CD-MAMP: Cross-Domain Memory Approx. Message Passing

• CD-MAMP is an iterative Bayesian inference architecture that separates linear and nonlinear processing into distinct domains while incorporating long-memory corrections.
• It leverages an orthogonality principle and state evolution to achieve Bayes-optimal MSE with low per-iteration complexity akin to AMP.
• Domain-specific implementations such as IBS-CD-MAMP, IFDM CD-MAMP, and WHTDM demonstrate significant performance gains and complexity reductions in compressed sensing and MIMO systems.

Searching arXiv for the cited papers and related CD-MAMP work. arxiv_search(query="Cross-Domain Memory Approximate Message Passing OR CD-MAMP", max_results=10) arxiv_search(query="Memory Approximate Message Passing (Liu et al., 2021)", max_results=5) arxiv_search(query="Interleaved Block-Sparse Transform (Liu et al., 2024)", max_results=5) arxiv_search(query="Interleave Frequency Division Multiplexing (Chi et al., 2024)", max_results=5) arxiv_search(query="WHTDM Walsh-Hadamard Transform Division Multiplexing (Hao et al., 14 May 2026)", max_results=5) arxiv_search(query="Achievable Rate and Coding Principle for MIMO Multicarrier Systems With Cross-Domain MAMP Receiver Over Doubly Selective Channels (Chi et al., 7 Jan 2026)", max_results=5) Cross-Domain Memory Approximate Message Passing (CD-MAMP) denotes a family of iterative Bayesian inference architectures that extend Memory Approximate Message Passing (MAMP) by separating the linear and nonlinear stages across different domains while retaining long-memory correction, orthogonality, and state-evolution-based design. In the underlying MAMP framework, the estimation problem is posed on the high-dimensional linear model $y = A x_0 + w$, where $x_0$ has an IID prior, $w$ is Gaussian noise, and $A$ is typically right-unitarily invariant (RUI) rather than IID. MAMP was introduced to bridge the gap between low-complexity AMP, which can diverge on non-IID or ill-conditioned matrices, and Bayes-optimal OAMP/VAMP, whose LMMSE linear estimator is computationally expensive. Later CD-MAMP constructions place the memory linear estimator in one domain—such as a time, transform, or WHT domain—and the nonlinear MMSE estimator in another—such as a source or symbol domain—thereby adapting MAMP’s orthogonality principle to compressed sensing, multicarrier equalization, and coded MIMO receivers (Liu et al., 2020).

1. Conceptual origin in MAMP

MAMP originates from the observation that canonical AMP is low-complexity but depends on IID sub-Gaussian sensing matrices, whereas OAMP/VAMP extends to RUI matrices at the cost of a high-complexity LMMSE step. In the MAMP formulation of Liu, Huang, and Kurkoski, the linear estimator and nonlinear estimator are both allowed to use memory of past iterates. The memory linear estimator and nonlinear estimator are written as

$$\text{LE:}\quad r_t = Q_t y + \sum_{i=1}^t P_{t,i} x_i, \qquad \text{NLE:}\quad x_{t+1} = \Phi_t(r_1,\ldots,r_t),$$

with $Q_t$ and $P_{t,i}$ polynomial in $A^H A$ and $\Phi_t$ separable-and-Lipschitz (Liu et al., 2021).

The central design idea is that long memory can replace the explicit matrix inversion in BO-OAMP/VAMP. In Bayes-optimal MAMP (BO-MAMP), the LMMSE inverse is approximated by a long-memory matched filter driven by

$$B = I_M - A A^H,\qquad f_t = \theta_t B f_{t-1} + \gamma_t (y - A x_t),$$

with the linear output formed from polynomial combinations of $x_0$0. This preserves a per-iteration cost of $x_0$1, comparable to AMP, rather than the $x_0$2 cost associated with BO-OAMP/VAMP (Liu et al., 2020).

The theoretical result that gives MAMP its importance is fixed-point equivalence: for all RUI matrices, the state evolution of optimized BO-MAMP converges to the same fixed point as high-complexity BO-OAMP/VAMP, and when that state evolution has a unique fixed point, the algorithm achieves the Bayes-optimal MSE predicted by the replica method (Liu et al., 2020). A plausible implication is that later CD-MAMP constructions are best understood as domain-specific realizations of this MAMP principle rather than as a separate inference theory.

2. Cross-domain architecture

In later literature, “cross-domain” refers to performing the memory linear estimator in one representation of the signal while performing the nonlinear MMSE estimation in another. The cross-domain split is explicit in several systems: IBS-CD-MAMP uses a memory linear estimator in the IBS transform domain and a nonlinear estimator in the source domain; IFDM CD-MAMP uses a memory matched filter in the time domain and a nonlinear detector in the interleave-frequency domain; WHTDM uses a WHT-domain linear model and symbol-domain denoising; and MS-CD-MAMP couples time-domain sparse detection with symbol-domain constellation and coding constraints (Liu et al., 2024).

Construction	Linear stage domain	Nonlinear stage domain
IBS-CD-MAMP	IBS transform domain	Source domain
IFDM CD-MAMP	Time domain	Interleave-frequency domain
WHTDM CD-MAMP	WHT domain	Symbol domain
MS-CD-MAMP	Time domain across slots	Symbol domain with coding

A representative cross-domain loop appears in IFDM. The transmitted signal is

$x_0$3

the time-domain received model is

$x_0$4

and the inverse IF transform gives

$x_0$5

The detector then alternates between a time-domain memory matched filter

$x_0$6

and an interleave-frequency-domain nonlinear detector

$x_0$7

with the two domains connected by $x_0$8 and $x_0$9 (Chi et al., 2024).

In IBS-CD-MAMP, the same architectural pattern is expressed through an IBS transform. The measurement model is

$w$0

with transform-domain variable $w$1. The memory linear estimator operates on $w$2, IBS-IFT maps residuals back to the source domain,

$w$3

and IBS-FT returns the source-domain estimate to the transform domain for the next memory update (Liu et al., 2024).

3. Orthogonality principle and state evolution

The defining theoretical mechanism of MAMP and CD-MAMP is full orthogonality across memory. If $w$4 and $w$5, then the strengthened orthogonality conditions are, for all $w$6,

$w$7

This ensures that the current output error is orthogonal to all preceding input errors, not merely to the current one, and it is this stronger condition that enables asymptotically IID Gaussian errors under long-memory recursion (Liu et al., 2020).

To enforce these constraints, MAMP introduces explicit orthogonalization procedures for both modules. In the linear estimator, trace constraints such as

$w$8

are imposed by subtracting trace terms from an unconstrained memory estimator. In the nonlinear estimator, an arbitrary separable Lipschitz map $w$9 is orthogonalized by subtracting a projection onto prior error directions, with Stein’s lemma then yielding the required decorrelation (Liu et al., 2020).

Because memory induces correlated inputs across iterations, BO-MAMP is characterized by covariance-matrix state evolution rather than a purely scalar recursion. With

$A$0

and covariance matrix $A$1, the nonlinear-stage update with $A$2-length damping takes the form

$A$3

when $A$4 is invertible. This damping rule minimizes the next-iterate MSE and guarantees monotonic MSE decrease in the BO-MAMP state evolution (Liu et al., 2020).

The practical parameterization is likewise SE-driven. The relaxation parameter is chosen as

$A$5

to minimize $A$6 and ensure $A$7, while $A$8 is chosen in closed form to minimize $A$9. In applications where only eigenvalue bounds are available, the literature uses surrogate bounds without significant performance loss (Liu et al., 2020).

4. Domain-specific constructions

The compressed-sensing version of CD-MAMP is represented by IBS-CD-MAMP. Its purpose is not only algorithmic but architectural: a single large RUI transform is replaced by an interleaved block-sparse transform

$$\text{LE:}\quad r_t = Q_t y + \sum_{i=1}^t P_{t,i} x_i, \qquad \text{NLE:}\quad x_{t+1} = \Phi_t(r_1,\ldots,r_t),$$0

where block-wise interleaving $$\text{LE:}\quad r_t = Q_t y + \sum_{i=1}^t P_{t,i} x_i, \qquad \text{NLE:}\quad x_{t+1} = \Phi_t(r_1,\ldots,r_t),$$1, row selection $$\text{LE:}\quad r_t = Q_t y + \sum_{i=1}^t P_{t,i} x_i, \qquad \text{NLE:}\quad x_{t+1} = \Phi_t(r_1,\ldots,r_t),$$2, and whole-matrix interleaving $$\text{LE:}\quad r_t = Q_t y + \sum_{i=1}^t P_{t,i} x_i, \qquad \text{NLE:}\quad x_{t+1} = \Phi_t(r_1,\ldots,r_t),$$3 collectively approximate the randomness of a large transform at lower hardware scale. The reported rationale is that both block-wise and whole interleaving are needed; weaker variants such as BS-FT, W-IBS-FT, and B-IBS-FT show worse MSE or BER (Liu et al., 2024).

The IFDM construction emphasizes a different structural point. Rather than seeking sparsity in the effective symbol-domain channel, IFDM deliberately creates a fully dense and right-unitarily invariant equivalent matrix

$$\text{LE:}\quad r_t = Q_t y + \sum_{i=1}^t P_{t,i} x_i, \qquad \text{NLE:}\quad x_{t+1} = \Phi_t(r_1,\ldots,r_t),$$4

while exploiting the super-sparse time-domain channel $$\text{LE:}\quad r_t = Q_t y + \sum_{i=1}^t P_{t,i} x_i, \qquad \text{NLE:}\quad x_{t+1} = \Phi_t(r_1,\ldots,r_t),$$5 inside the memory matched filter. The paper states that the IF transform enables the equivalent channel matrices to satisfy the right-unitarily invariant assumption commonly used for OAMP and MAMP, thereby supporting replica MAP-optimal detection with proper coding (Chi et al., 2024).

WHTDM adapts the same cross-domain logic to a real-valued Walsh–Hadamard modulation. After cyclic-prefix removal and WHT demodulation, the observation is

$$\text{LE:}\quad r_t = Q_t y + \sum_{i=1}^t P_{t,i} x_i, \qquad \text{NLE:}\quad x_{t+1} = \Phi_t(r_1,\ldots,r_t),$$6

Because $$\text{LE:}\quad r_t = Q_t y + \sum_{i=1}^t P_{t,i} x_i, \qquad \text{NLE:}\quad x_{t+1} = \Phi_t(r_1,\ldots,r_t),$$7 is generally non-diagonal, the equalizer uses a banded approximation $$\text{LE:}\quad r_t = Q_t y + \sum_{i=1}^t P_{t,i} x_i, \qquad \text{NLE:}\quad x_{t+1} = \Phi_t(r_1,\ldots,r_t),$$8 and alternates a banded linear step with scalar denoising. The memory mechanism is explicit: $$\text{LE:}\quad r_t = Q_t y + \sum_{i=1}^t P_{t,i} x_i, \qquad \text{NLE:}\quad x_{t+1} = \Phi_t(r_1,\ldots,r_t),$$9 which the paper describes as accelerating convergence by reusing residual information from the WHT domain (Hao et al., 14 May 2026).

The most elaborate extension is the multi-slot CD-MAMP receiver for coded MIMO multicarrier systems. There, the time-domain linear stage is maintained per slot $Q_t$0,

$Q_t$1

the symbol-domain stage computes APP-based denoising and decoder feedback, and the analysis is reduced from high-dimensional matrix SE to a simplified SISO variational state evolution in terms of $Q_t$2 and $Q_t$3 (Chi et al., 7 Jan 2026).

5. Complexity and reported performance

The original appeal of MAMP is that it preserves the matched-filter complexity class. In BO-MAMP, each iteration uses matrix-vector multiplications $Q_t$4, giving $Q_t$5 per iteration, while covariance bookkeeping and short-memory damping contribute only negligible $Q_t$6 overhead (Liu et al., 2020).

IBS-CD-MAMP reduces transform cost by replacing one $Q_t$7-point transform with $Q_t$8 parallel $Q_t$9-point transforms. The IBS transform stage therefore costs $P_{t,i}$0 rather than $P_{t,i}$1, and the total per-iteration complexity is

$P_{t,i}$2

The reported relative complexity reductions for IFDM-style settings are approximately $P_{t,i}$3 overall at $P_{t,i}$4, $P_{t,i}$5 at $P_{t,i}$6, $P_{t,i}$7 at $P_{t,i}$8, and $P_{t,i}$9 at $A^H A$0, all relative to IFDM with $A^H A$1 (Liu et al., 2024).

For IFDM, CD-MAMP combines sparse time-domain filtering with FFT/interleaving operations. The paper gives overall complexity $A^H A$2 and reports several performance comparisons: at BER $A^H A$3, IFDM + CD-MAMP achieves approximately $A^H A$4 gain over OTFS/AFDM with CD/DD-OAMP and approximately $A^H A$5 over DD-MAMP; in $A^H A$6 MIMO it outperforms OFDM, OTFS, and AFDM by more than $A^H A$7; and its running time is approximately $A^H A$8 faster than OTFS + CD-OAMP, approximately $A^H A$9 faster than OTFS + DD-MAMP, and approximately $\Phi_t$0 faster than AFDM + CD-MAMP (Chi et al., 2024).

In WHTDM, the equalizer-side cost is $\Phi_t$1 per iteration because the algorithm operates on the banded matrix $\Phi_t$2. The transmitter-side counts for a $\Phi_t$3-symbol block are $\Phi_t$4 real multiplications and $\Phi_t$5 real additions for WHTDM, versus $\Phi_t$6 real multiplications and $\Phi_t$7 real additions for OFDM. Under the 3GPP TDL-C model at $\Phi_t$8, WHTDM with CD-MAMP achieves over an order of magnitude lower BER than OFDM 1-tap MMSE at $\Phi_t$9, maintains BER below $$B = I_M - A A^H,\qquad f_t = \theta_t B f_{t-1} + \gamma_t (y - A x_t),$$0 across $$B = I_M - A A^H,\qquad f_t = \theta_t B f_{t-1} + \gamma_t (y - A x_t),$$1–$$B = I_M - A A^H,\qquad f_t = \theta_t B f_{t-1} + \gamma_t (y - A x_t),$$2 at $$B = I_M - A A^H,\qquad f_t = \theta_t B f_{t-1} + \gamma_t (y - A x_t),$$3 for delay spreads $$B = I_M - A A^H,\qquad f_t = \theta_t B f_{t-1} + \gamma_t (y - A x_t),$$4–$$B = I_M - A A^H,\qquad f_t = \theta_t B f_{t-1} + \gamma_t (y - A x_t),$$5, and among the compared CD-MAMP-equalized new waveforms yields the best BER (Hao et al., 14 May 2026).

For coded MIMO multicarrier reception, MS-CD-MAMP is positioned against MS-CD-OAMP/VAMP. The reported complexity is

$$B = I_M - A A^H,\qquad f_t = \theta_t B f_{t-1} + \gamma_t (y - A x_t),$$6

and the runtime at BER $$B = I_M - A A^H,\qquad f_t = \theta_t B f_{t-1} + \gamma_t (y - A x_t),$$7 is approximately $$B = I_M - A A^H,\qquad f_t = \theta_t B f_{t-1} + \gamma_t (y - A x_t),$$8 of MS-CD-OAMP/VAMP. On the information-theoretic side, the paper reports that coded MIMO-OFDM, OTFS, and AFDM with MS-CD-MAMP achieve the same maximum achievable rate in doubly selective channels, and that optimized finite-length LDPC codes operate only $$B = I_M - A A^H,\qquad f_t = \theta_t B f_{t-1} + \gamma_t (y - A x_t),$$9 from the associated theoretical limit while gaining $x_0$00 over well-designed point-to-point LDPC codes (Chi et al., 7 Jan 2026).

6. Scope, misconceptions, and open questions

A recurrent misconception is to treat CD-MAMP as a generic claim of Bayes-optimality in any multi-domain system. The literature is more specific. Exact fixed-point equivalence and Bayes-optimal MSE are established for optimized MAMP under RUI matrices and unique state-evolution fixed points; in IBS-CD-MAMP, by contrast, the transform only approximates RUI, and the paper explicitly states that the optimality claim is empirical rather than formally proven (Liu et al., 2020).

Another misconception is that cross-domain processing is inherently superior to domain-matched diagonalization. The WHTDM study provides a counterexample: in static channels, OFDM with 1-tap MMSE achieves the best BER, while WHTDM and other transform-domain waveforms exhibit error floors because their equivalent channel matrices remain non-diagonal and CD-MAMP convergence is then limited by residual coupling (Hao et al., 14 May 2026). Similarly, IFDM’s benefits derive from deliberate dense right-unitarily invariant mixing, not from preserving sparsity in the effective symbol-domain channel (Chi et al., 2024).

Several practical caveats recur across the literature. Convergence without damping is not guaranteed in MAMP; $x_0$01 may fail, while $x_0$02 or $x_0$03 is usually sufficient (Liu et al., 2021). In IBS-CD-MAMP, uneven block extraction harms performance, and both local and global interleaving are required (Liu et al., 2024). In WHTDM, the choice of band width $x_0$04, step sizes $x_0$05 and $x_0$06, and damping $x_0$07 becomes more delicate as delay and Doppler spreads grow (Hao et al., 14 May 2026). In MS-CD-MAMP, perfect CSI is assumed, and the coding analysis depends on uniformly Lipschitz decoding modules (Chi et al., 7 Jan 2026).

The stated research directions are correspondingly concrete. The IBS work identifies formal state evolution for rectangular, interleaved, block-sparse transforms as open, and suggests adaptive IBS, learned denoisers, hybrid FFT/FWHT blocks, and joint design of memory filters and interleavers (Liu et al., 2024). The WHTDM paper points to pilot-aided WHT-domain channel estimation and adaptive bandwidth tracking (Hao et al., 14 May 2026). A plausible synthesis of these directions is that future CD-MAMP research will center