# Memory Approximate Message Passing

arXiv: https://arxiv.org/abs/2106.02237 · HTML: https://mlsanity.com/paper/2106.02237/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 54 references that are in the map.
- First posted on arXiv 4 Jun 2021.

## Builds on (most cited first, 7 of 7)

- [Expectation Propagation for approximate Bayesian inference](https://mlsanity.com/paper/1301.2294/) (2013)
- [Approximate Message Passing algorithms for rotationally invariant matrices](https://mlsanity.com/paper/2008.11892/) (2020)
- [Multi-Layer Generalized Linear Estimation](https://mlsanity.com/paper/1701.06981/) (2017)
- [Asymptotics of MAP Inference in Deep Networks](https://mlsanity.com/paper/1903.01293/) (2019)
- [Expectation Propagation for Approximate Inference: Free Probability Framework](https://mlsanity.com/paper/1801.05411/) (2018)
- [Sparse Estimation with the Swept Approximated Message-Passing Algorithm](https://mlsanity.com/paper/1406.4311/) (2014)
- [Understanding the dynamics of message passing algorithms: a free probability heuristics](https://mlsanity.com/paper/2002.02533/) (2020)

## Built on by (most cited first, 4 of 4)

- [Estimation in Rotationally Invariant Generalized Linear Models via Approximate Message Passing](https://mlsanity.com/paper/2112.04330/) (2021)
- [Sufficient-Statistic Memory AMP](https://mlsanity.com/paper/2112.15327/) (2021)
- [Sufficient Statistic Memory Approximate Message Passing](https://mlsanity.com/paper/2206.11674/) (2022)
- [Capacity Optimality of OAMP in Coded Large Unitarily Invariant Systems](https://mlsanity.com/paper/2206.11680/) (2022)

## Abstract (arXiv)

Approximate message passing (AMP) is a low-cost iterative parameter-estimation technique for certain high-dimensional linear systems with non-Gaussian distributions. However, AMP only applies to independent identically distributed (IID) transform matrices, but may become unreliable for other matrix ensembles, especially for ill-conditioned ones. To handle this difficulty, orthogonal/vector AMP (OAMP/VAMP) was proposed for general right-unitarily-invariant matrices. However, the Bayes-optimal OAMP/VAMP requires high-complexity linear minimum mean square error estimator. To solve the disadvantages of AMP and OAMP/VAMP, this paper proposes a memory AMP (MAMP), in which a long-memory matched filter is proposed for interference suppression. The complexity of MAMP is comparable to AMP. The asymptotic Gaussianity of estimation errors in MAMP is guaranteed by the orthogonality principle. A state evolution is derived to asymptotically characterize the performance of MAMP. Based on the state evolution, the relaxation parameters and damping vector in MAMP are optimized. For all right-unitarily-invariant matrices, the optimized MAMP converges to OAMP/VAMP, and thus is Bayes-optimal if it has a unique fixed point. Finally, simulations are provided to verify the validity and accuracy of the theoretical results.