Neurobiological studies and representation learning have observed that representations of objects belonging to the same category in high-dimensional neural spaces exhibit low-dimensional object manifold characteristics, and different object manifolds are linearly separable in these neural spaces. However, these experimentally observed phenomena lack rigorous theoretical validation to date. This paper proposes a new stochastic separability theorem for embedding manifolds of two different object categories. First, we establish a projection measure concentration theorem for embedding manifolds under general conditions. We develop a new two-layer measure concentration analysis technique, which unifies two estimation bounds via the law of total expectation to derive measure concentration inequalities. Based on the measure concentration theorem, we further prove a stochastic separability theorem for embedding manifolds of two different object categories. If two datasets have distinct means and bounded total variances, their samples become linearly separable with high probability, provided that the projection direction satisfies a non-singularity condition. The main contributions of this paper are twofold: 1. We prove the projection concentration properties of embedding manifolds in high-dimensional spaces by using two-lawyer tail-bound inequalities. 2. We identify a non-singularity condition for the stochastic separability between embedding manifolds, and rigorously prove the stochastic projection separability theorem. The theorem not only uncovers geometric and statistical properties of the object embedding manifolds, but also provides a novel mechanism for representation learning in deep networks.
Several theoretical works have tried to explain the adversarial vulnerability of deep neural networks through properties of high-dimensional geometry. However, the assumptions underlying these works are rarely examined empirically, and systematic evidence remains limited. In this work, we present a systematic study of the role of input dimensionality in both the emergence and the targeted control of adversarial examples. We first analyse the scope and limitations of existing theoretical frameworks based on concentration of measure, showing that real image classes exhibit strong empirical localization, beyond what such theories typically assume. We then conduct an extensive empirical evaluation across hierarchical image datasets spanning a wide range of input dimensionalities and diverse neural architectures. Our results consistently show that adversarial examples become easier to construct as dimensionality increases. We also investigate how input dimensionality affects the additional difficulty of crafting targeted adversarial examples. In particular, we provide theoretical arguments showing that high-dimensional geometry implies that enforcing a specific target label entails only a limited additional distortion compared to untargeted attacks. We corroborate this insight through extensive experiments, demonstrating that the gap between targeted and untargeted perturbations remains small and further narrows as input dimensionality increases. While, taken together, our findings establish high input dimensionality as a fundamental factor underlying the emergence and targeted control of adversarial examples, whether this phenomenon primarily arises from the interplay between high-dimensional geometry and data distributions or from the architectural properties of deep neural networks remains an open question.