Compressed Sensing off the Grid (1207.6053v3)

Abstract: We consider the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized frequency domain [0,1]. We propose an atomic norm minimization approach to exactly recover the unobserved samples. We reformulate this atomic norm minimization as an exact semidefinite program. Even with this continuous dictionary, we show that most sampling sets of size O(s log s log n) are sufficient to guarantee the exact frequency estimation with high probability, provided the frequencies are well separated. Numerical experiments are performed to illustrate the effectiveness of the proposed method.

• The paper's main contribution is the use of an atomic norm minimization framework that directly addresses grid mismatch by operating on continuous frequencies.
• It reformulates frequency recovery as a semidefinite programming problem, enabling polynomial-time exact recovery with high probability.
• Numerical experiments demonstrate superior accuracy and efficiency over traditional grid-based methods, validating its practical utility in signal processing.

Compressed Sensing Off the Grid

"Compressed Sensing Off the Grid" by Gongguo Tang, Badri Narayan Bhaskar, Parikshit Shah, and Benjamin Recht investigates the problem of estimating frequency components of a mixture of complex sinusoids from a subset of regularly spaced samples. Distinctively, this work relaxes the common assumption in compressed sensing that frequencies lie on a fixed grid. Instead, it considers arbitrary frequencies in the normalized frequency domain [0, 1]. To address this challenge, the authors employ an atomic norm minimization framework, which they reformulate as a semidefinite programming (SDP) problem. This essay will provide an overview of the methodology, key results, and implications of this work.

Methodology and Contributions

The approach taken in this paper diverges from traditional compressed sensing methods that often discretize the frequency domain into fixed grids. This discretization can cause issues such as basis mismatch. The authors propose an innovative solution that avoids discretization by utilizing an atomic norm minimization technique directly on the continuous domain. The central hypotheses and innovations of this paper include:

1. Atomic Norm Minimization: The use of atomic norm minimization overcomes the basis mismatch problem inherent in discrete dictionaries. An atomic norm is analogous to the $\ell_1$ norm for sparse recovery but applied to continuous dictionaries.
2. Semidefinite Programming Reformulation: The minimization problem is reformulated as an exact semidefinite program, allowing polynomial-time complexity solutions via SDP solvers.
3. Random Sample Sufficiency: The paper shows that $O(s \log s \log n)$ random samples are sufficient to guarantee exact frequency localization with high probability, provided the frequencies are sufficiently separated.

Key Results and Proof Strategy

The key theorem states that one can recover the original frequencies and coefficients of sinusoids exactly under certain conditions. Here are crucial points of their theoretical findings:

• Threshold for Frequency Separation:
  • The frequencies should be well-separated, with a minimum distance $$\Delta_f \geq \frac{1}{\left\lfloor \frac{(n-1)}{4} \right\rfloor}$$ for exact recovery using a randomly chosen subset of samples.
• Sample Complexity:
  • The number of required samples $m$ to guarantee exact recovery satisfies:

  $$m \geq C \max \left\{ \log^2 \frac{n}{\delta}, s \log \frac{s}{\delta} \log \frac{n}{\delta} \right\},$$

  where $C$ is a numerical constant and $\delta$ controls the probability of successful recovery.

To construct the proof, the authors applied several advanced techniques:

1. Dual Certificate Construction: They construct a dual polynomial with specific properties to verify that the optimal atomic norm solution matches the true signal.
2. Random Kernel Approach: The random kernel is used to generate a polynomial whose coefficients align with the observed entries to achieve the required dual certificate properties.
3. Bernstein and Hoeffding Inequalities: These probabilistic tools are utilized to ensure that the constructed polynomial maintains the desired properties across the entire frequency domain.

Numerical Experiments

The authors conducted comprehensive numerical experiments to validate their theoretical guarantees. Key insights from these experiments include:

1. Comparison with Discrete Methods: When frequencies do not fall on a grid, traditional discrete basis pursuit (BP) methods had significant recovery errors. Higher discretization levels improved BP accuracy but increased computational burden, often exceeding that of the proposed SDP method for practical levels of discretization, such as 64 times denser grids.
2. Performance Profiles: The proposed SDP-based method consistently demonstrated higher accuracy and comparable or better running times than BP, particularly in scenarios with high levels of frequency sparsity.
3. Phase Transition Behavior: The experiments affirmed a clear phase transition from perfect recovery to failure, contingent on sufficient frequency separation and sample count, thus illustrating the practical efficacy of the theoretical bounds derived.

Implications and Future Work

The contributions of this paper have broad implications both in theory and practice:

1. Extended Capabilities in Signal Processing: By eliminating the need for grid-based discretization, this framework enhances the applicability of compressed sensing to real-world problems where frequencies may not align with a predefined grid.
2. Improved Sampling Strategies: The findings advocate the use of random sampling as a viable strategy for compressed signal acquisition, suggesting practical implementations in scenarios like radar, communications, and array processing.

Looking forward, several promising research directions emerge:

• Stability and Robustness Analysis: Exploring the robustness of this method in the presence of noise and incomplete data could provide further validation and utility.
• Algorithmic Refinements: Enhancing solver efficiency for large-scale problems through specialized optimization algorithms could facilitate the practical adoption in high-dimensional applications.

Overall, "Compressed Sensing Off the Grid" introduces a significant step forward in the field of signal processing by addressing fundamental limitations of grid-based frequency estimation and paving the way for more flexible and efficient compressed sensing techniques.