Local regression on path spaces with signature metrics
This addresses the problem of scalable and accurate analysis of sequential data for researchers and practitioners in fields like machine learning and statistics, though it is incremental as it builds on existing kernel regression and signature methods.
The paper tackles nonparametric regression and classification for path-valued data by introducing a functional Nadaraya-Watson estimator that combines the signature transform with local kernel regression, achieving competitive accuracy and computational efficiency in applications like time series classification.
We study nonparametric regression and classification for path-valued data. We introduce a functional Nadaraya-Watson estimator that combines the signature transform from rough path theory with local kernel regression. The signature transform provides a principled way to encode sequential data through iterated integrals, enabling direct comparison of paths in a natural metric space. Our approach leverages signature-induced distances within the classical kernel regression framework, achieving computational efficiency while avoiding the scalability bottlenecks of large-scale kernel matrix operations. We establish finite-sample convergence bounds demonstrating favorable statistical properties of signature-based distances compared to traditional metrics in infinite-dimensional settings. We propose robust signature variants that provide stability against outliers, enhancing practical performance. Applications to both synthetic and real-world data - including stochastic differential equation learning and time series classification - demonstrate competitive accuracy while offering significant computational advantages over existing methods.