Interference Alignment as a Rank Constrained Rank Minimization (1010.0476v3)

Abstract: We show that the maximization of the sum degrees-of-freedom for the static flat-fading multiple-input multiple-output (MIMO) interference channel is equivalent to a rank constrained rank minimization problem (RCRM), when the signal spaces span all available dimensions. The rank minimization corresponds to maximizing interference alignment (IA) so that interference spans the lowest dimensional subspace possible. The rank constraints account for the useful signal spaces spanning all available spatial dimensions. That way, we reformulate all IA requirements to requirements involving ranks. Then, we present a convex relaxation of the RCRM problem inspired by recent results in compressed sensing and low-rank matrix completion theory that rely on approximating rank with the nuclear norm. We show that the convex envelope of the sum of ranks of the interference matrices is the normalized sum of their corresponding nuclear norms and introduce tractable constraints that are asymptotically equivalent to the rank constraints for the initial problem. We also show that our heuristic relaxation can be tuned for the multi-cell interference channel. Furthermore, we experimentally show that in many cases the proposed algorithm attains perfect interference alignment and in some cases outperforms previous approaches for finding precoding and zero-forcing matrices for interference alignment.

• The paper reformulates the interference alignment problem as a rank constrained rank minimization problem, providing a new framework for maximizing degrees-of-freedom in wireless networks.
• It proposes using nuclear norm minimization as a convex relaxation heuristic to make the complex rank minimization problem computationally tractable.
• Numerical results demonstrate that the proposed algorithm achieves perfect interference alignment in many scenarios and outperforms existing methods, promising enhanced wireless network capacity.

Interference Alignment as a Rank Constrained Rank Minimization

The paper "Interference Alignment as a Rank Constrained Rank Minimization" by Dimitris S. Papailiopoulos and Alexandros G. Dimakis presents a novel perspective on tackling interference in wireless communication systems, specifically focusing on the multiple-input multiple-output (MIMO) interference channels. This work reinterprets the interference alignment (IA) problem as a rank constrained rank minimization (RCRM) problem, providing a new framework for maximizing the sum degrees-of-freedom (DoF) in wireless networks.

The authors demonstrate that by reformulating IA requirements as rank minimization tasks, one can address the fundamental challenge of interference efficiently. The rank minimization problem serves to identify precoding and zero-forcing strategies that ensure interference spans the lowest possible dimensional subspace while useful signal spaces span all available spatial dimensions. This novel approach leverages the concepts from compressed sensing and low-rank matrix completion theory, particularly focusing on nuclear norm minimization as a convex relaxation of the problem.

Key Contributions

The paper makes several significant contributions to the interference alignment domain:

1. Reformulation of IA: The paper introduces a reformulation of the IA problem, framing it as a rank constrained rank minimization (RCRM) problem, thereby extending the feasible scenarios for interference mitigation beyond typical settings.
2. Convex Relaxation: By employing nuclear norm minimization—a convex approximation of rank minimization—the authors propose a heuristic relaxation that improves computational tractability of the IA problem.
3. Algorithm Performance: Through experimental validation, it is shown that the proposed algorithm achieves perfect interference alignment in many scenarios, outperforming existing methods in certain cases.
4. Extension to Multi-cell Networks: The paper extends the approximation algorithm to multi-cell interference channels, providing tractable solutions for more complex cellular network configurations.

Theoretical and Practical Implications

The theoretical implication of this paper lies in the RCRM approach to interference alignment, which shifts from solving NP-hard bilinear equations to a more approachable rank minimization paradigm. This transformation not only expands the class of feasible solutions but also opens the possibility for more efficient heuristic methods like those based on nuclear norm minimization. Practically, this framework holds promise in enhancing wireless network capacity, allowing more users to operate interference-free, thus maximizing spectral efficiency in dense network environments.

Numerical Results

The paper presents numerical results showcasing the efficacy of the proposed algorithm. Across various MIMO system settings, including single antenna symbol extended setups and multi-user configurations, the algorithm exhibits substantial improvements in sum-rate and per-user DoF compared to existing techniques like interference leakage minimization and max-SINR strategies.

Future Directions

The paper suggests several avenues for future research:

• Exploring tighter relaxations or alternative heuristic approaches to further enhance DoF maximization in non-standard interference channels.
• Investigating theoretical bounds that characterize the performance of rank minimization strategies under different system constraints.
• Extending this approach to scenarios with more complex channel models, including frequency-selective or time-varying channels.

In summary, the work by Papailiopoulos and Dimakis provides a robust foundation for reconsidering the design of interference mitigation techniques in MIMO networks, with implications for both algorithmic advancement and practical deployment in next-generation wireless systems.