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Learning-to-Transition for Large-scale and High-Order MIMO Detection
Yubo Zhang, Yiyao Liu, Xiaodong Wang
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Abstract
Abstract:High-order multiple-input multiple-output (MIMO) detection requires efficient search over a large discrete symbol space while producing reliable soft information for channel decoding. This paper develops a learning-to-transition (L2T) framework that formulates MIMO detection as a stochastic sequence of complete-vector transitions. At each transition, a channel-coupled Transformer updates both the instance embedding and the sampling policy, while a blockwise autoregressive factorization captures inter-stream dependence with moderate sequential complexity. For hard-output detection, a transition network is applied recursively and trained through a residual-to-BER curriculum, which first learns the MIMO search geometry from the exact residual metric and then aligns the policy with transmitted-bit accuracy. For soft-output reception, the well-trained hard policy is cloned at the parameter level into every layer of an untied soft-input soft-output iterative detection and decoding (IDD) receiver. This tied-to-untied transfer preserves the learned zero-prior search dynamics while enabling layer- and round-specific specialization under decoder feedback. Within each IDD round, decoder priors tilt candidate generation according to Bayes' rule, and likelihood-weighted terminal hypotheses produce posterior and extrinsic log-likelihood ratios for LDPC decoding. A multi-stage training strategy further stabilizes the hard-to-soft transfer by progressively exposing the receiver to synthetic and in-loop decoder-generated priors.
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- Source: https://arxiv.org/abs/2608.14511v1
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Learning-to-Transition for Large-scale and High-Order MIMO Detection Yubo Zhang Yiyao Liu Xiaodong Wang Abstract High-order multiple-input multiple-output (MIMO) detection requires efficient search over a large discrete symbol space while producing reliable soft information for channel decoding. This paper develops a learning-to-transition (L2T) framework that formulates MIMO detection as a stochastic sequence of complete-vector transitions. At each transition, a channel-coupled Transformer updates both the instance embedding and the sampling policy, while a blockwise autoregressive factorization captures inter-stream dependence with moderate sequential complexity. For hard-output detection, a transition network is applied recursively and trained through a residual-to-BER curriculum, which first learns the MIMO search geometry from the exact residual metric and then aligns the policy with transmitted-bit accuracy. For soft-output reception, the well-trained hard policy is cloned at the parameter level into every layer of an untied soft-input soft-output iterative detection and decoding (IDD) receiver. This tied-to-untied transfer preserves the learned zero-prior search dynamics while enabling layer- and round-specific specialization under decoder feedback. Within each IDD round, decoder priors tilt candidate generation according to Bayes’ rule, and likelihood-weighted terminal hypotheses produce posterior and extrinsic log-likelihood ratios for LDPC decoding. A multi-stage training strategy further stabilizes the hard-to-soft transfer by progressively exposing the receiver to synthetic and in-loop decoder-generated priors. Index Terms: MIMO detection, high-order QAM, learning to optimize, Transformer, reinforcement learning, iterative detection and decoding. I Introduction Multiple-input multiple-output (MIMO) transmission improves spectral efficiency by multiplexing several data streams, but joint detection becomes increasingly difficult as the system loading and modulation order grows. With perfect channel state information (CSI), maximum-likelihood (ML) detection must search over QNtQ^N_t possible transmit vectors for NtN_t streams and a Q-ary constellation. This exponential complexity is prohibitive for large or square high-order MIMO systems, whereas low-complexity linear detectors may suffer substantial performance loss under strong inter-stream interference or ill-conditioned channels [1]. Approximate message passing (AMP), orthogonal AMP, expectation propagation (EP), and model-driven unfolding methods provide useful performance-complexity tradeoffs. Learned receivers, including OAMP-Net2, RE-MIMO, and the Soft Graph Transformer (SGT), further improve robustness or better capture dense inter-stream interactions [2, 3, 4]. Nevertheless, most existing neural detectors either inherit a prescribed inference recursion or directly map a received instance to symbol estimates or marginals. As a result, they do not explicitly learn how to traverse the discrete space of complete high-order MIMO transmit vectors. Related combinatorial and Ising formulations expose the same discrete structure, but typically rely on specialized optimization procedures rather than an amortized transition policy [5, 6]. Reinforcement learning for combinatorial optimization offers a complementary perspective. Constructive policies generate a solution one component at a time, whereas learned improvement heuristics usually apply a prescribed local modification [14, 18]. In contrast, we learn the stochastic transition rule itself. Across search steps, the same Transformer is applied recursively to map the current complete MIMO vector to a distribution over the next complete vector; within each step, the next vector is generated in a blockwise autoregressive manner across transmit streams. The sampled candidates are then evaluated by the exact channel metric. Therefore, the term “complete-vector transition” refers to the search state before and after each transition, while the decisions within a transition can still follow a structured autoregressive factorization. A second difficulty is objective alignment. The residual energy provides a model-based cost for every sampled vector, but minimizing the residual alone does not necessarily minimize the transmitted-bit BER. We therefore adopt a residual-to-BER curriculum for hard-detector training. The policy is first trained with a residual-driven policy-gradient objective, which teaches the coupled MIMO search geometry, and is then smoothly shifted to a transmitted-bit binary cross-entropy (BCE) objective, which directly aligns the learned policy with bit-error-rate performance [21]. Soft-output reception further requires reliable uncertainty quantification, not merely a strong hard estimate. Iterative detection and decoding (IDD) needs posterior and extrinsic soft information, together with competing hypotheses for both values of each coded bit. Classical SISO and variational detectors make this posterior/extrinsic distinction explicit [19, 20]. We follow this principle by transferring the hard policy to a prior-conditioned soft detector. Specifically, the well-trained tied hard policy is cloned at the parameter level into every transition layer of every IDD round, and the resulting copies are then optimized independently. This tied-to-untied transfer preserves the learned zero-prior search dynamics at initialization, while allowing each soft layer to specialize to its position in the search trajectory and to the decoder-prior distribution of its IDD round. During soft-output inference, decoder priors tilt candidate sampling according to Bayes’ rule; a likelihood-weighted terminal mixture produces detector posterior LLRs; and explicit prior subtraction yields the extrinsic LLRs passed to the decoder. The main contributions are summarized as follows: • We formulate high-order MIMO detection as a complete-vector transition process and develop a channel-coupled Transformer policy for this process. Instead of constructing a transmit vector component by component or applying a prescribed local update, the detector learns to map the current complete vector to a distribution over the next complete vector. The Transformer jointly updates the instance embedding and the sampling policy, while a blockwise autoregressive factorization captures inter-stream dependence within each transition. • We train the hard-output detector through trajectory-level reinforcement learning with a residual-to-BER curriculum. The residual-driven policy-gradient objective first exploits the exact channel metric to learn the coupled MIMO search geometry, and the later transmitted-bit BCE objective directly aligns the policy with bit-error-rate performance. • We extend the hard detector to a soft-input soft-output IDD receiver via tied-to-untied hard-to-soft transfer. The well-trained tied hard policy is cloned at the parameter level into every independently trainable soft-detector layer, providing a common zero-prior search initialization while allowing specialization across transition depth and IDD rounds. The soft receiver further incorporates Bayesian prior-tilted sampling, likelihood-weighted posterior LLR computation, explicit extrinsic LLR generation, and staged training for drifting decoder priors. I Problem Statement I-A MIMO Detection Consider a point-to-point spatial-multiplexing MIMO channel with NtN_t transmit antennas and NrN_r receive antennas. For each channel use, =+,∼(,σ2Nr), y= H x+ n, n ( 0,σ^2 I_N_r), (1) where ∈ℂNr×Nt H ^N_r× N_t is the channel matrix, following the distribution ∼pH H p_H, and σ2σ^2 is the noise variance at each receive antenna. Let QA_Q denote a unit-energy, Gray-coded quadrature-amplitude modulation (QAM) constellation with order Q. Each QAM symbol contains B=log2QB= _2Q coded bits, with B/2B/2 bits on each pulse-amplitude modulation (PAM) axis. The corresponding constellation mapping is depicted in Fig. 1, demonstrating how QAM scheme exploits the two-dimensional space to separate modulated symbols. Fig. 1: Constellation for Gray-coded data symbols. Let μQ:0,1B→Q _Q:\0,1\^B _Q be the prescribed Gray mapping. We collect the bits assigned to all transmit antennas and define the modulated symbol as follows: =ℳQ()=[μQ(1),…,μQ(Nt)], x=M_Q( b)= [ _Q( b_1),…, _Q( b_N_t) ] T, (2) where =[1,…,Nt] b=[ b_1 T,…, b_N_t T] T belongs to ℬ=0,1NtBB=\0,1\^N_tB. Traditional maximum-likelihood (ML) detection then adopts the following exhaustive search: ^ML(,)=argmin∈ℬ|−ℳQ()∥22. b_ML( y, H)= _ b | y- HM_Q( b)\|_2^2. (3) Note that the entire space now contains 2NtB2^N_tB feasible solutions, hence the ML detection in (3) becomes computationally prohibitive under the large system size (i.e., NtN_t increases) and high-order QAM (i.e., B increases). Moreover, it is inefficient to repeatedly solve similar detection problems as the channel H frequently varies. To tackle these issues, we aim to train a single neural network that solves the symbol detection problems for channel instances across the entire distribution ∼pH H p_H. This follows the essence of learning-to-optimize (L2O)/ amortized optimization, which shifts the computational burden from online iterative optimization to offline training. Instead of solving each problem instance from scratch, L2O learns an optimizer from the data, exploiting recurring task structures for fast inference. Specifically, denote the residual function of the symbol detection as follows: f(,,)=‖−ℳQ()‖22. f( b; y, H)=\| y- HM_Q( b)\|_2^2. (4) Then the L2O solver aims to minimize this metric in the population level: let ^(,) b_ θ( y, H) denotes its final output, where θ denotes the learnable parameters of the neural solver, the amortized learning problem becomes min∼pH[f(^(,),,)]. _ θ\ E_ H p_H [f ( b_ θ( y, H); y, H ) ]. (5) This formulation remains challenging, since symbol detection is inherently an NP-hard combinatorial optimization problem, which lacks a high-quality general-purpose solver. I-B Learning to Transition Based on the objective in (5), we seek to train a neural network that learns an effective transition policy over the entire solution space ℬB. Similar to the prior graph optimization work, learning an expressive channel embedding is crucial to the detection performance in this work. Note that the residual function in (4) can be rewritten as ‖−‖22=−2Re+‖22,\| y- H x\|_2^2= x H R x-2Re\ x H z\+\| y\|_2^2, (6) where ≜ℳQ()∈ℂNt x _Q( b) ^N_t represents the modulated symbol, ≜∈ℝNt×Nt R H H H ^N_t× N_t represents the Gram/correlation matrix of channel H, and ≜∈ℂNt z H H y ^N_t represents the corresponding matched filter output. Inspired by [13], the effective channel features can be learned by drawing an analogy with graph-embedding generation. Specifically, the matched-filter entries, i.e., z, and all diagonal entries of Gram matrix, i.e., diag()diag( R), form the node-wise inputs (each node relates to one transmit antenna); while all off-diagonal entries of Gram matrix, i.e., offdiag()offdiag( R), form the edge-wise inputs (which reflects the complex inter-stream couplings). Then a graph neural network (GNN) encoder G_ maps these sufficient input statistics to the initial channel embedding, i.e., (0)=G(,diag(),offdiag())∈ℝD×Nt, Z^(0)=G_ ( z,diag( R),offdiag( R) ) ^D× N_t, (7) where D is the feature dimension of each node (transmit side). To reduce the difficulty of learning the transition policy, inspired by the recursive optimization process in [11, 10], we consider a Learning-to-transition (L2T) scheme consisting of T updates, therefore amortizing the searching task across all steps. Importantly, a static embedding cannot reflect which parts of the search space have already been explored by the evolving candidate. We therefore keep updating the channel embedding across the transition steps, inspired by the works in [7, 16]. Assume that the transition process at each step is performed by a neural network with learnable parameters 0 θ_0. Hence the forward process at each transition step t is given by [(t),(t)]=Π0((t−1),(t−1)), [ p^(t), Z^(t) ]= _ θ_0 ( b^(t-1), Z^(t-1) ), (8) for t=1,…,Tt=1,…,T. The first output undergoes a bit-wise sampling process to obtain the next solution, i.e., (t)∼(t) b^(t) p^(t), while the second output is the updated feature embedding at transition step t. I Hard-Decision Receiver We now design a non-iterative receiver where the transmitted bits are directly obtained via hard decisions. Equivalently, a learning-based solver is trained to address the NP-hard combinatorial optimization problem in a population level. To achieve this, we first propose the policy network design and then provide the policy gradient-based training scheme. I-A Transformer-based Policy Network The transition policy Π0 _ θ_0 in (8) can be parameterized by a Transformer network [12]. In particular, the previous sampled solution is reshaped as (t−1)=[1(t−1),…,Nt(t−1)]∈ℝB×Nt b^(t-1)=[ b^(t-1)_1,…, b^(t-1)_N_t] ^B× N_t, and then concatenated with the corresponding channel embedding (t−1)∈ℝD×Nt Z^(t-1) ^D× N_t in a top-down manner, which yields the input token sequence to the next layer: (t)=[(t−1);(t−1)]∈ℝ(B+D)×Nt. C^(t)=[ b^(t-1); Z^(t-1)] ^(B+D)× N_t. (9) Let ¯(t)=LayerNorm((t)) C^(t)=LayerNorm( C^(t)). For each attention head h, h=1,…,Hh=1,…,H, the query, key, and value projections are (t,h) Q^(t,h) =(¯(t))⊤Q(h)∈ℝNt×Dk/H, =( C^(t)) W_Q^(h) ^N_t× D_k/H, (t,h) K^(t,h) =(¯(t))⊤K(h)∈ℝNt×Dk/H, =( C^(t)) W_K^(h) ^N_t× D_k/H, (t,h) V^(t,h) =(¯(t))⊤V(h)∈ℝNt×Dv/H. =( C^(t)) W_V^(h) ^N_t× D_v/H. (10) The corresponding attention output is (t,h)=softmax((t,h)(t,h)⊤Dk/H)(t,h). A^(t,h)=softmax\! ( Q^(t,h) K^(t,h) D_k/H ) V^(t,h). (11) The H attention outputs are concatenated and projected to (t)=o[(t,1),…,(t,H)]⊤=[p(t);f(t)], Y^(t)= W_o[ A^(t,1),…, A^(t,H)] =[ Y_p^(t); Y_f^(t)], (12) where o∈ℝ(B+D)×Dv W_o ^(B+D)× D_v, p(t)∈ℝB×Nt Y_p^(t) ^B× N_t represents the solution logits and f(t)∈ℝD×Nt Y_f^(t) ^D× N_t contains the channel embedding residuals. Then the solution at the next transition step is sampled as follows: (t)=sigmoid(p(t))=[1(t),…,Nt(t)], P^(t)=sigmoid( Y_p^(t))=[ p^(t)_1,…, p^(t)_N_t], i(t)∼Bernoulli(i(t)),i=1,…,Nt. b_i^(t) ( p_i^(t)),\ i=1,…,N_t. (13) In parallel, the channel embedding is updated through residual connections and an MLP: ~(t) Z^(t) =(t−1)+f(t), = Z^(t-1)+ Y_f^(t), (t) Z^(t) =~(t)+MLP(LayerNorm(~(t))). = Z^(t)+MLP\! (LayerNorm( Z^(t)) ). (14) (I-A) and (I-A) are the two output branches of the same Transformer layer. Unrolling this layer over T transitions recursively refines both the discrete solution and the continuous embedding. I-B Blockwise Autoregressive Sampling In principle, the mean-field sampling for each solution (i.e., every bit is sampled in parallel in the process (t)∼(t) b^(t) p^(t)) is fast but ignores conditional correlations among N bits. At the other extreme, bit-by-bit autoregressive sampling captures every conditional dependence but requires a large mount of sequential forward processes. We therefore introduce blockwise autoregressive sampling [17] as an efficient tradeoff between these two extreme cases, to generate the complete solution (t) b^(t) at each transition step. In this work, each block represents a B-bit QAM symbol from each transmit antenna. The dependence is retained across those NtN_t symbols because the dense off-diagonal entries in the Gram matrix R couple the symbol detections at different antennas. In contrast, the bits within each QAM symbol are generated in parallel. This mean-field factorization avoids a bit-by-bit chain of length NtBN_tB while preserving the dominant inter-symbol dependence. The block ordering n=1,2,…,Ntn=1,2,…,N_t may be randomized during training to avoid imposing a persistent ordering bias. Define mask b_mask as an all-zero vector with length B, which can be regarded as each undetermined data symbol during a sequential sampling process. Then we define the following partial solution before sampling each data symbol n(t) b^(t)_n at transition step t: <n(t)≜[1(t),…,n−1(t),mask,…,mask⏟Nt−n+1copies] b^(t)_<n [ b^(t)_1,…, b^(t)_n-1, b_mask,…, b_mask_N_t-n+1\ copies] (15) Conditioned on <n(t) b^(t)_<n, the complete prior solution (t−1) b^(t-1), and the current channel embedding n−1(t) Z^(t)_n-1, the policy samples all variables in the next block in parallel: [n(t),n(t)]=Π0(⋅∣<n(t),(t−1),n−1(t)), [ p^(t)_n, Z^(t)_n]= _ θ_0\! (· b^(t)_<n, b^(t-1), Z^(t)_n-1 ), n(t)∼parallelBernoulli(n(t)), b_n^(t)\ parallel \ Bernoulli( p^(t)_n), (16) with 0(t)=(t−1) Z^(t)_0= Z^(t-1) and (t)=Nt(t) Z^(t)= Z^(t)_N_t. After NtN_t blockwise samplings, we obtain the complete solution (t) b^(t). Note that the special case M=1M=1 reduces to a mean-field sampler, whereas M=NM=N reduces to a bit-by-bit autoregressive sampler. I-C Policy-Gradient Training Scheme Implementing the above T-step stochastic transition process yields a complete L2T solution trajectory τ≜((0),(1),…,(T))τ ( b^(0), b^(1),…, b^(T)). Its sampling probability can be computed as p0(τ∣,) p_ θ_0(τ y, H) =p((0))∏t=1(t)Π0((t)∣(t−1),(t−1);,). =p( b^(0)) _t=1^(t) _ θ_0 ( b^(t) b^(t-1), Z^(t-1); y, H ). (17) where p((0))p( b^(0)) does not depend on 0 θ_0. Denote the step-wise objective function as g((t),,)g( b^(t); y, H). Similar to the GCO paper, the following trajectory-wise objective function with entropy regularization is proposed: g(τ,,) g(τ; y, H) =∑t=1Tg((t),,), = _t=1^Tg( b^(t); y, H), Jλ(0,τ,,) J_λ( θ_0;τ, y, H) =g(τ,,)+λlogp0(τ∣,). =g(τ; y, H)+λ p_ θ_0(τ y, H). (18) Since τ[logp0(τ∣,)]E_τ[ p_ θ_0(τ y, H)] is the negative entropy term, the coefficient λ>0λ>0 encourages explorations at the early stage. The corresponding distribution-level learning objective is then given as follows: λ(0)=∼pHτ∼p0(⋅∣,)[Jλ(0;τ,,)],J_λ( θ_0)=E_ H p_HE_τ p_ θ_0(· y, H) [J_λ( θ_0;τ, y, H) ], (19) and the ultimate learning problem becomes min0λ(0), _ θ_0~J_λ( θ_0), (20) which is solved using a gradient-based method. Note that (19) contains two nested expectations. The outer expectation ∼pHE_ H p_H arises because we aim to learn a population-based solver whose policy generalizes across the channel distribution; The inner expectation τ∼p0(⋅∣,)E_τ p_ θ_0(· y, H) accounts for the stochasticity of the learned transition policy because each trajectory is sampled from the probability distributions output by the network. For each detection instance (,)( y, H), this expectation is approximated by averaging over multiple sampled trajectories τ1,…,τK\ _1,…, _K\. Although the ultimate interest is f((T),,)f( b^(T); y, H) at the final step, aggregating the objective values along the trajectory encourages progressive improvement throughout the search process. At inference, for a new instance (,)( y, H), we expect the recursive application of the trained policy network to produce a near-optimal result (T) b^(T). Since each update only requires a feedforward mapping and a sampling operation, the resulting solver can be executed repeatedly with low latency, making real-time deployment feasible. We next show how the step-wise objective function g((t),,)g( b^(t); y, H) is constructed. Note that the ultimate goal is to reduce the detection bit-error-rate (BER), and the true transmitted bits are assumed to be known during training. Hence the BER-oriented binary cross-entropy (BCE) can be regarded as an effective step-wise loss function: specifically, let Nb=NtBN_b=N_tB denote the entire codeword length; if the ground truth is bj∗b_j^* for every jthj^th bit, while the network produces probabilities pj(t)≜P0(r)(bj=1)p^(t)_j P^(r)_ θ_0(b_j=1) at transition step t, j=1,…,Nbj=1,…,N_b, the BER-oriented BCE loss for this codeword is ℓtBCE=−1Nb∑j=1Nb[bj∗logpj(t)+(1−bj∗)log(1−pj(t))]. ^BCE_t=- 1N_b _j=1^N_b [b_j^* p^(t)_j+(1-b_j^*) (1-p^(t)_j) ]. (21) In practice, the residual loss function in (4) is also utilized as a smooth surrogate in the early-stage training. Hence the following hybrid step-wise loss function is proposed g((t),,)=ρs⋅f((t),,)+(1−ρs)⋅ℓtBCE, g( b^(t); y, H)= _s· f( b^(t); y, H)+(1- _s)· ^BCE_t, (22) where the coefficient ρs _s gradually decreases from 1 to 0, explicitly reflecting a schedule shifts from the smooth residual loss to the BER-oriented BCE loss. As a result, substituting (22) into (I-C) yields the trajectory-wise loss function. According to (19), when computing the population-level loss, for each instance (,)( y, H), K solution trajectories are sampled, which accounts for the expectation term τ∼p0(⋅∣,)E_τ p_ θ_0(· y, H). The initializations of those K trajectories (denoted as 1(0),…,K(0) b_1^(0),…, b_K^(0)) are all randomly-generated valid solutions. Finally, the policy network Π0 _ θ_0 is trained according to (20). During inference, for each (,)( y, H), we recursively employ the trained policy network Π0∗ _ θ_0^* for T transitions based on the random initializations (1(0),…,K(0))( b_1^(0),…, b_K^(0)), and obtain the final solution set (1(T),…,K(T))( b_1^(T),…, b_K^(T)). The hard decision result is the final solution k∗(T) b_k^*^(T) with the smallest residual loss, i.e., k∗(T):k∗=argmink=1,…,Kf(k(T),,). b_k^*^(T):k^*= _k=1,…,Kf( b_k^(T); y, H). (23) IV Soft-Input Soft-Output Receiver IV-A Iterative Detection and Decoding Iterative detection and decoding (IDD) exchanges soft information between the MIMO detector and a soft-output low-density parity-check (LDPC) decoder. An interleaver distributes one LDPC codeword over a batch of MIMO observations; after detection, their LLRs are simply reassembled in codeword order before decoding. We use the convention L(b)=log(P(b=0)P(b=1))L(b)= ( P(b=0)P(b=1) ), so P(b=1)=sigmoid(−L(b))P(b=1)=sigmoid(-L(b)). Let Γ denote the code interleaver. At the mthm^th IDD round, the complete turbo exchange uses the LDPC belief-propagation (BP) mapping BPD_BP, given as follows: E,(m) L^E,(m) =P,(m)−A,(m), = L^P,(m)- L^A,(m), (24) Din,(m) L_D^in,(m) =Γ−1(E,(m)),DP,(m)=BP(Din,(m)), = ^-1( L^E,(m)),\ L_D^P,(m)=D_BP( L_D^in,(m)), (25) A,(m+1) L^A,(m+1) =Γ(DP,(m)−Din,(m)), = ( L_D^P,(m)- L_D^in,(m) ), (26) where the subscripts (A,P,E)(A,P,E) in the above equations represent the a-priori, posterior, and extrinsic LLR. The exchange is initialized with A,(0)= L^A,(0)= 0. The two subtractions are essential: each module passes only information newly created from its own observation or parity constraints, rather than feeding its input prior back as if it were new evidence. IV-B LLR Computations For a bit b, Bayes’ rule gives P(b∣ℱ,LA)∝Pnet(b∣ℱ)PA(b),P(b ,L^A) P_net(b )P^A(b), (27) where ℱF contains the detection instance and previously generated bits. With LA=log(PA(0)PA(1))L^A= ( P^A(0)P^A(1) ) and equal-prior LLR output by the policy network: anet=logPnet(b=1∣ℱ)Pnet(b=0∣ℱ),a^net= P_net(b=1 )P_net(b=0 ), (28) the posterior LLR and the sampling probability are given as P(b=1∣ℱ,LA)P(b=0∣ℱ,LA) P(b=1 ,L^A)P(b=0 ,L^A) =exp(anet−LA), = (a^net-L^A), (29) psample p^sample =sigmoid(anet−LA). =sigmoid(a^net-L^A). (30) Thus an a-priori LLR induces a bias into the sampling probabilities of the detector’s output. At A= L^A= 0, the policy reduces to the previous hard-decision detector. After the final transition, let qk,jq_k,j be trajectory k’s probability that bit j is one and let k(T) b_k^(T) be its sampled vector. We use the exact normalized residual to combine the K trajectories: wk w_k =exp[−f(k(T);ξ)/τw]∑k′=1Kexp[−f(k′(T);ξ)/τw], = [-f( b_k^(T);ξ)/ _w] _k =1^K [-f( b_k ^(T);ξ)/ _w], (31) q¯j q_j =∑k=1Kwkqk,j. = _k=1^Kw_kq_k,j. (32) Because f is normalized by Nrσ2N_rσ^2, the complex-Gaussian likelihood satisfies p(∣,)∝exp[−Nrf(,ξ)]p( y b, H) [-N_rf( b;ξ)]. Consequently, τw=1/Nr _w=1/N_r is the physically matched value in (31); any other value is an explicit likelihood-tempering choice rather than a new estimate of the noise variance. The detector posterior and extrinsic LLRs are LjP=log1−q¯jq¯j,LjE=LjP−LjA.L_j^P= 1- q_j q_j, L_j^E=L_j^P-L_j^A. (33) This is a Rao–Blackwellized terminal-mixture estimator: qk,jq_k,j retains the final head’s bit uncertainty instead of replacing it by the sampled bit, while the exact likelihood favors candidates that fit the observation. Only the terminal Bernoulli is Rao–Blackwellized: the mixture weight wkw_k is evaluated at the single sampled vector k(T) b_k^(T) rather than as an expectation under the terminal head, so (32) is a likelihood-weighted terminal-marginal mixture rather than a fully marginalized posterior; the two are equal when each trajectory is nearly deterministic. The extrinsic subtraction in (33) is exact for a single mean-field trajectory. With qk,j=sigmoid(ak,jnet−LjA)q_k,j=sigmoid(a^net_k,j-L^A_j), (29) gives log((1−qk,j)/qk,j)=LjA−ak,jnet ((1-q_k,j)/q_k,j)=L^A_j-a^net_k,j, so the per-trajectory extrinsic −ak,jnet-a^net_k,j carries only the network evidence and is free of the prior. After the residual-weighted mixture (32), however, the logit of a convex combination of prior-tilted marginals is no longer an affine function of LjAL^A_j, and the weights wkw_k themselves depend on LjAL^A_j through the sampled vectors. Consequently LjE=LjP−LjAL_j^E=L_j^P-L_j^A removes the bit’s own prior only approximately, with a residual dependence that shrinks as the trajectories agree. We therefore measure this residual prior dependence, rather than assume it is zero, through the prior-sensitivity diagnostic in Section V-B. IV-C IDD Training Scheme The main training difficulty introduced by the soft IDD learning scheme is that the detector input prior (i.e., the feedback from LDPC decoder) at each round is not stationary. As detector parameters evolve across the iterations, the distribution of the prior A,(m) L^A,(m) also drifts across IDD rounds. Note that the training scheme for the hard-decision receiver assumes equal prior for every transmitted bit and cannot adapt to the practical IDD settings; Moreover, starting from the random initializations (at all IDD rounds) usually renders unstable convergence and overconfident feedback. To address these issues, we propose the following multi-stage training scheme tailored to the soft IDD learning problem. IV-C1 Hard-decision initialization Note that for the hard-decision receiver in Sec. I, the same policy Π0∗ _ θ^*_0 is recursively adopted at all T transitions. This parameter sharing is useful for learning a stable transition rule, but it also constrains the overall expressive power. In contrast, soft-IDD receiver admits a more expressive construction. We unfold the detector across both the T transition steps and the M IDD rounds [8]. The resulting parameter family S=S(m,t)m,t _S=\ θ_S^(m,t)\_m,t is untied, so each layer can adapt to its position in the search and to the reliability of the decoder prior received in its round. Crucially, the hard policy provides the common initialization for every untied soft layer, i.e., [k(m,t),k(m,t)] [ P_k^(m,t), Z_k^(m,t) ] =ΠS(m,t)(k(m,t−1),k(m,t−1)), = _ θ_S^(m,t) ( b_k^(m,t-1), Z_k^(m,t-1) ), (34) S(m,t)|init . θ_S^(m,t) |_init =0⋆,m=0,…,M−1,t=1,…,T. = θ_0 , subarraycm=0,…,M-1,\\ t=1,…,T subarray. Parameter sharing is released only after initialization: subsequent soft training lets different IDD rounds adapt to increasingly informative decoder feedback. This gives a direct continuation from the stable, parameter-efficient hard-decision solver to the higher-capacity iterative soft receiver. IV-C2 Synthetic-prior training The detector is next trained without invoking the LDPC decoder. We draw synthetic priors from the classical Gaussian LLR model in [9]. For a sampled reliability σA _A and εj∼(0,1) _j (0,1), LjA L_j^A =(1−2bj∗)σA22+σAεj, =(1-2b_j^*) _A^22+ _A _j, LjA|bj∗ L_j^A b_j^* ∼((1−2bj∗)σA22,σA2). ((1-2b_j^*) _A^22, _A^2 ). (35) Hence the mean has the correct sign for the transmitted bit and its magnitude increases with reliability. In practice, we sample the target a priori mutual information and obtain σA=J−1(IA) _A=J^-1(I_A). Covering the required reliability range exposes the detector to uninformative, moderate, and confident priors without decoder-induced drift. IV-C3 In-loop training The fixed LDPC decoder is inserted in the forward pass, and the exchange in (24)-(26) is unrolled from A,(0)= L^A,(0)= 0. The resulting priors are exactly those encountered by the deployed receiver, closing the remaining gap from synthetic-prior training. For training, only the posterior probability from the final transition of each IDD-round network enters the loss. Let NcN_c be the number of coded bits in the interleaved codeword. The round-m posterior BCE is ℒpost(m)=−1Nc∑j=1Nc[bj∗logq¯m,j+(1−bj∗)log(1−q¯m,j)]. _post^(m)=- 1N_c _j=1^N_c [b_j^* q_m,j+(1-b_j^*) (1- q_m,j) ]. (36) The ultimate soft-IDD training loss is then given by ℒsoft=∑m=0M−1ωmℒpost(m),ωm≥0. _soft= _m=0^M-1 _mL_post^(m), _m≥ 0. (37) The forward pass uses the actual 5G NR LDPC decoder configured according to the Sionna link-level implementation [22]. The module passed to the next detector is the decoder’s extrinsic mapping ℰBP()=BP()−E_BP( l)=D_BP( l)- l. During backpropagation, the straight-through rule is applied to this composite mapping: ∂ℰBP()∂≈. _BP( l)∂ l≈ I. (38) The true decoder posterior and the subtraction are therefore both retained in the forward pass, whereas the backward pass copies the gradient through the whole extrinsic module. V Performance Evaluation V-A Simulation Settings We consider a downlink MIMO system with perfect CSI available at the receiver. Each channel sample follows the Rayleigh distribution, with []ij∼i.i.d.(0,1/Nt)[ H]_ij i.i.d. CN(0,1/N_t), and Gray-coded 64-QAM or 256-QAM symbols are normalized to unit average energy. Unless otherwise stated, the L2O framework uses T=8T=8 transitions, embedding dimension d=256d=256, eight attention heads, feed-forward dimension dff=256d_f=256, and K=16K=16 trajectories. One trajectory is initialized by the LMMSE estimate and the remaining trajectories by its randomized perturbations. Residual-only post-processing is disabled so that the reported gain is attributable to the learned transition policy. The hard detector is trained at 20 dB for 10410^4 AdamW updates with learning rate 10−410^-4. The residual-to-BER curriculum uses s0=103s_0=10^3 and s1=5×103s_1=5× 10^3, and the batch sizes are 64 and 16 for 64-QAM and 256-QAM, respectively. For soft reception, every unrolled detector layer is initialized from the same hard checkpoint according to (34) and then follows the synthetic-prior and decoder-in-the-loop stages in Section IV-C. Soft training uses 7×1037× 10^3 updates, learning rate 3×10−53× 10^-5, and batch size 16, with K=32K=32 for 64-QAM and K=16K=16 for 256-QAM. The coded link uses the 5G NR LDPC implementation in Sionna [22], code rate 1/21/2, ten belief-propagation iterations, and three IDD rounds. Each codeword packs P=8P=8 independently detected MIMO vectors, giving rate-matched codeword lengths of 384 and 512 bits for 64-QAM and 256-QAM, respectively. Every reported SNR point is averaged over at least five random seeds until either 200 frame errors or 10710^7 information bits are observed. For the hard detector, we compare LMMSE, QR-domain K-best, OAMP-Net2, RE-MIMO, and SGT [2, 3, 4]. For soft IDD, the comparisons are soft LMMSE, MMSE-PIC, K-best list-MAP, SGT, and DUIDD [8]. All methods use the same channel realizations, modulation, code, decoder iterations, and SNR definition. Uncoded BER is the primary hard-decision metric. Post-decoding BLER is the primary soft-receiver metric, with information-bit BER, generalized mutual information (GMI), and Brier score used to diagnose the quality of the detector LLRs. Parameter count and batch-one latency are measured under identical hardware, software, precision, and synchronization settings; detector and decoder latency are reported separately. V-B Ablation Studies We first isolate the residual-to-BER curriculum at one representative 64-QAM SNR near the BER waterfall. The proposed smooth schedule is compared with residual-only training, BCE-only training, and an abrupt residual-to-BCE switch. For each variant, we report final BER, normalized residual, and the standard deviation over five seeds. For the full model, BER, residual, and normalized Hamming distance between consecutive vectors are also plotted against the transition index. This compact experiment tests both roles of the curriculum: the residual phase should establish a stable search direction, whereas the BCE phase should align the terminal policy with the transmitted bits. Across transitions, a useful learned policy should reduce BER and residual without collapsing immediately to identical trajectories. The soft-receiver ablation is performed at one SNR near the coded waterfall and is summarized in a single table. Starting from the complete three-stage training scheme, we separately (i) remove the hard-checkpoint initialization, (i) retain hard-style parameter tying across all soft-detector layers after the common initialization, (i) remove prior tilting from candidate generation, (iv) omit the synthetic-prior stage, (v) omit decoder-in-the-loop fine-tuning, and (vi) stop the gradient at the LDPC decoder instead of using (38). We report BLER after each IDD round together with final-round GMI and Brier score. To check that prior subtraction in (33) does not leave excessive self-information, we additionally vary one bit’s input prior while holding the channel observation and all other priors fixed. We then report the sensitivity of that bit’s extrinsic LLR and the rate at which its sign changes. This targeted test diagnoses the approximation in (33) without requiring a full EXIT characterization. V-C Overall Performance The first main experiment reports uncoded BER versus SNR for 64-QAM and 256-QAM in two panels. This comparison determines whether the learned complete-vector transitions improve upon linear detection and existing neural detectors, and whether they approach the K-best performance obtained with an explicit tree search. The second main experiment reports post-LDPC BLER versus SNR for both modulation orders. 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