Random Projections and Natural Sparsity in Time-Series Classification: A Theoretical Analysis

Within this domain, the Rocket algorithm stands out as an elegantly straightforward yet robust approach that has demonstrated superior performance compared to existing methods. At its core, the algorithm transforms time-series data through randomly initialized convolutional kernels, followed by a non-linear transformation step. This structure mirrors a simplified convolutional neural network with a single hidden layer, but uniquely eliminates the computational burden of parameter optimization. While Rocket's practical effectiveness is well-documented, its theoretical underpinnings have remained largely unexplored. Our research addresses this gap by establishing a theoretical framework that connects Rocket's random convolution operations to compressed sensing principles, demonstrating how random projections maintain the distinctive patterns within time-series data. This theoretical analysis illuminates the connections between kernel configuration and signal properties, offering a more systematic approach to algorithm tuning. Furthermore, we demonstrate that its non-linear transformation component, which calculates the ratio of positive values post-convolution, effectively captures the inherent sparsity patterns in time-series data. Our mathematical investigation additionally proves that Rocket exhibits two essential properties for time-series classification: invariance to temporal shifts and resilience against noise. These insights not only enhance the algorithm's transparency - particularly valuable in regulated industries - but also provide practical guidelines for parameter selection in challenging scenarios, thus contributing to both the theoretical foundations and practical applications of time-series classification methods.