Annealed Langevin Dynamics for Massive MIMO Detection

Abstract

We propose a solution for linear inverse problems based on higher-order Langevin diffusion. More precisely, we propose pre-conditioned second-order and third-order Langevin dynamics that provably sample from the posterior distribution of our unknown variables of interest while being computationally more efficient than their first-order counterpart and the nonconditioned versions of both dynamics. Moreover, we prove that both pre-conditioned dynamics are well-defined and have the same unique invariant distributions as the non-conditioned cases. We also incorporate an annealing procedure that has the double benefit of further accelerating the convergence of the algorithm and allowing us to accommodate the case where the unknown variables are discrete. Numerical experiments in two different tasks (MIMO symbol detection and channel estimation) showcase the generality of our method and illustrate the high performance achieved relative to competing approaches (including learningbased ones) while having comparable or lower computational complexity.

Detection by Sampling: Massive MIMO Detector based on Langevin Dynamics

“…After that, it follows the score function of the joint posterior density of perturbed variables, starting with high σ 1,X and σ 1,H , progressively reducing to σ L,X ≈ σ L,H ≈ 0. At early noise levels, the likelihood term directs the dynamics toward an estimate mainly driven by the measurements, while in later noise levels, the prior refines the estimate, as explained further in [13]. Annealing benefits are threefold: it is used to train the score network via score-matching, it enhances dynamic mixing, and it allows for discrete-to-continuous variable approximation.…”

“…Intuitively, the approximation in (10) assumes that the annealing and measurement noise are independent. Although this approximation entails a worse symbol error rate (SER) than the one introduced in [13], it provides a good approximation that allows one to reduce the computational burden of the algorithm.…”

“…Hence, the complexity of computing the guidance term and the score prior is O(N 3 u + KNu). A deeper analysis of the complexity of this method can be found in [13]. For the case of H, the heavier computation is carried by the score network.…”

“…Leveraging their ability to generate high-dimensional data from unknown probability distributions, diffusion models have achieved state-of-the-art results in various areas, such as denoising and inpainting in image processing [11,12]. In wireless technology, they have been used to derive a massive MIMO detector based on Langevin diffusion [13,14], and for channel estimation [15], both of which have shown remarkable results. However, these methods are tailored for problems with known forward operators and may not apply to blind inverse problems, where the operator is unknown.…”

Unfolding WMMSE Using Graph Neural Networks for Efficient Power Allocation

“…Algorithm unfolding decouples the update steps of an iterative algorithm to create a cascade of hybrid layers that preserve the original update structure but introduces one or more learnable parameters from data. This form of domain-inspired learning has been extremely popular and effective in several application areas, including but not limited to non-negative matrix factorization [29], iterative soft thresholding [30], semantic image segmentation [31], blind deblurring [32], clutter suppression [33], particle filtering [34], symbol detection [35], link scheduling [36], energy-aware power allocation [37], and beamforming in wireless networks [21]- [23]. These algorithms use various neural layers to learn one or more parameters of the iterative algorithm being unfolded or to approximate certain computational steps in the algorithm to reduce complexity and speed-up processing.…”