We propose new scattering networks for signals measured on simplicial complexes, which we call Multiscale Hodge Scattering Networks (MHSNs). Our construction builds on multiscale basis dictionaries on simplicial complexes—namely, the κ -GHWT and κ -HGLET—which we recently developed for simplices of dimension κ ∈ N in a given simplicial complex by generalizing the node-based Generalized Haar–Walsh Transform (GHWT) and Hierarchical Graph Laplacian Eigen Transform (HGLET). Both the κ -GHWT and the κ -HGLET form redundant sets (i.e., dictionaries) of multiscale basis vectors and the corresponding expansion coefficients of a given signal. Our MHSNs adopt a layered structure analogous to a convolutional neural network (CNN), cascading the moments of the modulus of the dictionary coefficients. The resulting features are invariant to reordering of the simplices (i.e., node permutation of the underlying graphs). Importantly, the use of multiscale basis dictionaries in our MHSNs admits a natural pooling operation—akin to local pooling in CNNs—that can be performed either locally or per scale. Such pooling operations are more difficult to define in traditional scattering networks based on Morlet wavelets and in geometric scattering networks based on Diffusion Wavelets. As a result, our approach extracts a rich set of descriptive yet robust features that can be combined with simple machine learning models (e.g., logistic regression or support vector machines) to achieve high-accuracy classification with far fewer trainable parameters than most modern graph neural networks require. Finally, we demonstrate the effectiveness of MHSNs on three distinct problem types: signal classification, domain (i.e., graph/simplex) classification, and molecular dynamics prediction.

Integrate-and-fire is a resource efficient time-encoding mechanism that summarizes into a signed spike train those time intervals where a signal’s charge exceeds a certain threshold. We analyze the IF encoder in terms of a very general notion of approximate bandwidth, which is shared by most commonly-used signal models. This complements results on exact encoding that may be overly adapted to a particular signal model. We take into account, possibly for the first time, the effect of uncertainty in the exact location of the spikes (as may arise by decimation), uncertainty of integration leakage (as may arise in realistic manufacturing), and boundary effects inherent to finite periods of exposure to the measurement device. The analysis is done by means of a concrete bandwidth-based Ansatz that can also be useful to initialize more sophisticated model specific reconstruction algorithms, and uses the earth mover’s (Wasserstein) distance to measure spike discrepancy.

We construct smooth localized orthonormal bases compatible with anisotropic Triebel-Lizorkin and Besov type spaces on R d . The construction is based on tensor products of so-called univariate brushlet functions that are based on local trigonometric bases in the frequency domain, and the construction is painless in the sense that all parameters for the construction are explicitly specified. It is shown that the associated decomposition system form unconditional bases for the full family of Triebel-Lizorkin and Besov type spaces, including for the so-called α -modulation and α -Triebel-Lizorkin spaces. In the second part of the paper we study nonlinear m -term approximation with the constructed bases, where direct Jackson and Bernstein inequalities for m -term approximation with the tensor brushlet system in α -modulation and α -Triebel-Lizorkin spaces are derived. The inverse Bernstein estimates rely heavily on the fact that the constructed system is non-redundant.