In this paper, we attempt to enhance the theoretical understanding of convolutional neural networks (CNNs) as feature extractors in classification tasks by analyzing them through the lens of Cover's function-counting theory. Specifically, our focus lies on the notion of separation capacity, a combinatorial quantity derived from counting the number of realizable dichotomies (i.e., binary label assignments). Our contributions are threefold. First, we extend Cover's framework by establishing a conceptually insightful and practically useful formulation for the separation capacity. Second, leveraging this formulation, we identify the factors governing the separation capacity of feature extractors that employ a specific CNN architecture, so-called scattering networks, in terms of their network building blocks. Third, we provide practical insights for scattering network design.
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.
A foundational assumption in CNN interpretability -- that deep encoders suppress background pixels while classifiers merely select from a cleaned feature pool (the Spatial Funnel Hypothesis) -- remains untested due to spatial hallucinations in existing visualization tools. We address this by introducing a hallucination-free inversion framework built on magnitude-phase decoupling and Local Adjoint Correctors. Our method mathematically guarantees that the spatial gradient support of every reconstruction stems strictly from genuinely active channels. Using this framework as a geometric probe, we uncover the first pixel-level evidence of strong superposition in vision encoders. We show that per-channel inversions are uniformly holographic: positive and negative weight reconstructions are visually and energetically indistinguishable. However, their algebraic sum sharply concentrates on the foreground. This proves classification operates via destructive interference -- classifier weights cancel a shared background direction in pixel space and constructively assemble class-discriminative residuals, directly falsifying the Spatial Funnel Hypothesis. This interference model identifies the volume of the admissible interference subspace as the geometric quantity governing channel requirements. We prove this volume is dual to the GAP covariance determinant, yielding a covariance-volume channel selection algorithm with a (1−1/e) approximation guarantee. This algorithm mathematically reveals out-of-distribution (OOD) failure as a measurable collapse of the covariance volume essential for interference-based classification. Our framework extends seamlessly to attention-based heads without retraining.
Neural networks are increasingly employed to identify both well-defined and ambiguous concepts, yet output-level metrics reveal little about how those concepts are represented internally. Our study asks if these networks exhibit \textit{conceptual separation}: if examples of the same concept form coherent representations, and whether related concepts lie closer together in the representation space. We examine this conceptual organisation in Convolutional Neural Networks (CNNs) and Large Language Models (LLMs) through geometric and distributional analysis of their internal activations. In CNNs, familiar ImageNet concepts form coherent and semantically ordered representations, while this coherence weakens for unseen concepts and suffers within-class domain shift. In LLMs, clearly distinct domains remain well separated, related subdomains move closer together, and the distinction between ambiguous topics collapses at both the mean and covariance level. These results suggest that conceptual separation can reveal structure that output accuracy alone cannot, and may serve as a useful diagnostic of how robustly a model represents the concepts it is asked to identify. Code and data available on \href{https://github.com/JaeeRoshniCapstoneProject/Are-You-Thinking-What-I-m-Thinking-Examining-Conceptual-Separation-in-Neural-Architectures}{GitHub}.