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 fil…
The success of deep learning models in classification and regression is widely attributed to the low-dimensional structure that real-world data tend to exhibit, despite their high-dimensional representation. This work attempts to provide a mathematical framework for binary classi…
While autoregressive models optimize the exact data likelihood via the chain rule, diffusion models are typically trained with denoising objectives. We develop conservation laws based on generalized extrinsic information transfer (GEXIT) functions for a broad class of memoryless…
Linear attention promises constant-time recurrent inference but degrades sharply on associative recall. We formulate attention recall as a spherical-packing problem and introduce Kernelized Linear Attention Activations (KATA), a framework whose feature maps are derived from first…
We establish a $\widetildeΩ(d^{5/4}\sqrt T)$ lower bound on the minimax expected regret of stochastic bandit convex optimization of $1$-Lipschitz functions on the Euclidean ball. This presents the first nontrivial regret lower bound that grows faster than $d\sqrt{T}$ for this pro…
Cooperative multi-task semantic communication (CMT-SemCom) improves task execution performance by leveraging shared representations. However, as we demonstrated in [1], cooperative multi-tasking can be either constructive or destructive, depending on the semantic relationships am…