3})$ when $d=2$.

Recent work has developed a non-parametric Bayesian approach to the calibration of a computer model, which abstractly amounts to the inversion of a pushforward of stochastic input parameters by a smooth map. The framework has been used in several complex scientific applications, motivating our investigation on the continuity of the solution operator with respect to the distribution on the input parameters. We demonstrate that the solution operator for this approach is uniformly continuous in the total variation metric and weakly continuous for a broad class of distributions.

We propose a dense associative memory for empirical measures (weighted point clouds). Stored patterns and queries are finitely supported probability measures, and retrieval is defined by minimizing a Hopfield-style log-sum-exp energy built from the debiased Sinkhorn divergence. We derive retrieval dynamics as a spherical Hellinger Kantorovich (SHK) gradient flow, which updates both support locations and weights. Discretizing the flow yields a deterministic algorithm that uses Sinkhorn potentials to compute barycentric transport steps and a multiplicative simplex reweighting. Under local separation and PL-type conditions we prove basin invariance, geometric convergence to a local minimizer, and a bound showing the minimizer remains close to the corresponding stored pattern. Under a random pattern model, we further show that these Sinkhorn basins are disjoint with high probability, implying exponential capacity in the ambient dimension. Experiments on synthetic Gaussian point-cloud memories demonstrate robust recovery from perturbed queries versus a Euclidean Hopfield-type baseline.

We introduce Exponential Family Discriminant Analysis (EFDA), a unified generative framework that extends classical Linear Discriminant Analysis (LDA) beyond the Gaussian setting to any member of the exponential family. Under the assumption that each class-conditional density belongs to a common exponential family, EFDA derives closed-form maximum-likelihood estimators for all natural parameters and yields a decision rule that is linear in the sufficient statistic, recovering LDA as a special case and capturing nonlinear decision boundaries in the original feature space. We prove that EFDA is asymptotically calibrated and statistically efficient under correct specification, and we generalise it to $K \geq 2$ classes and multivariate data. Through extensive simulation across five exponential-family distributions (Weibull, Gamma, Exponential, Poisson, Negative Binomial), EFDA matches the classification accuracy of LDA, QDA, and logistic regression while reducing Expected Calibration Error (ECE) by $2$-$6\times$, a gap that is structural: it persists for all $n$ and across all class-imbalance levels, because misspecified models remain asymptotically miscalibrated. We further prove and empirically confirm that EFDA's log-odds estimator approaches the Cramér-Rao bound under correct specification, and is the only estimator in our comparison whose mean squared error converges to zero. Complete derivations are provided for nine distributions. Finally, we formally verify all four theoretical propositions in Lean 4, using Aristotle (Harmonic) and OpenGauss (Math, Inc.) as proof generators, with all outputs independently machine-checked by AXLE (Axiom).

Despite their dominance in vision and language, deep neural networks often underperform relative to tree-based models on tabular data. To bridge this gap, we incorporate five key inductive biases into deep learning: robustness to irrelevant features, axis alignment, localized irregularities, feature heterogeneity, and training stability. We propose LassoFlexNet, an architecture that evaluates the linear and nonlinear marginal contribution of each input via Per-Feature Embeddings, and sparsely selects relevant variables using a Tied Group Lasso mechanism. Because these components introduce optimization challenges that destabilize standard proximal methods, we develop a Sequential Hierarchical Proximal Adaptive Gradient optimizer with exponential moving averages (EMA) to ensure stable convergence. Across $52$ datasets from three benchmarks, LassoFlexNet matches or outperforms leading tree-based models, achieving up to a $10$% relative gain, while maintaining Lasso-like interpretability. We substantiate these empirical results with ablation studies and theoretical proofs confirming the architecture's enhanced expressivity and structural breaking of undesired rotational invariance.

We present a statistical analysis of a variant of the periodogram method that forms power spectral density estimates by cross-correlating the discrete Fourier transforms of adjacent time windows. The proposed estimator is closely related to cross-power spectral methods and to a technique introduced by Nelson, which has been observed empirically to improve detection of sinusoidal components in noise. We show that, under a white Gaussian noise model, the expected contribution of noise to the proposed estimator is zero and that the estimator is unbiased under certain window alignment conditions. This contrasts with classical estimators where averaging reduces variance but not expected noise. Moreover, we derive closed-form expressions for the variance and prove an upper bound on the expected magnitude of the estimator that decreases as the number of windows increases. This establishes that the proposed method achieves a noise floor that decays with averaging, unlike standard nonparametric spectral estimators. We further analyze the effect of taking the absolute value to enforce nonnegativity, providing bounds on the resulting bias, and show that this bias also decreases with the number of windows. Theoretical results are validated through numerical simulations. We demonstrate the potential sensitivity to phase misalignment and methods of realignment. We also provide empirical evidence that the estimator is robust to other types of noise.

Solving ill-posed inverse problems necessitates effective regularization strategies to stabilize the inversion process against measurement noise. While classical methods like Tikhonov regularization require heuristic parameter tuning, and standard deep learning approaches often lack interpretability and generalization across resolutions, we propose SC-Net (Spectral Correction Network), a novel operator learning framework. SC-Net operates in the spectral domain of the forward operator, learning a pointwise adaptive filter function that reweights spectral coefficients based on the signal-to-noise ratio. We provide a theoretical analysis showing that SC-Net approximates the continuous inverse operator, guaranteeing discretization invariance. Numerical experiments on 1D integral equations demonstrate that SC-Net: (1) achieves the theoretical minimax optimal convergence rate ($O(\delta^{0.5})$ for $s=p=1.5$), matching theoretical lower bounds; (2) learns interpretable sharp-cutoff filters that outperform Oracle Tikhonov regularization; and (3) exhibits zero-shot super-resolution, maintaining stable reconstruction errors ($\approx 0.23$) when trained on coarse grids ($N=256$) and tested on significantly finer grids (up to $N=2048$). The proposed method bridges the gap between rigorous regularization theory and data-driven operator learning.

The use of clean energy is a global trend, with solar photovoltaic plants serving as a cornerstone of this energy transition. To support this rapid growth, optimize energy utilization, and enable a wide range of applications and services, it is essential to have access to more sophisticated and detailed solar data. Specifically, existing datasets lack integration, contain significant gaps, and have limited geographic coverage. In contrast, this study proposes a reliable, standardized, and multidimensional dataset with a global scope. Through a reproducible methodology and automated processes, we have successfully collected, generated, and combined 27 attributes of geographic, topographic, logistical, climate, and power nature, which are critical for the study of photovoltaic plants worldwide. Based on descriptive statistical analysis of the 58,978 records comprising the compiled dataset, the raw data have been transformed into valuable information for the energy sector. This demonstrates the utility of this product as a source of knowledge discovery, publicly available to the academic and professional communities.

Uncovering causal relationships is a fundamental problem across science and engineering. However, most existing causal discovery methods assume acyclicity and direct access to the system variables – assumptions that fail to hold in many real-world settings. For instance, in genomics, cyclic regulatory networks are common, and measurements are often corrupted by instrumental noise. To address these challenges, we propose RECLAIM, a causal discovery framework that natively handles both cycles and measurement noise. RECLAIM learns the causal graph structure by maximizing the likelihood of the observed measurements via expectation-maximization (EM), using residual normalizing flows for tractable likelihood computation. We consider two measurement models: (i) Gaussian additive noise, and (ii) a linear measurement system with additive Gaussian noise. We provide theoretical consistency guarantees for both the settings. Experiments on synthetic data and real-world protein signaling datasets demonstrate the efficacy of the proposed method.

Denoising diffusion probabilistic models (DDPMs) have emerged as powerful generative models for complex distributions, yet their use in arbitrage-free derivative pricing remains largely unexplored. Financial asset prices are naturally modeled by stochastic differential equations (SDEs), whose forward and reverse density evolution closely parallels the forward noising and reverse denoising structure of diffusion models. In this paper, we develop a framework for using DDPMs to generate risk-neutral asset price dynamics for derivative valuation. Starting from log-return dynamics under the physical measure, we analyze the associated forward diffusion and derive the reverse-time SDE. We show that the change of measure from the physical to the risk-neutral measure induces an additive shift in the score function, which translates into a closed-form risk-neutral epsilon shift in the DDPM reverse dynamics. This correction enforces the risk-neutral drift while preserving the learned variance and higher-order structure, yielding an explicit bridge between diffusion-based generative modeling and classical risk-neutral SDE-based pricing. We show that the resulting discounted price paths satisfy the martingale condition under the risk-neutral measure. Empirically, the method reproduces the risk-neutral terminal distribution and accurately prices both European and path-dependent derivatives, including arithmetic Asian options, under a GBM benchmark. These results demonstrate that diffusion-based generative models provide a flexible and principled approach to simulation-based derivative pricing.

This study examined how student interns in Philippine higher education use intelligent tools during their OJT. Data were collected from 384 respondents using a structured questionnaire that asked about AI tool usage, task-specific applications, and perceptions of confidence, ethics, and support. Analysis of task-based usage identified four main purposes: productivity and report writing, communication and content drafting, technical assistance and code support, and independent task completion. ChatGPT was the most commonly used AI tool, followed by Quillbot, Canva AI, and Grammarly. Students reported moderate confidence in using AI and applied these tools selectively and ethically during OJT tasks. This indicate that AI tools assist student interns in various OJT activities related to work-readiness. The study suggests that higher education programs include AI literacy and onboarding. Clear policies and fair access to AI tools are important to support responsible use and prepare students for future careers.

Independent component analysis is a core framework within blind source separation for recovering latent source signals from observed mixtures under statistical independence assumptions. In this work, we propose PDGMM-VAE, a source-oriented variational autoencoder in which each latent dimension, interpreted explicitly as an individual source component, is assigned its own adaptive Gaussian mixture model prior. The proposed framework imposes heterogeneous per-dimension prior constraints, enabling different latent dimensions to model different non-Gaussian source marginals within a unified probabilistic encoder-decoder architecture. The parameters of these source-specific GMM priors are not fixed in advance, but are jointly learned together with the encoder and decoder under the overall training objective. Beyond the model construction itself, we provide a theoretical analysis clarifying why adaptive per-dimension prior design is meaningful in this setting. In particular, we show that heterogeneous per-dimension priors reduce latent permutation symmetry relative to homogeneous shared priors, and we further show that the KL regularization induced by the adaptive GMM prior creates source-specific attraction behavior that helps explain source-wise specialization during training. We also clarify the relation of the proposed model to the standard VAE and provide a weak recovery statement in an idealized linear low-noise regime. Experimental results on both linear and nonlinear mixing problems show that PDGMM-VAE can recover latent source signals and fit source-specific non-Gaussian marginals effectively. These results suggest that adaptive per-dimension mixture-prior design provides a principled and promising direction for VAE-based ICA and source-oriented generative modeling.

Despite the strong predictive performance achieved by machine learning models across many application domains, assessing their trustworthiness through reliable estimates of predictive confidence remains a critical challenge. This issue arises in scenarios where the likelihood of error inferred from learned representations follows a bimodal distribution, resulting from the coexistence of confident and ambiguous predictions. Standard regression approaches often struggle to adequately express this predictive uncertainty, as they implicitly assume unimodal Gaussian noise, leading to mean-collapse behavior in such settings. Although Mixture Density Networks (MDNs) can represent different distributions, they suffer from severe optimization instability. We propose a family of distribution-aware loss functions integrating normalized RMSE with Wasserstein and Cramér distances. When applied to standard deep regression models, our approach recovers bimodal distributions without the volatility of mixture models. Validated across four experimental stages, our results show that the proposed Wasserstein loss establishes a new Pareto efficiency frontier: matching the stability of standard regression losses like MSE in unimodal tasks while reducing Jensen-Shannon Divergence by 45% on complex bimodal datasets. Our framework strictly dominates MDNs in both fidelity and robustness, offering a reliable tool for aleatoric uncertainty estimation in trustworthy AI systems.

For decades, physics-based climate models have been used to provide insights for climate decision-making. Their application is, however, constrained by significant computational and technical demands. Machine learning (ML) emulators offer a way to reduce these high computational costs; yet, it remains challenging to use ML emulators effectively in climate research. In practice, climate scientists often bypass emulators altogether, and machine learning researchers frequently develop them as methodological showcases without proving their practical utility. The reasons are diverse, ranging from limited accessibility and a lack of specialized knowledge to broader concerns about the physical grounding of ML methods. Here, we discuss limitations and introduce a framework for guiding emulator development, considering both climate science and machine learning perspectives. We argue that designing easy-to-adopt emulators that address clearly defined tasks and demonstrate their reliability is essential. This offers a promising path towards making machine-learning approaches more relevant and usable for applied climate research.

Forecasting accuracy is bounded by the information available about the future. This paper makes that statement precise using information-theoretic tools. Under logarithmic loss, the expected performance of any probabilistic forecast decomposes into two parts: an irreducible component and an approximation component. The irreducible term is the conditional entropy of the future given the available information, while the approximation term is the divergence between the true conditional distribution and the forecasting method. The gap between this conditional-entropy limit and an unconditional baseline is exactly the mutual information between the future observation and the declared information set. This leads to a definition of forecastability as the maximum achievable reduction in expected log loss. Evaluated across horizons, forecastability forms a profile that describes how predictive information varies with lead time. This profile reflects the dependence structure of the process and need not be monotone: predictive information may be concentrated at particular lags, including seasonal horizons, even when intermediate horizons contain little useful signal. From this profile, the paper defines the informative horizon set: the horizons at which forecastability exceeds a practical threshold. At horizons not in this set, the achievable gain over the unconditional baseline is necessarily small, regardless of the forecasting method used. The framework therefore separates what is learnable from what is not, and distinguishes limits imposed by the data from errors introduced by modelling. The result is a pre-modelling diagnostic that identifies where meaningful prediction is feasible before any model is chosen, providing a principled basis for allocating modelling effort across forecast horizons.

Behavior cloning is a fundamental paradigm in machine learning, enabling policy learning from expert demonstrations across robotics, autonomous driving, and generative models. Autoregressive models like transformer have proven remarkably effective, from large language models (LLMs) to vision-language-action systems (VLAs). However, applying autoregressive models to continuous control requires discretizing actions through quantization, a practice widely adopted yet poorly understood theoretically. This paper provides theoretical foundations for this practice. We analyze how quantization error propagates along the horizon and interacts with statistical sample complexity. We show that behavior cloning with quantized actions and log-loss achieves optimal sample complexity, matching existing lower bounds, and incurs only polynomial horizon dependence on quantization error, provided the dynamics are stable and the policy satisfies a probabilistic smoothness condition. We further characterize when different quantization schemes satisfy or violate these requirements, and propose a model-based augmentation that provably improves the error bound without requiring policy smoothness. Finally, we establish fundamental limits that jointly capture the effects of quantization error and statistical complexity.