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Diffusion Denoiser Achievable Analysis for Finite Blocklength Unsourced Random Access
Yuming Han, Yuxin Long
Intelligence
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Summary
This paper introduces a decoder-compatible diffusion denoiser for the unsourced multiple access channel (MAC) problem in the finite blocklength regime. By integrating a score-based diffusion model into the joint decoding process, the authors derive a tighter random-coding achievable bound and demonstrate at least a 0.5 dB improvement in energy efficiency across various existing decoders like FASURA and MSUG-MRA.
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Relation Signals (3)
Diffusion Denoiser → appliedto → Unsourced Random Access (URA)
confidence 98% · we introduce a decoder compatible diffusion denoiser as a lightweight analysis within joint decoding for URA
Diffusion Denoiser → improvesperformanceof → FASURA
confidence 95% · Simulations on existing decoders, including FASURA... show consistent performance gains
Diffusion Denoiser → tightensbound → Polyanskiy Framework
confidence 92% · we derive a diffusion-denoiser random-coding achievable bound that is strictly tighter [than Polyanskiy's]
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Abstract
Abstract:Polyanskiy proposed a framework for the unsourced multiple access channel (MAC) problem where users employ a common codebook in the finite blocklength regime. However, existing approaches handle channel noise before the joint decoder. In this work, we introduce a decoder compatible diffusion denoiser as a lightweight analysis within joint decoding. The score network is trained on samples drawn from the channel output distribution, making the method easy to integrate with existing code designs. In our theoretical analysis, we derive a diffusion-denoiser random-coding achievable bound that is strictly tighter. Simulations on existing decoders, including FASURA, MSUG-MRA and pilot-based method, show consistent performance gains with at least a $0.5$ $\mathrm{dB}$ improvement in required $\mathrm{E_b/N_0}$ at a fixed error target.
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- Source: https://arxiv.org/abs/2604.09904v1
- Canonical: https://arxiv.org/abs/2604.09904v1
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Diffusion Denoiser Achievable Analysis for Finite Blocklength Unsourced Random Access Abstract Polyanskiy proposed a framework for the unsourced multiple access channel (MAC) problem where users employ a common codebook in the finite blocklength regime [16]. However, existing approaches handle channel noise before the joint decoder. In this work, we introduce a decoder compatible diffusion denoiser as a lightweight analysis within joint decoding. The score network is trained on samples drawn from the channel output distribution, making the method easy to integrate with existing code designs. In our theoretical analysis, we derive a diffusion-denoiser random-coding achievable bound that is strictly tighter. Simulations on existing decoders, including FASURA, MSUG-MRA and pilot-based method, show consistent performance gains with at least a 0.50.5 dBdB improvement in required Eb/N0E_b/N_0 at a fixed error target. Index Terms— Unsourced random access channel, Finite-blocklength, Diffusion models 1 Introduction MAC protocols have recently attracted many research efforts. Specially, a unsourced random access channel (URA) under a finite-blocklength (FBL) constraint was suggested in [16]. In URA, each user employs the same codebook and the task of the decoder is to recover the list of transmitted messages irrespective of the identity of the users. Recent works [13, 4, 9, 17, 10] have established fundamental limits for different relevant multiple access channel models. In [12], the authors studied multi-user interference (MUI) cancellation with and without channel state information. [15] considered both mis-detection and false alarms, which can serve as a benchmark for unsourced multiple access with random user activities. However, most recent work analyzes the Gaussian channel while decoding directly from raw outputs. In our work, we add a denoiser at the receiver side to give us a tighter random-coding achievable bound. Diffusion models gradually adds Gaussian noise to the available training data in the forward diffusion process until the data becomes pure noise. Then, in the reverse sampling process, it learns to recover the data from the noise, which match the denoising process in decoder. In [11], diffusion model is employed to generate the wireless channel for an end-to-end communications system, achieving almost the same performance as the channel-aware case. In [2], diffusion model with an adapted diffusion process is proposed for the decoding of algebraic block codes. [18] proposed the channel denoising diffusion models to eliminate the channel noise under Rayleigh fading channel and AWGN channel. To our best knowledge, there is no no theoretical analysis on diffusion model with achievable bound. Most efforts focus on applications and use the denoiser as a pre-processing module at the receiver. In this work, we embed a diffusion denoiser into the joint decoding analysis, and use it to derive a random-coding achievable bound that is strictly tighter than [16]. The score network is trained on samples from the channel output distribution so that the approach is highly compatible with existing code designs. Simulations on common decoders (FASURA, MSUG-MRA, pilot-based) confirm at least 0.50.5 dBdB improvements in required energy at the same target error rate. 2 System Model We consider an URA system model from [16] with KaK_a users (or devices). An active user is defined as a user that generates a packet during a given time slot. Each active user generates a packet of k bits. We will assume that an active user does not generate more than one packet at each (discrete) time. Under the power constraint ‖2≤nP\|c\|^2≤ nP, we define an (M,d,ϵ)(M,d,ε) code, where M=2kM=2^k is the size of the codebook, and the error probability ϵ∈(0,1)ε∈(0,1), consists of: • An encoder f:[M]→nf:[M] ^n that produces the transmitted codeword j∼(0,P′In)=f(Wj)c_j (0,P I_n)=f(W_j) of user j for a given message WjW_j uniformly distributed over [M][M]. • A decoder g:n→([M])g:Y^n→ D([M]), where ([M]) D([M]) is a set of indices in 1…M1… M; the decoder outputs W~1,…W~Ka=g(Ka,)\ W_1,… W_K_a\=g(K_a,y) the list of messages. Let g~(Ka,) g(K_a,y) be the multiset of messages. Then we define the j-th error event as follows E(Wj)= E(W_j)=\ Wj∈g~(Ka,)∩Wj∉g(Ka,), W_j∈ g(K_a,y)∩ W_j∉ g(K_a,y), (1) or Wj repeated in g~(Ka,). or $W_j$ repeated in $ g(K_a,y)$\. As in [16] there are two type of errors: if, for a message WjW_j transmitted by some user, Wj∉g(Ka,)W_j∉ g(K_a,y) an error happens. If two or more users transmit the same message with the same decoding time, a collision happens, which may be counted as an error. However, as in [16] the probability of collision for realistic parameters is extremely small and does not influence performance. We require that the steady state error is bounded as 1Ka∑Wj∈g~(Ka,)ℙ(E(Wj))≤ϵ. 1K_a _W_j∈ g(K_a,y)P(E(W_j))≤ε. Notice that we do not require the error probability at each active user to be bounded by ϵε, only the all users’ averaged error probability. In the remainder of this paper we exclusively focus on the Gaussian MAC (GMAC). The received signal is given by =∑i=1Kai+,∼(0,n).y\;=\; _i=1^K_ac_i\;+\;Z, (0,I_n). (2) Following [16] we only care about obtaining a list of messages, not from which transmitters they are transmitted. In our case, the list g(Ka,)g(K_a,y) consisting of estimates of messages transmitted with KaK_a active users. It should be noted that the receiver is assumed to know how many messages are generated at each point in time, but not which users generate them. Without this knowledge, the receiver would have to estimate the number of active users KaK_a at each point of time. This has been considered in a number of papers [15, 3, 14, 5]. 3 Diffusion Denoiser Achievable Bound 3.1 Achievable Bound We propose a diffusion denoiser model at the receiver within joint decoding. We give a random-coding achievable bound below. This bound is first established by upper-bounding the probability of t-misdecoded messages, followed by an information density analysis to further refine the error probability. Theorem 1. (Diffusion Denoiser Random-coding Bound): Fix P′<P <P, there exists an (M,d,ϵ)(M,d,ε) random-access code with constants J∗≥0J_*≥ 0 and KE≥0K_E≥ 0 satisfying the power constraint P and ϵ≤∑t=1KatKamin(q1(t),q2(t))+q0, ε≤ _t=1^K_a tK_a (q_1(t),q_2(t))+q_0, (3) where q0=1M(Ka2)+KaΓ(n,nP/P′)Γ(n), q_0= 1M K_a2+K_a (n,nP/P ) (n), (4) q1(t)=maxρ2,ρ3∈[0,1] q_1(t)= _ _2, _3∈[0,1] (Kat)ρ1(M−Kt)ρ0ρ1⋅ K_at _1 M-Kt _0 _1· exp(−ρ0ρ12tP′+16v∗)(1+tP′8v∗)−nρ0ρ1, (- _0 _12tP +16v_* ) (1+ tP 8v_* )^-n _0 _1, (5) q2(t)=infψℙ[It≤ψ]+expn(tR1+R2)/2−ψ,q_2(t)= _ψ \P[I_t≤ψ]+ \n(tR_1+R_2)/2-ψ \ \, (6) It=minS0⊂[Ka],|S0|=t I_t= _S_0⊂[K_a],|S_0|=t n2log(1+P′t)+‖−c(S0c)‖221+P′t n2 (1+P t)+ ||y-c(S_0^c)||^2_21+P t −‖−c(S0)−c(S0c)‖22, -||y-c(S_0)-c(S_0^c)||^2_2, (7) where v∗=1+J∗+KE2v_*=1+J_*+K_E^2, R1=1ntlog(M−Kat)R_1= 1nt M-K_at, R2=1nlog(Kat)R_2= 1n K_at, ∼(,(1+KaP′)n)y (0,(1+K_aP )I_n) and Γ(x,y)=∫y∞zx−1e−zz (x,y)= _y^∞z^x-1e^-zdz. As in [16], we analyze the probability of error after we replace the measure over which the expectation is taken by the one under which: i) all active users transmit distinct messages; i) each user transmits ic_i instead of i‖i‖22≤nPc_i 1\||c_i||_2^2≤ nP\ under the true measure. The total variation dTVd_TV between the original measure and the new one can be upper-bound by: dTV d_TV =ℙ[Wj repeated]+ℙ[‖i‖2>nP,∀i∈Ka] =P[$W_j$ repeated]+P[||c_i||^2>nP,∀ i∈ K_a] (8) ≤1M(Ka2)+KaΓ(n,nP/P′)Γ(n)=q0, ≤ 1M K_a2+K_a (n,nP/P ) (n)=q_0, where (8) holds since ‖i‖2||c_i||^2 follows the Gamma distribution with shape n and scale P′P . Now after the change of measure, the probability of error can be bounded as ϵ≤∑t=1KatKaℙ[t-misdecoded]+q0. ε≤ _t=1^K_a tK_aP[t-misdecoded]+q_0. (9) We then proceed to bound ℙ[t-misdecoded]P[t-misdecoded] using two distinct methods. The first approach stems from the encoder’s perspective, yielding the term q1(t)q_1(t) in (17). Meanwhile, the second method adopts the decoder’s viewpoint, giving rise to the term q2(t)q_2(t) in (19). First, we define the sum-codewords as c(S)=∑i∈Sic(S)= _i∈ Sc_i. We assume t of KaK_a messages are mis-decoded, which is the same event as when we let S0⊂[Ka]S_0⊂[K_a] of messages be replaced with S0′⊂Ka+1,⋯,MS_0 ⊂\K_a+1,·s,M\ and |S0|=|S0′|=t|S_0|=|S_0 |=t. Then let F(S0,S0′)F(S_0,S_0 ) denote the set of (S0,S0′)(S_0,S_0 ) such that ‖c(S0)−c(S0′)+‖<‖\||c(S_0)-c(S_0 )+Z||<||Z||\ holds. Now, we define the residual of our diffusion denoiser D as ≜D()−∑i=1Ka. D(y)\;-\; _i=1^K_ac_i. (10) We set s()≜∇logpY()s(y)\; \;∇ p_Y(y) be the score function of recieved signal. The residual can be decomposed as =+s()+d(),R=Z+s(y)+d(y), where d()≜D()−(+s())d(y) \;D(y)- (y+s(y) ) is a mismatch term. Then, we rewrite the pairwise error event F(S0,S0′) F(S_0,S_0 ) =‖c(S0)−c(S0′)+‖<‖ =\||c(S_0)-c(S_0 )+R||<||R||\ =2hT+‖h‖2, =\2h^TR+||h||^2\, (11) where h=c(S0)−c(S0′)h=c(S_0)-c(S_0 ). We rewrite the error event as a union of F(S0,S0′)F(S_0,S_0 ) to have ℙ[t-misdecoded]≤ℙ[⋃S0∈(Kat)⋃S0′∈(M−Kat)F(S0,S0′)]. [t-misdecoded] [ _S_0∈ K_at _S_0 ∈ M-K_atF(S_0,S_0 ) ]. (12) By exponential Markov’s inequality, for any γ>0γ>0, we have ℙ \! [F(S0,S0′)|h,]≤e−γ‖h‖2 [F(S_0,S_0 )\, |\,h,Z ]≤ e^-γ\|h\|^2\, ([e−4γh⊤])1/2([e−4γh⊤s()])1/2([e−4γh⊤d()])1/2. (E[e^-4γ h Z ])^1/2 (E[e^-4γ h s(y) ])^1/2 (E[e^-4γ h d(y) ])^1/2. (13) BY direct computing with moment-generating function, there exist finite constants J∗≥0J_*≥ 0 and KE≥0K_E≥ 0 such that ℙ(F(S0,S0′)|h,)≤exp(−γ‖h‖2+4γ2v∗‖h‖2), \! (F(S_0,S_0 )\, |\,h,Z )≤ \! (-γ\|h\|^2+4γ^2v_*\|h\|^2 ), (14) where v∗= 1+J∗+KE2v_*\;=\;1+J_*+K_E^2. Optimizing over γ gives γ∗=1/(8v∗)γ^*=1/(8v_*) and the pairwise bound would be ℙ(F(S0,S0′)|h,)≤exp(−‖h‖216v∗).P\! (F(S_0,S_0 )\, |\,h,Z )\;≤\; \! (- \|h\|^216v_* ). (15) Since h|t∼(0,2tP′In)h\,|\,t (0,2tP I_n), we have ℙ(F(S0,S0′)|h,)≤exp(−12tP′+16v∗)(1+tP′8v∗)−n.P\! (F(S_0,S_0 )\, |\,h,Z )\;≤\; (- 12tP +16v_* ) (1+ tP 8v_* )^-n. (16) Using Gallager’s ρ-trick for any ρ0∈[0,1],ρ1∈[0,1] _0∈[0,1], _1∈[0,1] to get ℙ[t [t -misdecoded]≤(Kat)ρ1(M−Kat)ρ0ρ1⋅ -misdecoded]≤ K_at _1 M-K_at _0 _1· exp(−ρ0ρ12tP′+16v∗)(1+tP′8v∗)−nρ0ρ1=q1(t) (- _0 _12tP +16v_* ) (1+ tP 8v_* )^-n _0 _1=q_1(t) (17) Second, we introduce an alternative bound for ℙ[t-misdecoded]P[t-misdecoded]. The information density can be defined as it(c(S0);|c(S0c))= i_t(c(S_0);y|c(S_0^c))= 1nlog(1+P′t)+‖−c(S0c)‖221+P′t 1n (1+P t)+ ||y-c(S_0^c)||^2_21+P t −‖−c(S0)−c(S0c)‖22. -||y-c(S_0)-c(S_0^c)||^2_2. (18) Note that F(S0,S0′)F(S_0,S_0 ) is equivalent to it(c(S0′);|c(S0c))>it(c(S0);|c(S0c))\i_t(c(S_0 );y|c(S_0^c))>i_t(c(S_0);y|c(S_0^c))\. Let It=minS0it(c(S0);|c(S0c))I_t= _S_0i_t(c(S_0);y|c(S_0^c)). For a fixed ψ, we have ℙ[t-misdecoded] [t-misdecoded] ≤infψ(ℙ[It≤ψ] ≤ _ψ(P[I_t≤ψ] (19) +(Kat)(M−Kat)e−ψ)=q2(t). + K_at M-K_ate^-ψ)=q_2(t). This completes the proof of Theorem 1. Now we have a achievable bound for random-coding with constants J∗J_* and KEK_E. In practice, we need to train a score-diffusion network sθ()s_θ(y). The denoiser samples the trained score network to give an optimization of J∗J_* and KEK_E as follows. 3.2 Diffusion Model Diffusion models define a generative framework by coupling a forward diffusion process with a learned reverse denoising process. In discrete time, the forward process gradually perturbs a clean data point x0∼pdatax_0 p_data using a predefined variance schedule βll=1T\ _l\_l=1^T: q(xl∣xl−1)=(1−βlxl−1,βlI),l=1,…,T,q(x_l x_l-1)=N\! ( 1- _l\,x_l-1,\, _lI ), l=1,…,T, (20) which admits the closed form q(xl∣x0)=(α¯lx0,(1−α¯t)I),α¯l=∏s=1l(1−βs).q(x_l x_0)=N\! ( α_l\,x_0,\,(1- α_t)I ), α_l= _s=1^l(1- _s). (21) As l→Tl→ T, the distribution approaches an isotropic Gaussian, effectively destroying the input signal. An equivalent continuous-time formulation expresses the forward dynamics as a stochastic differential equation (SDE): dxl=f(xl,l)dl+u(t)dwl,l∈[0,T],dx_l=f(x_l,l)\,dl+u(t)\,dw_l, l∈[0,T], (22) where wlw_l is a standard Wiener process. The corresponding reverse-time SDE is given by dxl=[f(xl,l)−u(t)2∇xllogql(xl)]dl+u(l)dw¯l,dx_l= [f(x_l,l)-u(t)^2 _x_l q_l(x_l) ]\,dl+u(l)\,d w_l, (23) where l∈[T,0]l∈[T,0] and w¯l w_l denotes a reverse-time Wiener process, and ∇xllogql(xl) _x_l q_l(x_l) is the score function guiding the generative dynamics. Since the true score is intractable, it is approximated by a neural network sθ(xl,l)s_θ(x_l,l) trained via denoising score matching: ℒscore(θ)=t∼[0,T] _score(θ)=E_t [0,T]\, x0∼pdata,xt∼qt(xt|x0)[λ(t) _x_0 p_data,\,x_t q_t(x_t|x_0) [λ(t) (24) ∥sθ(xt,t)−∇xtlogqt(xt|x0)∥2], \,\|s_θ(x_t,t)- _x_t q_t(x_t|x_0)\|^2 ], where λ(l)λ(l) is a time-dependent weighting term. In practice, this objective reduces to predicting the Gaussian noise that was added during the forward process. Writing xl=α¯lx0+1−α¯lϵ,ϵ∼(0,I),x_l= α_l\,x_0+ 1- α_l\,ε,ε (0,I), as the loss becomes the simplified noise-prediction objective: ℒnoise(θ)=x0,ϵ,l[‖ϵ−ϵθ(xl,l)‖2].L_noise(θ)=E_x_0,ε,l\! [ \|ε- _θ(x_l,l) \|^2 ]. (25) Thus, training a diffusion model amounts to teaching the network to recover the injected noise ϵε from a corrupted sample (xl,l)(x_l,l). During inference, sampling begins with Gaussian noise xT∼(0,I)x_T (0,I), and the learned reverse dynamics are applied iteratively to denoise lT→l0l_T→ l_0. Equivalent ODE-based formulations provide both stochastic and deterministic generation procedures. In our model, we set the observation with the diffusion state at level l∗l^* with sθ()≜sθ(,l∗)s_θ(y) s_θ(y,l^*). J∗J_* and KEK_E characterize the concentration of the score function and the variance of the denoiser mismatch, respectively. Both parameters are estimated empirically from finite samples of the score network outputs and residuals. To approximate the spectral bound of the score covariance, we first sample N of the channel outputs y to have (k)y^(k). For each sample, the trained score network produces sθ(k)=sθ((k))s_θ^(k)=s_θ(y^(k)). The empirical second moment is then given by J^=1N∑k=1Nsθ(k)sθ(k)T J= 1N _k=1^Ns_θ^(k)s_θ^(k)T. Then, we extract the maximum eigenvalue via power iteration as J∗=λmax(J^).J_*= _ ( J). This procedure ensures that the largest variance direction of the score distribution is captured reliably. For the denoiser residual KEK_E, we employ a similar process. We can compute the mismatch residual using sθ()s_θ(y) as Eθ()=D()−(+sθ()).E_θ(y)=D(y)-(y+s_θ(y)). By sampling (k)∼y^(k) , we calculate the residuals r(k)=Eθ((k))r^(k)=E_θ(y^(k)). Then, we can have the empirical covariance matrix as KE=1N∑k=1Nr~(k)r~(k)TK_E= 1N _k=1^N r^(k) r^(k)T, where r~=r(k)−1N∑kr(k) r=r^(k)- 1N _kr^(k). Here the top eigenvalue of KEK_E provides a conservative estimate of the residual’s directional sub-Gaussian parameter. Our J∗J_* and KEK_E require only forward evaluation of the trained score network and the denoiser. These constants can then be directly integrated into the finite-blocklength bound. 4 SIMULATION RESULTS Fig. 1: The required energy per bit to achieve ϵ=0.001ε=0.001 as a function of the number of active users for Theorem 1 and different achievable schemes. First, we plot numerical results for the bounds developed above in Figure. 1. We compare various strategies in the following settings. Each active users is transmitting k=100k=100 bits of information. The frame length is n=30000n=30000 and the target per-user probability of error is 0.0010.001. We first compare our achievable bound against the classical result of [16]. Our bound reduces the required energy by approximately 0.80.8 dBdB. We then instantiate Theorem 1 for two representative schemes, RCU-TIN and SPARC [6]. In each case, the required energy drops by about 0.50.5 dBdB at the same error target. Fig. 2: The bounds on the probability of error ϵε as a function of Eb/N0E_b/N_0 for Theorem 1 and different achievable schemes. Second, we compare our approach with [8, 1, 7] across a range of target error probabilities in Figure. 2. For each decoder, augmenting with Theorem 1 reduces the required energy to meet a given error target by approximately 0.50.5 dBdB relative to the original scheme. This consistent gain demonstrates that our theorem is compatible and effective across state-of-the-art coding designs. 5 CONCLUSIONS We presented a decoder compatible diffusion denoiser that is integrated into the joint decoding for URA in the finite-blocklength regime. First, we derived a diffusion denoiser achievable bound that is strictly tighter than [16]. 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