# Stable Recovery Of Sparse Vectors From Random Sinusoidal Feature Maps

## TL;DR

This paper introduces a stable method for reconstructing sparse vectors from random sinusoidal features, extending their use from inference to accurate data recovery with theoretical guarantees and practical algorithms.

## Contribution

It presents a novel stable reconstruction algorithm for sparse vectors from sinusoidal features, with analysis of sample complexity and extensions to structured inverse problems.

## Key findings

- Reconstruction is possible with a mild increase in embedding dimension.
- The proposed algorithm is numerically stable and effective.
- Numerical experiments validate the approach.

## Abstract

Random sinusoidal features are a popular approach for speeding up kernel-based inference in large datasets. Prior to the inference stage, the approach suggests performing dimensionality reduction by first multiplying each data vector by a random Gaussian matrix, and then computing an element-wise sinusoid. Theoretical analysis shows that collecting a sufficient number of such features can be reliably used for subsequent inference in kernel classification and regression. In this work, we demonstrate that with a mild increase in the dimension of the embedding, it is also possible to reconstruct the data vector from such random sinusoidal features, provided that the underlying data is sparse enough. In particular, we propose a numerically stable algorithm for reconstructing the data 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 the efficacy of our approach via numerical experiments.