We demonstrate a compactness result holding broadly across supervised learning with a general class of loss functions: Any hypothesis class $\mathcal{H}$ is learnable with transductive sample complexity $m$ precisely when all of its finite projections are learnable with sample complexity $m$. We prove that this exact form of compactness holds for realizable and agnostic learning with respect to all proper metric loss functions (e.g., any norm on $\mathbb{R}^d$) and any continuous loss on a compact space (e.g., cross-entropy, squared loss). For realizable learning with improper metric losses, we show that exact compactness of sample complexity can fail, and provide matching upper and lower bounds of a factor of 2 on the extent to which such sample complexities can differ. We conjecture that larger gaps are possible for the agnostic case. Furthermore, invoking the equivalence between sample complexities in the PAC and transductive models (up to lower order factors, in the realizable case) permits us to directly port our results to the PAC model, revealing an almost-exact form of compactness holding broadly in PAC learning.

We develop a general theory of semantic dynamics for large language models by formalizing them as Continuous State Machines (CSMs): smooth dynamical systems whose latent manifolds evolve under probabilistic transition operators. The associated transfer operator $P: L^2(M,μ) \to L^2(M,μ)$ encodes the propagation of semantic mass. Under mild regularity assumptions (compactness, ergodicity, bounded Jacobian), $P$ is compact with discrete spectrum. Within this setting, we prove the Semantic Characterization Theorem (SCT): the leading eigenfunctions of $P$ induce finitely many spectral basins of invariant meaning, each definable in an o-minimal structure over $\mathbb{R}$. Thus spectral lumpability and logical tameness coincide. This explains how discrete symbolic semantics can emerge from continuous computation: the continuous activation manifold collapses into a finite, logically interpretable ontology. We further extend the SCT to stochastic and adiabatic (time-inhomogeneous) settings, showing that slowly drifting kernels preserve compactness, spectral coherence, and basin structure.

Deep learning models are used in safety-critical tasks such as automated driving and face recognition. However, small perturbations in the model input can significantly change the predictions. Adversarial attacks are used to identify small perturbations that can lead to misclassifications. More powerful black-box adversarial attacks are required to develop more effective defenses. A promising approach to black-box adversarial attacks is to repeat the process of extracting a specific image area and changing the perturbations added to it. Existing attacks adopt simple rectangles as the areas where perturbations are changed in a single iteration. We propose applying superpixels instead, which achieve a good balance between color variance and compactness. We also propose a new search method, versatile search, and a novel attack method, Superpixel Attack, which applies superpixels and performs versatile search. Superpixel Attack improves attack success rates by an average of 2.10% compared with existing attacks. Most models used in this study are robust against adversarial attacks, and this improvement is significant for black-box adversarial attacks. The code is avilable at https://github.com/oe1307/SuperpixelAttack.git.

Thank you for the review. The results are shown in Figure 1. Thank you for the review. To clarify the point "the author uses the structural properties in factored MDP, which is well-studied": note that our Reviewer 3: Thank you for the review. We will add this in the updated paper.

Constructive solid geometry (CSG) is a classical CAD representation; it models a 3D shape as a recursive assembly of solid primitives, e.g., cuboids, cylinders, etc., through Boolean operations

We demonstrate that supervised learning in the transductive model enjoys such structure.

In this paper, we propose to use the concept of local fairness for auditing and ranking redistricting plans.

Wasserstein distributionally robust estimators have emerged as powerful models for prediction and decision-making under uncertainty. These estimators provide attractive generalization guarantees: the robust objective obtained from the training distribution is an exact upper bound on the true risk with high probability. However, existing guarantees either suffer from the curse of dimensionality, are restricted to specific settings, or lead to spurious error terms. In this paper, we show that these generalization guarantees actually hold on general classes of models, do not suffer from the curse of dimensionality, and can even cover distribution shifts at testing. We also prove that these results carry over to the newly-introduced regularized versions of Wasserstein distributionally robust problems.