Iterative Hard Thresholding for Compressed Sensing

Abstract: Compressed sensing is a technique to sample compressible signals below the Nyquist rate, whilst still allowing near optimal reconstruction of the signal. In this paper we present a theoretical analysis of the iterative hard thresholding algorithm when applied to the compressed sensing recovery problem. We show that the algorithm has the following properties (made more precise in the main text of the paper) - It gives near-optimal error guarantees. - It is robust to observation noise. - It succeeds with a minimum number of observations. - It can be used with any sampling operator for which the operator and its adjoint can be computed. - The memory requirement is linear in the problem size. - Its computational complexity per iteration is of the same order as the application of the measurement operator or its adjoint. - It requires a fixed number of iterations depending only on the logarithm of a form of signal to noise ratio of the signal. - Its performance guarantees are uniform in that they only depend on properties of the sampling operator and signal sparsity.

• The paper introduces IHT by proving near-optimal error bounds in compressed sensing recovery through rigorous theoretical analysis.
• It demonstrates that IHT achieves efficient recovery using minimal observations and low per-iteration computational complexity.
• The study highlights IHT's robustness against noise, ensuring reliable performance in practical signal reconstruction scenarios.

Iterative Hard Thresholding for Compressed Sensing: An Analytical Overview

Thomas Blumensath and Mike E. Davies present an in-depth theoretical analysis of the Iterative Hard Thresholding (IHT) algorithm applied to the compressed sensing recovery problem. This paper elucidates the robust performance guarantees of IHT, positioning it as a viable alternative within the compressed sensing computational landscape.

Compressed Sensing Framework

Compressed sensing leverages the inherent sparsity of signals to sample below the Nyquist rate, enabling efficient signal reconstruction. Traditional signal acquisition systems, governed by the Nyquist-Shannon theorem, necessitate sampling at twice the signal bandwidth. However, compressed sensing exploits sparsity in transform domains to reduce the requisite number of observations substantially. This paper builds upon seminal contributions from Candes, Romberg, Tao, and Donoho, which demonstrated that linear programming algorithms could reconstruct the original signal with high accuracy under certain conditions.

IHT Algorithm and Properties

The primary contribution of this paper lies in the theoretical validation of the IHT algorithm for the compressed sensing recovery problem. The authors systematically establish that IHT possesses the following characteristics:

• Error Guarantees: The algorithm ensures near-optimal error bounds, achieving an estimation error guaranteed to be within a constant factor of the best attainable reconstruction.
• Noise Robustness: IHT maintains performance in the presence of observation noise.
• Observation Efficiency: The algorithm succeeds with a minimal number of observations, adhering to the restricted isometry property (RIP) requirements.
• Computational Efficiency: Per iteration, the computational complexity of IHT is equivalent to the application of the measurement operator or its adjoint.
• Uniform Performance: Performance guarantees are independent of the size and distribution of the signal's largest non-zero coefficients, relying solely on the sampling operator properties and signal sparsity.
• Logarithmic Iteration Dependency: The number of iterations required for desired accuracy depends logarithmically on a form of signal-to-noise ratio.

Theoretical Results

The authors present rigorous proofs to support their claims, drawing upon lemma and corollary-based arguments. The key results are summarized as follows:

• For a noisy observation $x = \Phi y + e$, where $y$ is an arbitrary vector, IHT will recover an approximation within $6\epsilon_s$ after a fixed number of iterations $k$, where $k = \log_2 (\|y^s\|_2 / \epsilon_s)$.
• The main theorem states that if $\beta_{3s} < 1/8$, IHT reduces the error in each iteration, achieving a final error within a factor of $$\epsilon_s = \|y - y^s\|_2 + \frac{1}{\sqrt{s}} \|y - y^s\|_1 + \|e\|_2$$.

Comparison to CoSaMP

A comparative analysis between IHT and CoSaMP reveals:

• Isometry Constant: IHT requires $\beta_{3s} \leq 0.125$, while CoSaMP requires $\delta_{4s} \leq 0.1$. This translates to $\delta_{2s} \leq 0.0222$ for IHT and $\delta_{2s} \leq 0.025$ for CoSaMP post-relaxation, showcasing a marginal advantage for CoSaMP.
• Error Bounds: IHT provides an optimal error guarantee up to a constant, achieving an error bound of $6\epsilon_s$ compared to CoSaMP's $20\epsilon_s$.

Practical and Theoretical Implications

The paper presents IHT as a computationally efficient algorithm with solid theoretical underpinnings, offering significant potential for practical applications in compressed sensing. Given the linear convergence and minimal storage requirements, IHT is particularly suitable for large-scale problems where rapid approximation is necessary.

Future research may explore enhancements in average-case performance, potentially integrating adaptive thresholding strategies or hybrid approaches to further optimize IHT in practical scenarios. Additionally, expanding the theoretical framework to encompass broader classes of signals and noise models would enhance the robustness and applicability of IHT in diverse domains.

Conclusion

Blumensath and Davies' comprehensive analysis affirms the viability of Iterative Hard Thresholding as a compelling method within the compressed sensing paradigm. By establishing concrete performance bounds and computational advantages, this paper contributes significantly to the ongoing development and refinement of sparse recovery algorithms. Researchers and practitioners alike may find IHT to be a valuable tool, warranting further investigation and integration into compressed sensing applications.