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In a data-rich digital world, finding sparse, efficient representations - especially for multi-exponential models - has become critical, particularly when measurements are costly or noisy. These models, which involve complex or real exponents, underpin key processes in signal processing, relaxation dynamics, chemical reactions, heat transfer, and fluid dynamics, with widespread real-world impact. The challenge lies at the intersection of several computational disciplines: structured matrices, rational approximation, sparse interpolation, quadrature, tensor decompositions, and subdivision methods - each offering potential pathways to more robust and efficient algorithms. Multi-exponential analysis is foundational across engineering and industry, enabling advances in DOA estimation, remote sensing, MRI, superresolution, seismology, radio astronomy, and telecommunications - areas vital to energy, health, transportation, and space research. This Dagstuhl Seminar "From Sparse Interpolation to Signal Processing: New Synergies" (25281) brought together experts from computational harmonic analysis, numerical linear algebra, computer algebra, signal processing, approximation theory, and engineering applications to foster cross-disciplinary collaboration and accelerate innovation in this dynamic field.