Asaconsequence, removing theerror P would reduce theloss more than removing the error N. Moreover, it is clear that this difference in error weighing increases withthelevelofimbalance between theclasses.

Figure 3: Reconstruction (top) anddivergence (bottom) losscomparisonforJSG (left) andJSG (right) against KL(q(z|x)kp(z))(VAE) and KL(p(z)kq(z|x))onthe MNISTdataset.

The analysis and control of stochastic dynamical systems rely on probabilistic models such as (continuous-space) Markov decision processes, but large or continuous state spaces make exact analysis intractable and call for principled quantitative abstraction. This work develops a unified theory of such abstraction by integrating category theory, coalgebra, quantitative logic, and optimal transport, centred on a canonical $\varepsilon$-quotient of the behavioral pseudo-metric with a universal property: among all abstractions that collapse behavioral differences below $\varepsilon$, it is the most detailed, and every other abstraction achieving the same discounted value-loss guarantee factors uniquely through it. Categorically, a quotient functor $Q_\varepsilon$ from a category of probabilistic systems to a category of metric specifications admits, via the Special Adjoint Functor Theorem, a right adjoint $R_\varepsilon$, yielding an adjunction $Q_\varepsilon \dashv R_\varepsilon$ that formalizes a duality between abstraction and realization; logically, a quantitative modal $μ$-calculus with separate reward and transition modalities is shown, for a broad class of systems, to be expressively complete for the behavioral pseudo-metric, with a countable fully abstract fragment suitable for computation. The theory is developed coalgebraically over Polish spaces and the Giry monad and validated on finite-state models using optimal-transport solvers, with experiments corroborating the predicted contraction properties and structural stability and aligning with the theoretical value-loss bounds, thereby providing a rigorous foundation for quantitative state abstraction and representation learning in probabilistic domains.

A common way to estimate an unknown convex regression function $f_0: Ω\subset \mathbb{R}^d \rightarrow \mathbb{R}$ from a set of $n$ noisy observations is to fit a convex function that minimizes the sum of squared errors. However, this estimator is known for its tendency to overfit near the boundary of $Ω$, posing significant challenges in real-world applications. In this paper, we introduce a new estimator of $f_0$ that avoids this overfitting by minimizing a penalty on the subgradient while enforcing an upper bound $s_n$ on the sum of squared errors. The key advantage of this method is that $s_n$ can be directly estimated from the data. We establish the uniform almost sure consistency of the proposed estimator and its subgradient over $Ω$ as $n \rightarrow \infty$ and derive convergence rates. The effectiveness of our estimator is illustrated through its application to estimating waiting times in a single-server queue.

State-of-the-art neural networks can be trained to become remarkable solutions to many problems. But while these architectures can express symbolic, perfect solutions, trained models often arrive at approximations instead. We show that the choice of regularization method plays a crucial role: when trained on formal languages with standard regularization ($L_1$, $L_2$, or none), expressive architectures not only fail to converge to correct solutions but are actively pushed away from perfect initializations. In contrast, applying the Minimum Description Length (MDL) principle to balance model complexity with data fit provides a theoretically grounded regularization method. Using MDL, perfect solutions are selected over approximations, independently of the optimization algorithm. We propose that unlike existing regularization techniques, MDL introduces the appropriate inductive bias to effectively counteract overfitting and promote generalization.

--The recent proliferation of photorealistic AIgenerated images (AIGI) has raised urgent concerns about their potential misuse, particularly on social media platforms. Current state-of-the-art AIGI detection methods typically rely on large, deep neural architectures, creating significant computational barriers to real-time, large-scale deployment on platforms like social media. T o challenge this reliance on computationally intensive models, we introduce LAID, the first framework--to our knowledge--that benchmarks and evaluates the detection performance and efficiency of off-the-shelf lightweight neural networks. In this framework, we comprehensively train and evaluate selected models on a representative subset of the GenImage dataset across spatial, spectral, and fusion image domains. Our results demonstrate that lightweight models can achieve competitive accuracy, even under adversarial conditions, while incurring substantially lower memory and computation costs compared to current state-of-the-art methods. This study offers valuable insight into the trade-off between efficiency and performance in AIGI detection and lays a foundation for the development of practical, scalable, and trustworthy detection systems. The source code of LAID can be found at: https://github.com/nchivar/LAID. The rapid advancement of deep generative models such as Diffusion Models (DMs), Generative Adversarial Networks (GANs), and V ariational Autoencoders (V AEs) has enabled the generation of highly photorealistic synthetic imagery.

In this paper, we present a novel funnel-based tracking control algorithm for robotic systems with unknown dynamics and prescribed input constraints. The Euler-Lagrange formulation, a common modeling approach for robotic systems, has been adopted in this study to address the trade-off between performance and actuator safety. We establish feasibility conditions that ensure tracking errors evolve within predefined funnel bounds while maintaining bounded control efforts, a crucial consideration for robots with limited actuation capabilities. We propose two approximation-free control strategies for scenarios where these conditions are violated: one actively corrects the error, and the other stops further deviation. Finally, we demonstrate the robust performance and safety of the approach through simulations and experimental validations. This work represents a significant advancement in funnel-based control, enhancing its applicability to real-world robotics systems with input constraints.