We survey the field of nonparametric inference under shape constraints, providing a historical overview and a perspective on its current state. An outlook and some open problems offer thoughts on future directions. 1 Introduction. Traditionally, we think of statistical methods as being divided into parametric approaches, which can be restrictive, but where estimation is typically straightforward (e.g. using maximum likelihood), and nonparametric methods, which are more flexible but often require careful choices of tuning parameters. Nonparametric inference under shape constraints sits somewhere in the middle, seeking in some ways the best of both worlds. The origins of the field are often traced to Grenander (1956), who proved that there exists a unique maximum likelihood estimator (MLE) of a decreasing density on the non-negative half-line (and was able to characterise it explicitly).

How can we generate samples from a conditional distribution that we never fully observe? This question arises across a broad range of applications in both modern machine learning and classical statistics, including image post-processing in computer vision, approximate posterior sampling in simulation-based inference, and conditional distribution modeling in complex data settings. In such settings, compared with unconditional sampling, additional feature information can be leveraged to enable more adaptive and efficient sampling. Building on this, we introduce Conditional Generator using MMD (CGMMD), a novel framework for conditional sampling. Unlike many contemporary approaches, our method frames the training objective as a simple, adversary-free direct minimization problem. A key feature of CGMMD is its ability to produce conditional samples in a single forward pass of the generator, enabling practical one-shot sampling with low test-time complexity. We establish rigorous theoretical bounds on the loss incurred when sampling from the CGMMD sampler, and prove convergence of the estimated distribution to the true conditional distribution. In the process, we also develop a uniform concentration result for nearest-neighbor based functionals, which may be of independent interest. Finally, we show that CGMMD performs competitively on synthetic tasks involving complex conditional densities, as well as on practical applications such as image denoising and image super-resolution.

Within the PAC-Bayesian framework, the Gibbs classifier (defined on a posterior $Q$) and the corresponding $Q$-weighted majority vote classifier are commonly used to analyze the generalization performance. However, there exists a notable lack in theoretical research exploring the certified robustness of majority vote classifier and its interplay with generalization. In this study, we develop a generalization error bound that possesses a certified robust radius for the smoothed majority vote classifier (i.e., the $Q$-weighted majority vote classifier with smoothed inputs); In other words, the generalization bound holds under any data perturbation within the certified robust radius. As a byproduct, we find that the underpinnings of both the generalization bound and the certified robust radius draw, in part, upon weight spectral norm, which thereby inspires the adoption of spectral regularization in smooth training to boost certified robustness. Utilizing the dimension-independent property of spherical Gaussian inputs in smooth training, we propose a novel and inexpensive spectral regularizer to enhance the smoothed majority vote classifier. In addition to the theoretical contribution, a set of empirical results is provided to substantiate the effectiveness of our proposed method.

Feature selection is crucial for fuzzy decision systems (FDSs), as it identifies informative features and eliminates rule redundancy, thereby enhancing predictive performance and interpretability. Most existing methods either fail to directly align evaluation criteria with learning performance or rely solely on non-directional Euclidean distances to capture relationships among decision classes, which limits their ability to clarify decision boundaries. However, the spatial distribution of instances has a potential impact on the clarity of such boundaries. Motivated by this, we propose Spatially-aware Separability-driven Feature Selection (S$^2$FS), a novel framework for FDSs guided by a spatially-aware separability criterion. This criterion jointly considers within-class compactness and between-class separation by integrating scalar-distances with spatial directional information, providing a more comprehensive characterization of class structures. S$^2$FS employs a forward greedy strategy to iteratively select the most discriminative features. Extensive experiments on ten real-world datasets demonstrate that S$^2$FS consistently outperforms eight state-of-the-art feature selection algorithms in both classification accuracy and clustering performance, while feature visualizations further confirm the interpretability of the selected features.

Diffusion large language models (dLLMs) have shown great potential in large-scale language modeling, and there is an increasing interest in further improving the capacity to solve complex problems by guiding the reasoning process step by step. Common practice for autoregressive language models typically learns a process reward model with dense annotation for each intermediate step. However, this is challenging for dLLMs where the generation is in an any-order fashion and intermediate states are partially masked sentences. To this end, in this paper, we propose reward-free guidance (RFG), a principled method for guiding the reasoning trajectory of dLLMs without explicit process reward. The key idea of RFG is to parameterize the process reward by log-likelihood ratios of the enhanced and reference dLLMs, where the enhanced model can be easily obtained by any off-the-shelf dLLM that has been post-trained with reinforcement learning (RL) or supervised fine-tuning (SFT). We provide theoretical justification that RFG induces the reward-guided sampling distribution with no additional reward. We conduct comprehensive experiments on four challenging mathematical reasoning and code generation benchmarks using a diverse suite of dLLMs enhanced with various post-training methods. RFG consistently yields significant improvements across all tasks and model types, achieving accuracy gains of up to 9.2%. These findings establish RFG as a general training-free framework that scales test-time reasoning without reliance on external reward models. By scaling up mask-predict pretraining on large-scale corpora through bidirectional computation, dLLMs have shown surprisingly competitive or even superior performance over autoregressive (AR) model baselines (Prabhudesai et al., 2025). Despite the impressive advancements, the current success of dLLMs is primarily limited to pre-training or continue-training on a specific domain, with limited exploration in test-time computation and alignment.

Equation discovery provides a grey-box approach to system identification by uncovering governing dynamics directly from observed data. However, a persistent challenge lies in ensuring that identified models generalise across operating conditions rather than over-fitting to specific datasets. This work investigates this issue by applying a Bayesian relevance vector machine (RVM) within a multi-task learning (MTL) framework for simultaneous parameter identification across multiple datasets. In this formulation, responses from the same structure under different excitation levels are treated as related tasks that share model parameters but retain task-specific noise characteristics. A simulated single degree-of-freedom oscillator with linear and cubic stiffness provided the case study, with datasets generated under three excitation regimes. Standard single-task RVM models were able to reproduce system responses but often failed to recover the true governing terms when excitations insufficiently stimulated non-linear dynamics. By contrast, the MTL-RVM combined information across tasks, improving parameter recovery for weakly and moderately excited datasets, while maintaining strong performance under high excitation. These findings demonstrate that multi-task Bayesian inference can mitigate over-fitting and promote generalisation in equation discovery. The approach is particularly relevant to structural health monitoring, where varying load conditions reveal complementary aspects of system physics.

We introduce learning to condition (L2C), a scalable, data-driven framework for accelerating Most Probable Explanation (MPE) inference in Probabilistic Graphical Models (PGMs), a fundamentally intractable problem. L2C trains a neural network to score variable-value assignments based on their utility for conditioning, given observed evidence. To facilitate supervised learning, we develop a scalable data generation pipeline that extracts training signals from the search traces of existing MPE solvers. The trained network serves as a heuristic that integrates with search algorithms, acting as a conditioning strategy prior to exact inference or as a branching and node selection policy within branch-and-bound solvers. We evaluate L2C on challenging MPE queries involving high-treewidth PGMs. Experiments show that our learned heuristic significantly reduces the search space while maintaining or improving solution quality over state-of-the-art methods.

Modifications to test-time sampling have emerged as an important extension to diffusion algorithms, with the goal of biasing the generative process to achieve a given objective without having to retrain the entire diffusion model. However, generating jointly correlated samples from multiple pre-trained diffusion models while simultaneously enforcing task-specific constraints without costly retraining has remained challenging. To this end, we propose Projected Coupled Diffusion (PCD), a novel test-time framework for constrained joint generation. PCD introduces a coupled guidance term into the generative dynamics to encourage coordination between diffusion models and incorporates a projection step at each diffusion step to enforce hard constraints. Empirically, we demonstrate the effectiveness of PCD in application scenarios of image-pair generation, object manipulation, and multi-robot motion planning. Our results show improved coupling effects and guaranteed constraint satisfaction without incurring excessive computational costs.

We introduce Flower, a solver for inverse problems. It leverages a pre-trained flow model to produce reconstructions that are consistent with the observed measurements. Flower operates through an iterative procedure over three steps: (i) a flow-consistent destination estimation, where the velocity network predicts a denoised target; (ii) a refinement step that projects the estimated destination onto a feasible set defined by the forward operator; and (iii) a time-progression step that re-projects the refined destination along the flow trajectory. We provide a theoretical analysis that demonstrates how Flower approximates Bayesian posterior sampling, thereby unifying perspectives from plug-and-play methods and generative inverse solvers. On the practical side, Flower achieves state-of-the-art reconstruction quality while using nearly identical hyperparameters across various inverse problems.

Group-fairness metrics (e.g., equalized odds) can vary sharply across resamples and are especially brittle under distribution shift, undermining reliable audits. We propose a Wasserstein distributionally robust framework that certifies worst-case group fairness over a ball of plausible test distributions centered at the empirical law. Our formulation unifies common group fairness notions via a generic conditional-probability functional and defines $\varepsilon$-Wasserstein Distributional Fairness ($\varepsilon$-WDF) as the audit target. Leveraging strong duality, we derive tractable reformulations and an efficient estimator (DRUNE) for $\varepsilon$-WDF. We prove feasibility and consistency and establish finite-sample certification guarantees for auditing fairness, along with quantitative bounds under smoothness and margin conditions. Across standard benchmarks and classifiers, $\varepsilon$-WDF delivers stable fairness assessments under distribution shift, providing a principled basis for auditing and certifying group fairness beyond observational data.