SlowGuess/NeurIPS_2023_OCR huggingface dataset:
To implement the LMMSE estimator described above, one simply requires the covariances  $C_{yy}, C_{ss}, C_{bb}$ . Under the problem formulation, one could potentially obtain the sample covariance, whose precision scales with the number of samples used. In the case of the synthetic signals, we assume oracle knowledge of the covariance when computing this baseline, so as to eliminate confounding effects from poor estimation of these statistics. However, for CommSignal2, we assume that the signal is zero-mean (validated with the empirical mean), and compute the autocorrelation matrix for time windows of length 2560, across 10,000 sample realizations drawn randomly from the frames provided in the dataset. The empirical covariance estimate that we used will be provided in the accompanying code/data repository.
Note that in this problem, the interference also has random time offsets, which has to be accounted for when computing or estimating the covariance of our interference signals. The works in [43, 12] address the effects of such random time offsets in detail, and discuss the suboptimality of the above LMMSE in such a setting. We acknowledge that without an additional time synchronization step, the LMMSE estimator adopted herein may be modeling the interference as time series from a stationary process.
We remark that the LMMSE estimator is optimal if the components were Gaussian. However, as digital communication signals contain some underlying discreteness and undergo unknown timeshifts, these signals are typically non-Gaussian (and often, even far from Gaussian). Hence, better performance can generally be obtained through nonlinear methods.
F BASIS for RF Source Separation
We use the BASIS method [5] as a learning-based benchmark for SCSS in our experiments. Similar to our method, BASIS leverages pre-trained generative models as statistical priors and does not rely on end-to-end (supervised) training with paired data. The manner in which these priors are used for separation, however differs:
1. Soft constraint vs. hard constraint: In the context of our problem formulation, we are interested in decomposing a mixture  $\mathbf{y} = \mathbf{s} + \kappa \mathbf{b}$  into its constituent components. As described in §3.2 our proposed method recovers the components by extending a MAP estimation rule with a hard constraint given by,
$$
\widehat {\mathbf {s}} = \underset {\mathbf {s} \in S \text {s . t} \mathbf {y} = \mathbf {s} + \kappa \mathbf {b}} {\arg \max } p _ {\mathbf {s} | \mathbf {y}} (\mathbf {s} | \mathbf {y}), \tag {68}
$$
where the resulting estimate of  $\mathbf{b}$  is  $\widehat{\mathbf{b}} = (\mathbf{y} - \widehat{\mathbf{s}}) / \kappa$ . The result of imposing this constraint is a MAP objective that leverages two priors but only requires optimization with a single
Algorithm 2 BASIS Separation (original, as proposed in [5])
1: function SEPARATION(y,  $\kappa$ , N,  $\{\sigma_t^2\}_{t=1}^T$ ,  $\psi_s^{(0)}$ ,  $\psi_b^{(0)}$ ) ▷ Specially tuned noise schedule  $\{\sigma_t\}_{t=1}^T$   
2: for  $t \gets 1, T$  do  
3:  $\psi_s^{(t)} \gets \psi_s^{(t-1)}$ ,  $\psi_b^{(t)} \gets \psi_b^{(t-1)}$ ,  $\eta_t \gets f(\sigma_t^2)$   
4: for  $i \gets 0, N-1$  do  
5:  $\epsilon_s, \epsilon_b \sim \mathcal{N}(0, \mathbf{I})$   
6:  $\widehat{\mathbf{z}}_s, \widehat{\mathbf{z}}_b = r_{\phi_s}\left(\psi_s^{(t)}, t\right), r_{\phi_b}\left(\psi_t^{(t)}, t\right)$   
7:  $\psi_s^{(t)} \gets \psi_s^{(t)} - \eta_t\underbrace{(\widehat{\mathbf{z}}_s / \sigma_t)}_{\text{approx. score}} - \eta_t\left(\mathbf{y} - \psi_s^{(t)} - \kappa \psi_b^{(t)}\right) / \sigma_t^2 + \sqrt{2\eta_t} \epsilon_s$   
8:  $\psi_b^{(t)} \gets \psi_b^{(t)} - \eta_t(\widehat{\mathbf{z}}_b / \sigma_t) - \eta_t\kappa\left(\mathbf{y} - \psi_s^{(t)} - \kappa \psi_b^{(t)}\right) / \sigma_t^2 + \sqrt{2\eta_t} \epsilon_b$   
9: end for  
10: end for  
11: return  $\widehat{\mathbf{s}} = \psi_s^{(T)}$ ,  $\widehat{\mathbf{b}} = \psi_b^{(T)}$   
12: end function
variable,
$$
\begin{array}{l} \underset {\mathbf {s} \in \mathcal {S} \text {s . t} \mathbf {y} = \mathbf {s} + \kappa \mathbf {b}} {\arg \max } p _ {\mathbf {s} | \mathbf {y}} (\mathbf {s} | \mathbf {y}) = \underset {\mathbf {s} \in \mathcal {S} \text {s . t} \mathbf {y} = \mathbf {s} + \kappa \mathbf {b}} {\arg \max } p _ {\mathbf {y} | \mathbf {s}} (\mathbf {y} | \mathbf {s}) P _ {\mathbf {s}} (\mathbf {s}) (69a) \\ = \underset {\mathbf {s} \in \mathcal {S}} {\arg \min } - \log P _ {\mathbf {s}} (\mathbf {s}) - \log p _ {\mathbf {b}} ((\mathbf {y} - \mathbf {s}) / \kappa). (69b) \\ \end{array}
$$
In contrast, BASIS strives to estimate the underlying components by sampling from the posterior. In particular, they do not enforce a hard constraint during sampling and instead assume a likelihood of the form
$$
p _ {\mathbf {y} | \mathbf {s}, \mathbf {b}} ^ {\mathrm {B A S I S}} (\mathbf {y} | \mathbf {s}, \mathbf {b}) = \mathcal {N} (\mathbf {y}; \mathbf {s} + \kappa \mathbf {b}, \gamma^ {2} \mathbf {I}), \tag {70}
$$
which actually corresponds to an alternate mixture model with auxiliary noise  $\mathbf{w} \sim \mathcal{N}(0, \gamma^2\mathbf{I})$  such that,
$$
\mathbf {y} = \mathbf {s} + \kappa \mathbf {b} + \mathbf {w}. \tag {71}
$$
Thus, the estimates  $\widehat{\mathbf{s}}^{\mathrm{BASIS}}$  and  $\widehat{\mathbf{b}}^{\mathrm{BASIS}}$  are obtained by sampling from the posterior,
$$
p _ {\mathbf {s}, \mathbf {b} | \mathbf {y}} ^ {\text {B A S I S}} (\mathbf {s}, \mathbf {b} | \mathbf {y}) = p _ {\mathbf {y} | \mathbf {s}, \mathbf {b}} ^ {\text {B A S I S}} (\mathbf {y} | \mathbf {s}, \mathbf {b}) P _ {\mathbf {s}} (\mathbf {s}) p _ {\mathbf {b}} (\mathbf {b}). \tag {72}
$$
Notice that sampling from this posterior requires computing two separate estimates due to the soft constraint in (71).
2. Annealed Langevin dynamics vs. randomized levels of Gaussian smoothing: The aforementioned BASIS estimates satisfy  $\mathbf{y} = \widehat{\mathbf{s}}^{\mathrm{BASIS}} + \kappa \widehat{\mathbf{b}}^{\mathrm{BASIS}}$  only when  $\gamma \rightarrow 0$  in the soft constraint in (71) enforced via the likelihood term. As shown in Algorithm 2, the idea behind BASIS is to leverage annealed Langevin dynamics sampling to sample from the posterior by slowly decreasing the value of  $\gamma$  towards zero by tuning a specially chosen noise schedule. While generally a separate noise schedule is required per prior, the authors suggest sharing the same schedule across both priors. Apart from tuning the noise schedule, the learning rate schedule is another hyperparameter that must also be tuned. In contrast, our algorithm leverages the existing noise schedule from pre-trained diffusion models in a randomized fashion without any additional tuning.
3. Optimization via sampling vs. optimization via gradient descent: Starting with initializations  $\psi_s^{(0)}$  and  $\psi_b^{(0)}$  for the SOI and interference respectively, BASIS refines the estimate such that  $\psi_s^{(t)}$  approximates a sample drawn from the distribution of the smoothened source  $p_{\tilde{s}_t}$ . Langevin dynamics leverages the approximate score of  $p_{\tilde{s}_t}$ , to update the estimate by sampling from higher density regions. A similar update is performed for the interference. The estimates are constrained to satisfy (71) through the Gaussian likelihood.
In constraint, as described in Algorithm 1 (see §3.2.2), we obtain an estimate  $\pmb{\theta}^{(t)}$  of the unsmoothened SOI for each randomly drawn noise level. The hard constraint allows us to readily get an estimate for the interference as  $(y - \pmb{\theta}^{(t)}) / \kappa$ . As described in §3.2.2, we then update the SOI estimate using gradient descent, with a rule guided by (69b), using the gradients of the log densities with respect to  $\pmb{\theta}$  (which is functionally equivalent to the scaled scores). The ideal update gradient is,
$$
\nabla_ {\boldsymbol {\theta}} \mathcal {L} (\boldsymbol {\theta}) := - \mathbb {E} _ {\mathrm {t}, z _ {t}} \left[ \sqrt {\alpha_ {t}} S _ {\tilde {\mathbf {s}} _ {t}} \left(\tilde {\mathbf {s}} _ {t} (\boldsymbol {\theta})\right) \right] + \frac {\omega}{\kappa} \mathbb {E} _ {\mathrm {u}, z _ {u}} \left[ \sqrt {\alpha_ {u}} S _ {\tilde {\mathbf {b}} _ {u}} \left(\tilde {\mathbf {b}} _ {u} (\boldsymbol {\theta}, \mathbf {y})\right) \right]. \tag {73}
$$
We first implement BASIS in its original form, with our best attempt at finding an appropriate learning rate. As shown in Figure 13, on QPSK (SOI) + OFDM (QPSK) mixture, this leads to worse performance than matched filtering—likely due to poor matching of the observation model in (71). Note that we are limited in the annealing schedule by the set of possible noise levels, based on the training setup of our diffusion models. Thus, we use the diffusion model training noise schedule discretized into  $T = 50$  timesteps (see §5.2) and additionally use  $N = 100$ . We seek to use the largest set of noise levels possible, consistent with approaches in most Langevin sampling based works. We tuned the learning rate schedule to the best of our abilities and found that  $\eta_{t} = f(\sigma_{t}^{2}) = (2\mathrm{e} - 8)\sigma_{t}^{2} / \sigma_{1}^{2}$ , gave the best results. All curves use an analytical score model for the SOI.
To make an even more conservative comparison (w.r.t. our proposed method), we replace the posterior using (72), which corresponds to the formulation in (71), with a hard constraint as in our setup. This
Figure 13: Comparing BASIS and Modified BASIS against conventional RF source separation methods. The vanilla BASIS algorithm performs worse than MF, while the Modified BASIS algorithm, which is inspired by our usage of an  $\alpha$ -posterior, is able to beat LMMSE. Still, our method outperforms all baselines.
Algorithm 3 BASIS Separation, using (69b)  
1: function SEPARATION(y,  $\kappa$ , N,  $\{\sigma_t^2\}_{t=1}^T$ ,  $\pmb{\theta}^{(0)}$ ) ▷ Specially tuned noise schedule  $\{\sigma_t\}_{t=1}^T$   
2: for  $t \gets 1, T$  do  
3:  $\pmb{\theta}^{(t)} \gets \pmb{\theta}^{(t-1)}$ ,  $\eta_t \gets f(\sigma_t^2)$   
4: for  $i \gets 0, N-1$  do  
5:  $\epsilon \sim \mathcal{N}(0, \mathbf{I})$   
6:  $\widehat{\mathbf{z}}_s, \widehat{\mathbf{z}}_b = r_{\phi_s}\left(\pmb{\theta}^{(t)}, t\right)$ ,  $r_{\phi_b}\left(\left(\mathbf{y} - \pmb{\theta}^{(t)}\right) / \kappa, t\right)$   
7:  $\pmb{\theta}^{(t)} \gets \pmb{\theta}^{(t)} \underbrace{-\eta_t(\widehat{\mathbf{z}}_s / \sigma_t) + \frac{1}{\kappa}\eta_t(\widehat{\mathbf{z}}_b / \sigma_t)}_{\text{from MAP objective}} + \sqrt{2\eta_t}\epsilon$   
8: end for  
9: end for  
10: return  $\widehat{s} = \pmb{\theta}^{(T)}$ ,  $\widehat{b} = \left(\mathbf{y} - \pmb{\theta}^{(T)}\right) / \kappa$   
11: end function
Algorithm 4 Modified BASIS Separation by introducing  $\alpha$ -posterior  
1: function SEPARATION(y,  $\kappa$ , N,  $\{\sigma_t^2\}_{t=1}^T$ ,  $\pmb{\theta}^{(0)}$ ) ▷ Specially tuned noise schedule  $\{\sigma_t\}_{t=1}^T$   
2: for  $t \gets 1, T$  do  
3:  $\pmb{\theta}^{(t)} \gets \pmb{\theta}^{(t-1)}, \eta_t \gets f(\sigma_t^2)$   
4: for  $i \gets 0, N-1$  do  
5:  $\epsilon \sim \mathcal{N}(0, \mathbf{I})$   
6:  $\widehat{\mathbf{z}}_s, \widehat{\mathbf{z}}_b = r_{\phi_s}\left(\pmb{\theta}^{(t)}, t\right), r_{\phi_b}\left(\left(\mathbf{y} - \pmb{\theta}^{(t)}\right) / \kappa, t\right) \triangleright$  Compute scores at noise level  $\sigma_t$   
7:  $\pmb{\theta}^{(t)} \gets \pmb{\theta}^{(t)} \underbrace{-\eta_t(\widehat{\mathbf{z}}_s / \sigma_t) + \frac{\omega}{\kappa} \eta_t(\widehat{\mathbf{z}}_b / \sigma_t)}_{\text{from MAP objective with } \alpha\text{-posterior}} + \sqrt{2\eta_t} \epsilon$   
8: end for  
9: end for  
10: return  $\widehat{\mathbf{s}} = \pmb{\theta}^{(T)}, \widehat{\mathbf{b}} = (\mathbf{y} - \pmb{\theta}^{(T)}) / \kappa$   
11: end function
method, as further described in Algorithm 3, leads to better performance, even outperforming LMMSE in some cases. We also experiment with  $N = 400$  (for a total of 20,000 iterations), and do not observe significant gains. Empirically we found this method to be typically stuck in local optimum which is in line with our characterization earlier in §4.
Finally, we augment the BASIS algorithm with an  $\alpha$ -posterior with  $\alpha = \omega$ , as described in Algorithm 4. As shown in Figure 13, the  $\alpha$ -posterior augmented method obtains better performance than vanilla BASIS, but is generally unable to reach the performance of our method. We emphasize again that in contrast to our method, the aforementioned BASIS-based methods follow a prescribed annealing
schedule, that can require cumbersome tuning for optimal performance. The implementations of Algorithm 3 and Algorithm 4 both use a learning rate schedule of  $\eta_t = f(\sigma_t^2) = (2\mathrm{e} - 6)\sigma_t^2 /\sigma_1^2$
G Complete Results
In this section we will provide the complete results of our study. We first present another baseline based on simulating the reverse diffusion process [21] as a denoiser, that complements the baselines already described in §5.2.
Reverse Diffusion. Given a mixture  $\mathbf{y}$ , we interpret the SOI as (scaled) additive noise on top of the interference  $\mathbf{b}$ . Since the reverse diffusion process can be interpreted as an iterative denoiser (see §2.2), we run a small chain of reverse diffusion for 10 timesteps starting at a noise variance of 0.005 and ending with variance of 0.0001, which was chosen based on different trials. Similar ideas have been used in prior works involving inverse problems for images [31, 51]. We note that it is generally cumbersome to find the optimal reverse diffusion noise range and it might in fact be dependent on the specific mixture's SIR.
Extracting the underlying bits. Our primary metric for assessing the quality of the estimated SOI is through the BER between the estimated bits and the truly encoded bits. The estimate of the SOI typically equals the true SOI with a very small amount of noise due to the continuous nature of the optimization problem. Thus, we can expect the underlying constellation to contain some symbols concentrated around, but not exactly at the constellation points. This can be modeled as the true SOI corrupted with a small amount of Gaussian noise, for which MF is the optimal technique to decode the bits by mapping the symbols to the closest constellation point (see Section E.1).
The analytical SOI score. We use an RRC-QPSK signal as the SOI across all our source separation experiments. As detailed in Section B.2, the score for a smoothened QPSK source can be analytically computed via Proposition 3, where it is more amenable to model and compute it in the symbol space (i.e., as i.i.d. symbols). Nevertheless, for the problem of separating the SOI from an interference source, we have to consider the components jointly in the time domain. Thus, we relate the time-domain representation to the symbols via  $\mathbf{s} = \mathbf{H}\mathbf{a}$ , where  $\mathbf{H}$  represents the RRC filter matrix and  $\mathbf{a}$  is a vector of symbols. To compute this analytical score, we smooth the symbols via a Gaussian smoothing model at noise level corresponding to timestep  $t$ ,
$$
\tilde {\mathbf {s}} _ {t} = \boldsymbol {H} \tilde {\mathbf {a}} _ {t} := \boldsymbol {H} \left(\gamma_ {t} \mathbf {a} + \sigma_ {t} \mathbf {z} _ {s}\right). \tag {74}
$$
With this relation, we express the score of the smoothened source as,
$$
\begin{array}{l} \nabla_ {\tilde {\mathbf {s}} _ {t}} \log p _ {\tilde {\mathbf {s}} _ {t}} (\tilde {\mathbf {s}} _ {t}) = \boldsymbol {H} \cdot \nabla_ {\tilde {\mathbf {a}} _ {t}} \log p _ {\tilde {\mathbf {a}} _ {t}} (\tilde {\mathbf {a}} _ {t}) (75) \\ = \boldsymbol {H} \cdot \left(\frac {1}{\sigma_ {t} ^ {2}} \left(- \tilde {\mathbf {a}} _ {t} + \sum_ {\mathbf {a} \in \mathcal {A} ^ {K}} \mathbf {a} \odot \phi_ {t} (\mathbf {a}; \tilde {\mathbf {a}} _ {t})\right)\right) (76) \\ \approx \frac {1}{\sigma_ {t} ^ {2}} \left(- \tilde {\mathbf {s}} _ {t} + \boldsymbol {H} \sum_ {\mathbf {a} \in \mathcal {A} ^ {K}} \mathbf {a} \odot \phi_ {t} \left(\mathbf {a}; \boldsymbol {H} ^ {\dagger} \tilde {\mathbf {s}} _ {t}\right)\right). (77) \\ \end{array}
$$
where  $\odot$  represents element-wise product,  $\phi_t$  is the softmax-like operator defined in (34) that is applied element-wise here, and  $H^\dagger$  is the pseudo-inverse of  $H$ . Note that (76) is obtained by applying (33) to a vector of i.i.d. smoothened QPSK symbols. In our implementation, the estimate of these smoothened symbols  $\tilde{\mathbf{a}}_t$  is obtained by reversing the RRC filter using  $H^\dagger$ , as in (77).
On the other hand, our RRC-QPSK diffusion model is trained directly on the time-domain waveform, and can be used directly to separate the SOI waveform from the mixture without conversion to the symbol domain. This once again sheds light on the practicality of using data-driven methods to circumvent otherwise computationally challenging statistical modeling technical problems.
Results. Figures 14 and 15 show the complete source separation results for mixtures with OFDM and CommSignal2 as interference, respectively. Our model that uses an analytical SOI score for the SOI and a diffusion-based score for the interference generally performs the best and outperforms all baselines. Furthermore, we show that using a learned SOI score still outperforms all baselines in
Figure 14: Comparing our method against various baselines on separating (top) RRC-QPSK + OFDM (BPSK) mixtures and (bottom) RRC-QPSK + OFDM (QPSK) mixtures. Our model that uses an analytical SOI score outperforms all baselines in terms of BER. Reverse diffusion is competitive at low SIRs since the interference dominates the mixture in this regime and hence iterative denoising to separate the SOI is effective. The trained SOI diffusion models significantly outperform all baselines at high SIR. The reason the analytical SOI performs slightly worse is due to the approximations we make in (77).
Figure 15: Comparing our method against various baselines on separating RRC-QPSK + CommSignal2 mixtures. Our method significantly outperforms all baselines in terms of BER. The large MSE at low SIRs suggests that there is background noise present in the CommSignal2 source, which is amplified at lower SIR (large  $\kappa$ ). We try to estimate the amount of noise and model it as additive Gaussian noise. Accounting for this noise could presumably lead to a lower bound on the BER, shown by dotted black line on the left.
terms of BER, despite the slight degradation. The trained SOI score models consistently outperform all methods at high SIR in terms of BER since the SOI diffusion model was trained with RRC-QPSK samples as opposed to the approximations made in (77) in implementing the analytical score.
Our method also outperforms baselines in terms of MSE for OFDM mixtures as shown in Figure 14. The performance at low SIR in the context of CommSignal2 mixtures is not the same. As shown in Figure 15, the large MSE at low SIRs suggest that there is background noise present in the CommSignal2 sources that is amplified for large values of  $\kappa$ , i.e., the interference is actually of the form  $\mathbf{b} + \mathbf{w}$  for some background noise  $\mathbf{w}$ . We validated that this was indeed the case, by visualizing
samples in both the time-domain and frequency domain using the RF challenge demo notebook<sup>10</sup>. As shown in Figure 16, we noticed segments of lower magnitude at the start and end, which we believe to be background noise that is not part of intended communication signal. During source separation, this presumably results in a noisier estimate of the SOI in comparison to mixtures with no additional background noise. We estimated the SNR to be  $16.9\mathrm{dB}$  by averaging across multiple samples. The dotted black curve on the left of Figure 15 is a presumable lower bound on the BER by accounting for the magnitude of the background noise and modeling it as additive white Gaussian noise.
Figure 16: Time-domain and frequency-domain plots of a single frame from the CommSignal2 dataset. Left: In the time-domain we observe segments of lower magnitude at the start and end, which we believe to be background noise. Right: In the frequency-domain we observe the bandlimited communication signal with spectrally flat features in the regions we identified as background noise.
Nevertheless, across all experiments, we demonstrate that our method sets a new state-of-the-art for applications such as interference mitigation where the interference can be learned from data recordings and the desired information can be decoded with knowledge of the SOI demodulation pipeline.
Figure 17: Top: BER and MSE versus  $\omega/\kappa^2$  for different SIR levels. Evidently, a good choice of  $\omega$ , on average, across different noise levels is  $\omega = \kappa^2$ . Bottom: By looking at individual SIRs we see that the minimum BER and MSE is achieved when  $\omega$  increases with increasing  $\kappa^2$  (decreasing SIR), visualized using a log scale on the x-axis.
$^{10}$ https://github.com/RFChallenge/rfchallenge_singlechannelstarter/blob/main/ notebook/Demo.ipynb
Figure 18: Comparing our method against a supervised learning setup that trains a UNet on paired data samples with an  $\ell_2$  loss. We show that we are able to achieve competitive BERs, and even outperform the supervised method at certain SIRs. Our method is particularly competitive in the challenging strong interference regime (low SIR), demonstrating the fidelity of our trained interference diffusion models. The gap is larger for CommSignal2 mixtures as the supervised method is presumably able to leverage knowledge of the joint statistics between the SOI and interference to effectively deal with the background noise in the interference signal.
Choice of  $\omega$ . We numerically verify that  $\omega = \kappa^2$  is a good choice of the  $\alpha$ -posterior term. To this end, we first find a suitable order of magnitude for  $\omega$ , by varying  $\omega / \kappa^2$  between  $10^{-2}$  and  $10^2$  across different noise levels. As shown in the top row in Figure 17, on average, the minimum BER and MSE is achieved when  $\omega = \kappa^2$ . We additionally validate that it is beneficial to adapt  $\omega$  as the SIR changes by varying  $\omega$  and studying the BER and MSE curves at individual noise levels. As shown in the bottom row of Figure 17, we observe that it  $\omega$  should increase with  $\kappa^2$  to achieve good results.
Comparing with supervised methods. As motivated in §1, we are interested in leveraging independently trained priors in our source separation setup. Nevertheless, we compare against a supervised setup that learns to separate mixtures end-to-end by training a UNet on paired data. We train three supervised models based on the recent work in [43], using their open-sourced training code<sup>11</sup>. These models are trained with an  $\ell_2$  loss. As such, we should expect the MSE performance to be better than our unsupervised approach, which is indeed the case across all the mixtures as shown in Figure 18. However, we notice that our method, which uses an analytical SOI score, is able to perform similarly to the UNet and even better in terms of BER for some SIRs, especially in the OFDM interference setting. Furthermore, in the challenging low SIR (strong interference regime) our method performs
<sup>11</sup>https://github.com/RFChallenge/SCSS_CSGaussian
well, showing that our interference diffusion models were able to learn the underlying statistical structures. Thus, if the mixture model changes in the future we can re-use our priors whereas the supervised approached could require an entire change to the architecture and additional training. We can potentially drive the BER in the CommSignal2 setting by modeling the background noise in these signals either through a different mixture model or through additional priors might help in driving down the BER. This will be a focus of future work.