The fractional Fourier transform generalizes the ordinary Fourier transform by a continuous parameter: instead of rotating the signal by exactly 90° from time to frequency, you rotate by any angle. This is useful for signals that aren't stationary — their frequency content changes over time. On graphs, the graph Fourier transform uses the Laplacian's eigenvectors as a frequency basis. The fractional graph Fourier transform combines both: a tunable rotation over an irregular topology.
Previous fractional graph Fourier transforms used a single parameter for both spatial and temporal dimensions. A graph signal — say, temperature at weather stations over time — has spatial structure (which stations are connected) and temporal structure (how readings evolve). Forcing both dimensions to use the same fractional order is like requiring the x and y axes of a photograph to have the same zoom level.
Li et al. (arXiv:2603.01484) introduce the bi-fractional Fourier transform for spatiotemporal graph signals, with independent fractional parameters for space and time. The spatial parameter controls how much the analysis emphasizes low-frequency (smooth) vs. high-frequency (oscillatory) spatial patterns. The temporal parameter does the same for time. Decoupling them lets the analysis adapt to signals where spatial and temporal dynamics operate at different scales.
A complication: with two independent parameters, the temporal transform can become ambiguous — multiple representations produce the same result. The fix is geodesic coupling: constructing a unified basis that interpolates between the spatial and temporal eigenvectors along the shortest path in a matrix manifold. The coupling preserves invertibility while eliminating the redundancy.
The framework learns its own parameters from data. Applied to dynamic networks and image restoration, it outperforms both classical fractional methods and some deep learning baselines. The advantage is interpretability: the learned parameters tell you the spatial and temporal scales at which the signal has structure, rather than hiding this information in network weights.