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.
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…
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 separatio…
Suppose we observe two sets of $n$ Gaussian vectors in $\mathbb{R}^d$, with the promise that, after applying a permutation of $[n]$ and a rotation of $\mathbb{R}^d$, the two sets are $ρ$-correlated. The Procrustes matching problem asks us to recover the unknown permutation of $[n…
The informativeness of a training set is as consequential as its size, yet most sampling strategies remain agnostic to the intrinsic geometry of the data distribution. We introduce CuBAS (Curvature-Based Adaptive Sampling), an information-geometric framework for adaptive data sel…
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…