We study a nonsmooth nonconvex optimization problem defined over nonconvex constraints, where the feasible set is given by the intersection of the closure of an open set and a smooth manifold. By endowing the open set with a Riemannian metric induced by a barrier function, we obtain a Riemannian subgradient flow formulated as a differential inclusion, which remains strictly within the interior of the feasible set. This continuous dynamical system unifies two classes of iterative optimization methods, namely the Hessian barrier method and mirror descent scheme, by revealing that these methods can be interpreted as discrete approximations of the continuous flow. We explore the long-term behavior of the trajectories generated by this dynamical system and show that the existing deficient convergence properties of the Hessian barrier and mirror descent scheme can be unifily and more insightfully interpreted through these of the continuous trajectory. For instance, the notorious spurious stationary points \cite{chen2024spurious} observed in Hessian barrier method and mirror descent scheme are interpreted as stable equilibria of the dynamical system that do not correspond to real stationary points of the original optimization problem. We provide two sufficient condition such that these spurious stationary points can be avoided if the strict complementarity conditions holds. In the absence of these regularity condition, we propose a random perturbation strategy that ensures the trajectory converges (subsequentially) to an approximate stationary point. Building on these insights, we introduce two iterative Riemannian subgradient methods, form of interior point methods, that generalizes the existing Hessian barrier method and mirror descent scheme for solving nonsmooth nonconvex optimization problems.

Large Language Models (LLMs) perform best with well-crafted prompts, yet prompt engineering remains manual, inconsistent, and inaccessible to non-experts. We introduce Promptomatix, an automatic prompt optimization framework that transforms natural language task descriptions into high-quality prompts without requiring manual tuning or domain expertise. Promptomatix supports both a lightweight meta-prompt-based optimizer and a DSPy-powered compiler, with modular design enabling future extension to more advanced frameworks. The system analyzes user intent, generates synthetic training data, selects prompting strategies, and refines prompts using cost-aware objectives. Evaluated across 5 task categories, Promptomatix achieves competitive or superior performance compared to existing libraries, while reducing prompt length and computational overhead making prompt optimization scalable and efficient.

Pairwise Euclidean distance calculation is a fundamental step in many machine learning and data analysis algorithms. In real-world applications, however, these distances are frequently distorted by heteroskedastic noise$\unicode{x2014}$a prevalent form of inhomogeneous corruption characterized by variable noise magnitudes across data observations. Such noise inflates the computed distances in a nontrivial way, leading to misrepresentations of the underlying data geometry. In this work, we address the tasks of estimating the noise magnitudes per observation and correcting the pairwise Euclidean distances under heteroskedastic noise. Perhaps surprisingly, we show that in general high-dimensional settings and without assuming prior knowledge on the clean data structure or noise distribution, both tasks can be performed reliably, even when the noise levels vary considerably. Specifically, we develop a principled, hyperparameter-free approach that jointly estimates the noise magnitudes and corrects the distances. We provide theoretical guarantees for our approach, establishing probabilistic bounds on the estimation errors of both noise magnitudes and distances. These bounds, measured in the normalized $\ell_1$ norm, converge to zero at polynomial rates as both feature dimension and dataset size increase. Experiments on synthetic datasets demonstrate that our method accurately estimates distances in challenging regimes, significantly improving the robustness of subsequent distance-based computations. Notably, when applied to single-cell RNA sequencing data, our method yields noise magnitude estimates consistent with an established prototypical model, enabling accurate nearest neighbor identification that is fundamental to many downstream analyses.

We study a privacy-preserving data-sharing setting where a privatizer transforms private data into a sanitized version observed by an authorized reconstructor and two unauthorized adversaries, each with access to side information correlated with the private data. The reconstructor is evaluated under a distortion function, while each adversary is evaluated using a separate loss function. The privatizer ensures the reconstructor distortion remains below a fixed threshold while maximizing the minimum loss across the two adversaries. This two-adversary setting models cases where individual users cannot reconstruct the data accurately, but their combined side information enables estimation within the distortion threshold. The privatizer maximizes individual loss while permitting accurate reconstruction only through collaboration. This echoes secret-sharing principles, but with lossy rather than perfect recovery. We frame this as a constrained data-driven minimax optimization problem and propose a data-driven training procedure that alternately updates the privatizer, reconstructor, and adversaries. We also analyze the Gaussian and binary cases as special scenarios where optimal solutions can be obtained. These theoretical optimal results are benchmarks for evaluating the proposed minimax training approach.

In recent years, Compressed Sensing (CS) has gained significant interest as a technique for acquiring high-resolution sensory data using fewer measurements than traditional Nyquist sampling requires. At the same time, autonomous robotic platforms such as drones and rovers have become increasingly popular tools for remote sensing and environmental monitoring tasks, including measurements of temperature, humidity, and air quality. Within this context, this paper presents, to the best of our knowledge, the first investigation into how the structure of CS measurement matrices can be exploited to design optimized sampling trajectories for robotic environmental data collection. We propose a novel Monte Carlo optimization framework that generates measurement matrices designed to minimize both the robot's traversal path length and the signal reconstruction error within the CS framework. Central to our approach is the application of Dictionary Learning (DL) to obtain a data-driven sparsifying transform, which enhances reconstruction accuracy while further reducing the number of samples that the robot needs to collect. We demonstrate the effectiveness of our method through experiments reconstructing $NO_2$ pollution maps over the Gulf region. The results indicate that our approach can reduce robot travel distance to less than $10\%$ of a full-coverage path, while improving reconstruction accuracy by over a factor of five compared to traditional CS methods based on DCT and polynomial dictionaries, as well as by a factor of two compared to previously-proposed Informative Path Planning (IPP) methods.

Designing industrial systems, such as building, improving, and automating distribution centers and manufacturing plants, involves critical decision-making with limited information in the early phases. The lack of information leads to less accurate designs of the systems, which are often difficult to resolve later. It is effective to use simulators to model the designed system and find out the issues early. However, the modeling time required by conventional simulators is too long to allow for rapid model creation to meet decision-making demands. In this paper, we propose a Rapid Modeling Architecture (RMA) for a lightweight industrial simulator that mitigates the modeling burden while maintaining the essential details in order to accelerate and improve decision-making. We have prototyped a simulator based on the RMA and applied it to the actual factory layout design problem. We also compared the modeling time of our simulator to that of an existing simulator, and as a result, our simulator achieved a 78.3% reduction in modeling time compared to conventional simulators.

Identifying the factors driving data shifts in tabular datasets is a significant challenge for analysis and decision support systems, especially those focusing on healthcare. Privacy rules restrict data access, and noise from complex processes hinders analysis. To address this challenge, we propose SIFOTL (Statistically-Informed Fidelity-Optimization Method for Tabular Learning) that (i) extracts privacy-compliant data summary statistics, (ii) employs twin XGBoost models to disentangle intervention signals from noise with assistance from LLMs, and (iii) merges XGBoost outputs via a Pareto-weighted decision tree to identify interpretable segments responsible for the shift. Unlike existing analyses which may ignore noise or require full data access for LLM-based analysis, SIFOTL addresses both challenges using only privacy-safe summary statistics. Demonstrating its real-world efficacy, for a MEPS panel dataset mimicking a new Medicare drug subsidy, SIFOTL achieves an F1 score of 0.85, substantially outperforming BigQuery Contribution Analysis (F1=0.46) and statistical tests (F1=0.20) in identifying the segment receiving the subsidy. Furthermore, across 18 diverse EHR datasets generated based on Synthea ABM, SIFOTL sustains F1 scores of 0.86-0.96 without noise and >= 0.75 even with injected observational noise, whereas baseline average F1 scores range from 0.19-0.67 under the same tests. SIFOTL, therefore, provides an interpretable, privacy-conscious workflow that is empirically robust to observational noise.

We introduce a reinforcement learning (RL) based adaptive optimization algorithm for aerodynamic shape optimization focused on dimensionality reduction. The form in which RL is applied here is that of a surrogate-based, actor-critic policy evaluation MCMC approach allowing for temporal 'freezing' of some of the parameters to be optimized. The goals are to minimize computational effort, and to use the observed optimization results for interpretation of the discovered extrema in terms of their role in achieving the desired flow-field. By a sequence of local optimized parameter changes around intermediate CFD simulations acting as ground truth, it is possible to speed up the global optimization if (a) the local neighbourhoods of the parameters in which the changed parameters must reside are sufficiently large to compete with the grid-sized steps and its large number of simulations, and (b) the estimates of the rewards and costs on these neighbourhoods necessary for a good step-wise parameter adaption are sufficiently accurate. We give an example of a simple fluid-dynamical problem on which the method allows interpretation in the sense of a feature importance scoring.

--Semantic communications focus on prioritizing the understanding of the meaning behind transmitted data and ensuring the successful completion of tasks that motivate the exchange of information. However, when devices rely on different languages, logic, or internal representations, semantic mismatches may occur, potentially hindering mutual understanding. This paper introduces a novel approach to addressing latent space misalignment in semantic communications, exploiting multiple-input multiple-output (MIMO) communications. Specifically, our method learns a MIMO precoder/decoder pair that jointly performs latent space compression and semantic channel equalization, mitigating both semantic mismatches and physical channel impairments. We explore two solutions: (i) a linear model, optimized by solving a biconvex optimization problem via the alternating direction method of multipliers (ADMM); (ii) a neural network-based model, which learns semantic MIMO pre-coder/decoder under transmission power budget and complexity constraints. Numerical results demonstrate the effectiveness of the proposed approach in a goal-oriented semantic communication scenario, illustrating the main trade-offs between accuracy, communication burden, and complexity of the solutions.

Large Language Models (LLMs) have shown remarkable reasoning ability through explicit Chain-of-Thought (CoT) prompting, but generating these step-by-step textual explanations is computationally expensive and slow. To overcome this, we aim to develop a framework for efficient, implicit reasoning, where the model "thinks" in a latent space without generating explicit text for every step. We propose that these latent thoughts can be modeled as temporally-extended abstract actions, or options, within a hierarchical reinforcement learning framework. To effectively learn a diverse library of options as latent embeddings, we first introduce the Variational Markovian Option Critic (VMOC), an off-policy algorithm that uses variational inference within the HiT-MDP framework. To provide a rigorous foundation for using these options as an abstract reasoning space, we extend the theory of continuous MDP homomorphisms. This proves that learning a policy in the simplified, abstract latent space, for which VMOC is suited, preserves the optimality of the solution to the original, complex problem. Finally, we propose a cold-start procedure that leverages supervised fine-tuning (SFT) data to distill human reasoning demonstrations into this latent option space, providing a rich initialization for the model's reasoning capabilities. Extensive experiments demonstrate that our approach achieves strong performance on complex logical reasoning benchmarks and challenging locomotion tasks, validating our framework as a principled method for learning abstract skills for both language and control.