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

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 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.

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

Finding "true" clusters in a data set is a challenging problem. Clustering solutions obtained using different models and algorithms do not necessarily provide compact and well-separated clusters or the optimal number of clusters. Cluster validity indices are commonly applied to identify such clusters. Nevertheless, these indices are typically relative, and they are used to compare clustering algorithms or choose the parameters of a clustering algorithm. Moreover, the success of these indices depends on the underlying data structure. This paper introduces novel absolute cluster indices to determine both the compactness and separability of clusters. We define a compactness function for each cluster and a set of neighboring points for cluster pairs. This function is utilized to determine the compactness of each cluster and the whole cluster distribution. The set of neighboring points is used to define the margin between clusters and the overall distribution margin. The proposed compactness and separability indices are applied to identify the true number of clusters. Using a number of synthetic and real-world data sets, we demonstrate the performance of these new indices and compare them with other widely-used cluster validity indices.