Title: BDN: Blaschke Decomposition Networks
Primary area: learning on time series and dynamical systems
Keywords: Blaschke product, unwinding series, decomposition, Blaschke roots

Abstract: We introduce the Blashke Decomposition Network (BDN), a novel neural network architecture for analyzing continuous real-valued or complex-valued 1-D and 2-D signals - data types that existing architectures, such as transformers or recurrent networks, are not designed to model. These signals are common in medicine, biology, and other scientific domains, yet their analytic structure is often underutilized in machine learning. Our approach is based on the Blaschke decomposition, which ``unwinds" a signal into a sequence of factors determined by its roots - the points in the complex unit disk where the analytic continuation of the signal vanishes. By iteratively peeling off these factors, the decomposition isolates oscillatory components of the signal and produces a compact representation. BDNs are trained to predict these roots directly, and we show that they provide powerful and interpretable representations for downstream tasks. We first design the architecture for 1-D signals and then extend it to 2-D using a wedge-based factorization, enabling the same framework to handle images and other spatially varying signals. Experiments on sensor-derived biomedical data, including electrocardiograms and phase holographic microscopy, show that BDNs achieve strong predictive performance while using fewer parameters than transformers, convolutional, or recurrent networks. Our code is available at: https://anonymous.4open.science/r/BDN-5603

TL;DR: We propose a novel neural architecture that advances the integration of complex analysis into machine learning by leveraging the Blaschke unwinding series.