This paper introduces a method for estimating the frequencies of a mixture of complex sinusoids from a random subset of regularly spaced samples, without assuming that the frequencies lie on a grid. Unlike traditional compressed sensing methods, which rely on discrete dictionaries, this approach uses a continuous dictionary of complex exponentials. The method is based on atomic norm minimization, which is reformulated as a semidefinite program. The key idea is that the atomic norm can recover the true frequencies and amplitudes of the sinusoids with high probability, provided the frequencies are sufficiently separated. The paper shows that only $ O(s \log s \log n) $ random samples are needed to guarantee exact recovery, where $ s $ is the number of sinusoids and $ n $ is the total number of samples. The method is validated through extensive numerical experiments, demonstrating its effectiveness in recovering frequencies from sparse, random samples. The paper also discusses the theoretical foundations of the atomic norm, its relationship to other optimization techniques, and its application to line spectral estimation. The results show that the proposed method outperforms traditional discretization-based approaches, offering a more flexible and accurate way to estimate frequencies in continuous domains.This paper introduces a method for estimating the frequencies of a mixture of complex sinusoids from a random subset of regularly spaced samples, without assuming that the frequencies lie on a grid. Unlike traditional compressed sensing methods, which rely on discrete dictionaries, this approach uses a continuous dictionary of complex exponentials. The method is based on atomic norm minimization, which is reformulated as a semidefinite program. The key idea is that the atomic norm can recover the true frequencies and amplitudes of the sinusoids with high probability, provided the frequencies are sufficiently separated. The paper shows that only $ O(s \log s \log n) $ random samples are needed to guarantee exact recovery, where $ s $ is the number of sinusoids and $ n $ is the total number of samples. The method is validated through extensive numerical experiments, demonstrating its effectiveness in recovering frequencies from sparse, random samples. The paper also discusses the theoretical foundations of the atomic norm, its relationship to other optimization techniques, and its application to line spectral estimation. The results show that the proposed method outperforms traditional discretization-based approaches, offering a more flexible and accurate way to estimate frequencies in continuous domains.