Relocation of compact sets in $\mathbb{R}^n$ by diffeomorphisms and linear separability of datasets in $\mathbb{R}^n$

Neural networks build representations through affine maps and nonlinear activations. Injective affine maps preserve linear separability, raising the problem of how they prepare data for nonlinear improvement and how much gain can be guaranteed before complete separation. We introduce the directional linear separability measure (D-LSM), which quantifies unavoidable competing-sample intrusion over affine halfspaces retaining every target sample, characterize its supporting geometry, and prove invariance under injective affine embeddings. For gated activations including ReLU, GELU, and SiLU, pre-activation projection bounds yield sufficient conditions for preserving all previous exclusions and recovering additional samples, with a gain bound determined by the certified recovery count. Under an aggregate-tube condition, an explicit affine construction realizes recovery with sufficient width, scaling conditions, and simultaneous multiclass guarantees through a shared layer. Exact controlled experiments compare certified and realized gains, assess certificate coverage, and exhibit bound attainment before complete separation and in affine-tube constructions. In learned Vision Transformer (ViT) representations, a feasible lower-bound estimator yields earlier post-GELU saturation certificates of exact separability, while boundary transport numerically supports affine invariance.

We aim to identify scattering network architectures that maximize the separation capacity on data with low intrinsic dimension. The networks we consider employ a fixed monomial nonlinearity and no pooling, so that the only design variable is the frame generated by the network filters. For data modeled as rectifiable sets, we first characterize and bound the separation capacity of general feature extractors in terms of the geometry of the dataset. We then particularize to scattering networks and obtain two design criteria: (i) the filters should meet the data on sufficiently many frequencies, and (ii) the matrices coupling the frame to the geometry of the data should be well-conditioned.

We study binary classification problems whose decision sets are given by definable sets in o-minimal expansions of the real field. Motivated by cell decomposition of definable sets, we introduce traceable sets as a classical proxy for definable decision regions and analyze their approximation by ReLU neural networks. Under uniform bounds on the number of connected components and suitable $C^m$ extensions for the boundary functions, we prove that characteristic functions of traceable subsets of $[-1/2,1/2]^n$ can be approximated in $L^p$ to accuracy $\varepsilon>0$ by ReLU neural networks of size $\mathcal{O}(\varepsilon^{-p(n-1)/m})$, with depth independent of $\varepsilon$ and polynomially bounded weights. This establishes quantitative approximation rates for certain definable collections in o-minimal structures using ReLU neural networks. The same approach also yields the stated approximation rates for a subclass of definable maps $[-1/2,1/2]^n \to \mathbb{R}$. We then combine the approximation capabilities with entropy estimates for ReLU neural network classes to obtain statistical learning rates for empirical risk minimization with hinge loss. For $N$ uniformly distributed samples, the resulting classifiers achieve expected misclassification error of order $N^{-m/(m+pn-p)}$ up to an arbitrarily small polynomial loss.