Robust Recovery of Signals From a Structured Union of Subspaces (0807.4581v2)

Abstract: Traditional sampling theories consider the problem of reconstructing an unknown signal $x$ from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that $x$ lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which $x$ lies in a union of subspaces. In this paper we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which $x$ lies in a sum of $k$ subspaces, chosen from a larger set of $m$ possibilities. The samples are modelled as inner products with an arbitrary set of sampling functions. To derive an efficient and robust recovery algorithm, we show that our problem can be formulated as that of recovering a block-sparse vector whose non-zero elements appear in fixed blocks. We then propose a mixed $\ell_2/\ell_1$ program for block sparse recovery. Our main result is an equivalence condition under which the proposed convex algorithm is guaranteed to recover the original signal. This result relies on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. Based on RIP we also prove stability of our approach in the presence of noise and modelling errors. A special case of our framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Adapting our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently.

• The paper introduces a novel block-sparse recovery approach that transforms signal recovery from a union of subspaces into a tractable convex optimization problem using a mixed ℓ2/ℓ1 norm.
• It establishes a block-RIP condition as an equivalence criterion, which guarantees robust recovery even in noisy environments and under modeling errors.
• Numerical experiments demonstrate that the proposed method outperforms standard ℓ1 minimization, offering enhanced recovery rates and stability for multiple measurement vectors.

Robust Recovery of Signals From a Structured Union of Subspaces

The paper by Yonina C. Eldar and Moshe Mishali addresses the problem of recovering signals that lie in a structured union of subspaces. The motivation for this work stems from the limitations of traditional sampling theories, which assume that the unknown signal resides in a single, known subspace. Their work proposes a framework that generalizes this model to settings where the signal lies in a union of several subspaces, providing robust and efficient recovery methods under these conditions.

Problem Formulation

Eldar and Mishali consider the scenario where an unknown signal $x$ can be decomposed into a sum of vectors from $k$ subspaces chosen out of $m$ possible subspaces. More formally, they aim to recover $x$ from a set of samples modeled as inner products with sampling functions. The challenge arises because $x$ could belong to any combination of these subspaces, making the problem a combinatorial one.

Block-Sparse Recovery

The authors propose casting the problem of signal recovery in this union of subspaces framework as a block-sparse recovery problem. In this context, the signal $x$ is transformed into a block-sparse vector $\theta$, where the non-zero elements are confined to certain blocks. They derive an efficient recovery algorithm using a mixed $\ell_2/\ell_1$ norm, designed to exploit the block-sparsity inherent in the signal model.

Theoretical Contributions

The central theoretical contribution is an equivalence condition under which the proposed convex optimization algorithm recovers the original signal. This condition employs the block-restricted isometry property (RIP), a generalization of the standard RIP from compressed sensing. The block-RIP ensures that if the measurement matrix satisfies this property, then the algorithm can recover the signal robustly even in the presence of noise and modeling errors.

Practical Implications

1. Multiple Measurement Vectors (MMV): One of the notable applications of their framework is in the recovery of MMV problems. Here, multiple measurement vectors share a joint sparsity pattern, and the authors adapt their block-sparse recovery results to this context, offering new recovery methods and equivalence conditions.
2. Convex Optimization: The proposed method leverages convex optimization techniques such as second-order cone programming (SOCP) to facilitate signal recovery. This makes the approach computationally feasible and robust across various scenarios.
3. Random Measurement Matrices: The paper also establishes that random measurement matrices, particularly from Gaussian or Bernoulli ensembles, are likely to satisfy the block-RIP with high probability. This result provides a practical guideline for designing measurement matrices in real-world applications.

Numerical Results and Insights

The authors provide substantial numerical evidence demonstrating the superior performance of their method compared to standard $\ell_1$ minimization techniques, especially in cases where the non-zero elements are structured in blocks. Their experiments illustrate that the proposed mixed $\ell_2/\ell_1$ approach achieves higher recovery rates and is more stable under noisy conditions.

Future Directions

The paper hints at several avenues for future research, including:

• Extending the framework to infinite union of subspaces and infinite-dimensional settings.
• Exploring the application of the theoretical results and recovery algorithms in more complex signal models and real-world applications.
• Further refining the block-RIP conditions to encompass even broader classes of measurement matrices.

Conclusion

Eldar and Mishali's work significantly advances the understanding and methodologies for signal recovery in structured union of subspaces. Their contributions in formulating the block-sparse recovery problem, establishing robust recovery conditions, and providing practical algorithms have set a new benchmark in the field. These results have profound implications for compressed sensing, signal processing, and broader areas where structured signal models are prevalent.