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Random sinusoidal features are a popular approach for speeding upkernel-based inference in large datasets. Prior to the inference stage, theapproach suggests performing dimensionality reduction by first multiplying eachdata vector by a random Gaussian matrix, and then computing an element-wisesinusoid. Theoretical analysis shows that collecting a sufficient number ofsuch features can be reliably used for subsequent inference in kernelclassification and regression.  In this work, we demonstrate that with a mild increase in the dimension ofthe embedding, it is also possible to reconstruct the data vector from suchrandom sinusoidal features, provided that the underlying data is sparse enough.In particular, we propose a numerically stable algorithm for reconstructing thedata vector given the nonlinear features, and analyze its sample complexity.Our algorithm can be extended to other types of structured inverse problems,such as demixing a pair of sparse (but incoherent) vectors. We support theefficacy of our approach via numerical experiments.

Stable Recovery Of Sparse Vectors From Random Sinusoidal Feature Maps