Distributed Basis Pursuit (1009.1128v3)

Abstract: We propose a distributed algorithm for solving the optimization problem Basis Pursuit (BP). BP finds the least L1-norm solution of the underdetermined linear system Ax = b and is used, for example, in compressed sensing for reconstruction. Our algorithm solves BP on a distributed platform such as a sensor network, and is designed to minimize the communication between nodes. The algorithm only requires the network to be connected, has no notion of a central processing node, and no node has access to the entire matrix A at any time. We consider two scenarios in which either the columns or the rows of A are distributed among the compute nodes. Our algorithm, named D-ADMM, is a decentralized implementation of the alternating direction method of multipliers. We show through numerical simulation that our algorithm requires considerably less communications between the nodes than the state-of-the-art algorithms.

• The paper presents D-ADMM, a novel decentralized adaptation of ADMM that solves the least ℓ1-norm solution in underdetermined systems.
• It details both row and column partitioning strategies to handle scenarios where nodes do not have full access to matrix A.
• Simulations demonstrate that the algorithm reduces communication load while maintaining reliable convergence in resource-constrained environments.

Distributed Basis Pursuit

The paper presents a novel distributed algorithm for solving the Basis Pursuit (BP) problem on a decentralized platform. BP seeks the least $\ell_1$-norm solution to an underdetermined linear system, a task pivotal in compressed sensing applications. The authors introduce D-ADMM, a decentralized adaptation of the Alternating Direction Method of Multipliers (ADMM), designed to function in distributed networks such as sensors where communication between nodes is minimized and no central processor exists.

Key Contributions

The algorithm operates under two main scenarios where matrix $A$'s dimensions are distributed among nodes—either by columns or rows. The authors focus on solving the BP problem in such a manner that not all nodes have a complete view of the matrix $A$ at any point, reflecting the real-world constraints in distributed systems like sensor networks.

1. Algorithm Design:
   • D-ADMM effectively partitions tasks among nodes and uses local computations along with limited inter-node communication to approach solutions.
   • It capitalizes on the network being connected and static but foregoes the need for a central processing node or a fully distributed matrix at any node.
2. Row vs. Column Partitioning:
   • The paper considers partitioning $A$ into rows for situations where each node receives some measurements (analogous to sensing environments), and columns for cases like seismic processing where distributed sources are used.
3. Numerical Efficiency:
   • Simulations indicate that D-ADMM drastically reduces the communication load compared to existing methods while maintaining convergence reliability.
   • The method is particularly efficient in energy-constrained and large-scale computing environments due to its reduced communication requirements.

Technical Insights

• Mathematical Formulation:

D-ADMM modifies the standard ADMM method to work in asynchronous distributed systems by employing a dual update strategy that does not require consistent global state across nodes. This allows local updates to proceed independently, providing a robust framework that tolerates network latencies and variabilities.

• Graph-Theoretical Applications:

The approach relies on graph coloring to partition the nodes, ensuring no neighboring nodes process concurrently, which strategically minimizes the risk of communication collision and hence conserves network bandwidth and node power.

Implications and Future Directions

The implications of the proposed method extend both practically and theoretically. Practically, D-ADMM is positioned as a highly efficient method for distributed computation in dynamic and resource-limited environments like wireless sensor networks or edge computing frameworks. Theoretically, the advancements in decentralized optimization methods challenge traditionally centralized frameworks, suggesting potential for more decentralized decision-making architectures.

Future research can explore adaptive tuning of algorithm parameters like $\rho$, which would enhance convergence rates and extend applicability flexibility in varying network conditions. Additionally, experimentation with different network topologies and node heterogeneity could further validate model robustness and unveil novel insights into distributed optimization dynamics.

By innovating a robust mechanism to handle BP in distributed settings, the paper positions D-ADMM as a method that not only addresses present challenges in distributed optimization but also sets a precedent for solving more complex large-scale optimization problems beyond the constraints of central control structures.