1 Introduction
Density estimation is the fundamental statistical task of estimating the density of a distribution given a finite set of measurements. However, in many scientific fields (see examples in Carroll et al., 2006), one only has access to a noisecorrupted set of measurements. Given knowledge of the statistics of the noise, density deconvolution methods attempt to recover the density function of the unobserved noisefree samples rather than the noisy measurements.
In this work we consider the problem of additive noise, where observed samples are produced by adding independent noise to unobserved values ,
(1) 
We also assume that the density function of the noise is perfectly known for every observation. The density function of observations is then a convolution
(2) 
When the noise distribution is constant, we could estimate the density of the observations with any density estimator and then solve Equation 2
, e.g. with a kernel density estimator using Fourier transforms
(e.g., Liu & Taylor, 1989; Carroll & Hall, 1988; Fan, 1991; Devroye, 1989) or wavelet decompositions (Pensky & Vidakovic, 1999).When the noise distribution is different for each observation, we only have one sample from each convolved density. This extreme deconvolution setting (Bovy et al., 2011) has previously been tackled by fitting a Gaussian Mixture Model (GMM) to the underlying density . When the noise distributions are all Gaussian, the marginal likelihood is tractable, and the GMM can be fitted by ExpectationMaximisation (EM, Bovy et al., 2011)
or Stochastic Gradient Descent
(SGD, Ritchie & Murray, 2019).Given enough mixture components, any density function can be approximated arbitrarily closely using GMMs. In practice, however, other representations of densities can be easier to fit, and often generalize better. There is growing interest in normalizing flows (Tabak & VandenEijnden, 2010; Tabak & Turner, 2013), a class of methods that transform a simple source density into a complex target density. Normalizing flows are an efficient alternative to GMMs (e.g., Rezende & Mohamed, 2015), providing both good scalability and high expressivity, and have shown promise in applications similar to density deconvolution (Cranmer et al., 2019).
In this work, we model the underlying density with a normalizing flow. The marginal likelihood is intractable, so we resort to approximate inference. We use amortized variational inference (Section 2), closely following Variational AutoEncoders (VAEs, Kingma & Welling, 2014; Rezende et al., 2014). Unlike for VAEs, in our framework, the model between the latent and observed vectors is fixed.
In this proof of concept, we use a fixed Gaussian noise distribution, but our approach would also allow us to use arbitrary and varying noise distributions, as found in realistic applications (e.g., Anderson et al., 2018). In a setting wellsuited to the existing Gaussian mixture approach, we find that fitting flows is harder (Section 4.1), possibly motivating further work on approximate inference in this setting. Nevertheless, on real data, we demonstrate that flows can already outperform GMMs for density deconvolution (Section 4.2).
2 Methods
We take a variational approach (Jordan et al., 1999) to density deconvolution. Introducing an approximate posterior gives a lower bound to the logmarginal likelihood
(3)  
(4)  
(5) 
where is the evidence lower bound (ELBO, see Appendix A). Our approximate posterior , a recognition network, represents beliefs about an underlying value given an observation and the parameters of the noise.
The ELBO gives a unified objective for both the parameters of the model and the parameters
of the recognition network. Stochastic gradient descent only needs unbiased estimates of the ELBO, which we obtain by Monte Carlo
(6) 
where Monte Carlo samples are simulated .
Our variational approach follows that of Variational AutoEncoders (VAEs, Kingma & Welling, 2014; Rezende et al., 2014), which provide a framework for amortized variational inference in graphical models. The focus of VAEs, however, is usually to build generative models matching the observations . In contrast, our main target is estimating an underlying density function . For certain applications, e.g. denoising a noisy measurement, we may also be interested in the approximate posterior . Another difference between our method and VAEs is that we do not learn a likelihood model between the latent variables and the observations . Instead, our “likelihood model” is fullycharacterized by the problem itself as .
We model both and as normalizing flows. Normalizing flows model probability density functions by transforming a source density into a target density using an invertible, differentiable transformation
(7) 
The density of can be computed using the changeofvariable formula
(8) 
In particular, we model with Masked Autoregressive Flows (MAF, Papamakarios et al., 2017), and with Inverse Autoregressive Flows (IAF, Kingma et al., 2016). For an extensive review on normalizing flows, we refer the reader to Papamakarios et al. (2019).
3 Related Work
Importance weighting:
The Importance Weighted Autoencoder
(Burda et al., 2015) has the same architecture as the standard VAE, however, it is trained on a lower bound that is tighter than the standard ELBO. Applying this idea to our model results in the following lower bound:(9)  
(10) 
It can be shown (Cremer et al., 2017) that, in expectation, is equivalent to with an implicit, more expressive approximate posterior. A theoretical advantage of over is that the former is consistent (under some mild boundedness assumptions, Burda et al., 2015, Theorem 1), i.e., .
In fact, it is possible to construct an unbiased estimator of using with finite (Luo et al., 2020), in combination with a Russian Roulette Estimator (Kahn, 1955)
. A drawback of this approach is that there is no guarantee that the variance of the estimator is finite.
Inference suboptimality: When using the ELBO, or with finite , we only have a bound on the marginal loglikelihood . This bound is loose when the approximate posterior is incorrect, which happens either because the form of the posterior cannot be represented, or because the recognition network does not produce good variational parameters for all data points. Both issues can be overcome by choosing the approximate posterior from an expressive variational family (Cremer et al., 2018), which is why we use a flow.
Expressive priors for representation learning: The standard VAE has a fixed prior, usually a multivariate standard normal distribution, but VAEs with more expressive priors have been proposed. Expressive priors are particularly useful when the distribution of the latent variables is used for representation learning.
A simple generalization for the prior is a learnable GMM (e.g Nalisnick et al., 2016; Dilokthanakul et al., 2016), which in our context would result in the existing extreme deconvolution model (Bovy et al., 2011), with no need for variational inference. Another approach is to model the prior with a collection of categorical distributions (e.g. van den Oord et al., 2017), which would be appropriate if an observation is wellmodeled as a composition of prototype sources. We use MAF, because for the applications we have in mind (e.g., Anderson et al., 2018), we want to use a flexible, continuous prior representation. VAEs have also used autoregressive flow priors before (Chen et al., 2017).
4 Experiments
In this section, we compare our method to a baseline of GMMs fitted with the Extreme Deconvolution (XD) model (Bovy et al., 2011), as by Ritchie & Murray (2019).
4.1 Mixture of Gaussians
In this synthetic task, the target underlying density is a mixture of Gaussians. We fit the models from observations that include additional noise from a Gaussian with fixed covariance. Figure 1 shows density plots of both the latent data and the observed data .
The exact posterior for a latent datapoint given a noisy observation is itself a mixture of Gaussians, and for some observations the components may be highly isolated. We have deliberately picked an example where the prior should be reasonably easy for a flow to model, but the posterior may cause issues for flows, as they are known to have trouble fitting mixture of Gaussians when the components are wellseparated (e.g., Jaini et al., 2019). Full experimental details are reported in Appendix B.1
Table 1 reports test average negative loglikelihood on both and , referred to as and , respectively. We estimate using . The GMM, the true model class, has the best results for both and . The flows give quite close estimates for when for both and , but show very high variance in their estimates of relative to the variance for .
The top row of Figure 2 shows example density plots using samples from the priors of our fitted models. The fitted GMM has matched the ground truth GMM closely. The flows recover the broad shape of the ground truth model, but those trained with and put too much mass in the centre. The flow trained with matched the Gaussian mixture model on the run shown, but the results had high variance, and other runs do not look as good.
The bottom row of Figure 2 shows example posteriors for the fitted models. The exact posterior for the GMM has two isolated modes, and is a close match to the ground truth posterior. The approximate posteriors for the flows trained with and are not good matches to the GMM posterior, as neither have isolated modes, but are reasonably consistent with their corresponding priors. The approximate posterior for the flow trained with uses samples drawn from with samplingimportanceresampling (e.g., Rubin, 1988). The resampling reflects this recognition network’s role as an adaptive proposal distribution under the objective rather than a direct approximation to the posterior Cremer et al. (2017). This reweighted approximation is a much better match to the ground truth posterior, but as with the prior, the variance across training runs was high, and other examples are qualitatively worse.
Method  K  
XDGMM  –  
Flow ()  1  
10  
25  
50  
Flow ()  10  
25  
50 
Test average negative loglikelihood for the Gaussian mixture toy dataset. Average over five runs with standard deviation.
To establish whether the inability of our procedure to consistently recover the correct ground truth model is a problem with the prior flow itself, the approximate posterior, the interaction of both, or the training objectives, we tried various methods of training each part separately. All results for these experiments are summarized in Table 2. Additional density plots are also available in Appendix C.
The flow was pretrained directly on the noise free samples underlying the training set via maximum likelihood. The test average negative loglikelihood on is much closer to the GMM. Therefore, while model mismatch is a slight disadvantage for the flows here, it is not entirely responsible for their worse results. Similarly, we trained the recognition network directly on noisy samples from the training set, paired with samples from the exact ground truth posterior. When combined with the pretrained prior, the test likelihood on was much closer to that of the GMM, suggesting that the flows can represent useful posteriors.
We then fitted the models with the objective, but initialized with the pretrained prior and posterior. The variational objective was significantly improved by moving to a model with similar as the GMM, however, doing so made worse. Despite using fairly flexible flows, the variational bound is not tight, and biases us towards worse prior models.
Finally we tried fitting a GMM with the objective rather than by maximizing the loglikelihood directly, using both samples from the exact posterior and samples from the conditional flow approximate posterior. When using exact samples, the variational bound is tight, but we experience the noisier gradients of variational fitting: the GMM still recovers the same result as before. However, using posterior samples from the flow causes a similar bias to before, showing that the flows are not learning good enough posteriors for variational inference to be accurate.
Model  
Pretrained flows  
After variational fitting  
GMM, flow posterior  
GMM, exact posterior 
4.2 UCI datasets
We now compare the two methods on three small UCI datasets (Dua & Graff, 2017) that are difficult to fit with GMMs (Uria et al., 2013). We discarded discretevalued attributes and normalized the data. The datasets are then split into training and testing sets; 90% are used for training and 10% are used for testing. We subsequently add noise from a zeromean independent normal distribution with diagonal covariance matrix to each noisefree training point to generate the observations .
For our method, we use the objective as it yielded the best Test average negative loglikelihood in the previous section. We note, however, that this might not be the best choice as the highvariance pattern for (see Table 1) might persist. The test results are reported in Table 3; full experimental details can be found in Section B.2. Flows outperformed GMMs in all three cases.
As an ablation, we tried fitting conventional flows to the noisy observations , without correcting for the noise in any way (Table 4). These flows beat the GMMs on one dataset, showing the importance of using good representations, however, the results are still significantly worse than flows with deconvolution (right column Table 3).
Dataset  XDGMM  Flow 
White wine  
Red wine  
Boston housing 
5 Conclusion
In this preliminary work, we have outlined an approach for density deconvolution using normalizing flows and arbitrary noise distributions by turning the deconvolution problem into an approximate inference problem. Our experiments on a toy setup have shown that variational inference with an inaccurate posterior can prevent the model prior from learning the underlying noisefree density. For future work, we are planning to experiment with unbiased inference, e.g., using Markov chain Monte Carlo methods
(e.g., Glynn & Rhee, 2014; Qiu et al., 2020) or importance weighting in combination with the Russian Roulette Estimator (Luo et al., 2020). Nevertheless, we have already been able to demonstrate that normalizing flows fitted with our approach can beat GMMs for density deconvolution on (small) realworld datasets, which indicates that further research on how to fit normalizing flows in this context is worth pursuing.Acknowledgements
We are grateful to Artur Bekasov and Conor Durkan for discussions around choices of normalizing flow for this application and use of their flows library (Durkan et al., 2019). Our experiments also made use of corner.py (ForemanMackey, 2016), Matplotlib (Hunter, 2007), NumPy (Oliphant, 2006), Pandas (McKinney, 2010)
and PyTorch
(Paszke et al., 2019). This work was supported in part by the EPSRC Centre for Doctoral Training in Data Science, funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016427/1) and the University of Edinburgh. Resources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute. We gratefully acknowledge funding support from NSERC and the Canada CIFAR AI Chairs Program.
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Appendix A The Evidence Lower Bound
The KL divergence between the variational distribution and the posterior can be written as
(11)  
(12) 
The joint distribution of
and can be computed as(13)  
(14)  
(15) 
where is the Dirac delta distribution. Hence,
(16)  
(17) 
Appendix B Details for Experiments
All code used to run the experiments is available:
https://github.com/bayesiains/densitydeconvolution
b.1 Mixture of Three Gaussians
Latent datapoints were drawn from a mixture of 3 Gaussians, with equal mixture weights, means
(18) 
and covariances
(19) 
Zeromean Gaussian noise with covariance
(20) 
was added to each to produce . Training and test sets consisted of 50 000 samples each, whilst the validation set had 12 500 samples.
The prior
was modelled with a standard normal base distribution with 5 layers of an affine Masked Autoregressive Flow (MAF) interspersed with linear transforms parameterized by an LUdecomposition and a random permutation matrix fixed at the start of training, following
Durkan et al. (2019). A residual network (He et al., 2016a) was used within each MAF layer, with 2 preactivation residual blocks (He et al., 2016b). Each block used two dense layers with 128 hidden features each. Masking of the residual blocks was done using the ResMADE architecture (Durkan & Nash, 2019).The recognition network used a setup adapted from Durkan et al. (2019), where the same flow configuration as the prior modeled the inverse transformation, making it an Inverse Autoregressive Flow (IAF, Kingma et al., 2016). Conditioning was done by concatenating to a flattened Cholesky decomposition of the noise covariance S and applying a 2 block residual network to produce a 64dimensional embedding vector. This vector was then concatenated directly onto the input of the neural network in every IAF layer. Whilst conditioning on the noise covariance S was not strictly necessary, because it was fixed for this experiment, we included it so that our implementation could handle the Extreme Deconvolution case where each observation has its own associated noise covariance .
We trained with Adam (Kingma & Ba, 2015), with initial learning rate , other parameters set to defaults, a minibatch size of 512, and with dropout (Srivastava et al., 2014)
probability 0.2. We trained for 300 epochs, and reduced the learning rate by a factor of
if there was no improvement in validation loss for 20 epochs.b.2 UCI datasets
Dataset  
White wine  
Red Wine  
Boston housing 
Since the datasets are relatively small, we tune the hyperparameters of the models using 5fold crossvalidation and grid search; the parameters of the grid search are reported in
Table 5. Once the hyperparameter values had been determined, we trained the models using a tenth of the training data for earlystopping and measured their performance on the 10% heldout test data.In contrast to the setup in Section B.1, we used simple dense layers rather than residual layers within each MAF layer. Masking of the dense layers was done using the standard MADE architecture (Germain et al., 2015). A further difference is that we conditioned the recognition network only on .
Training ran until no improvement in validation loss was observed for 30 epochs. For this experiment, we did not apply any Dropout. The minibatch size was chosen to be 100.
Hyperparameters  Tested values 
Fixed learning rate  0.001^{*†}, 0.0005^{$}, 0.0001 
MAF layers ()  3^{*}, 4^{$}, 5^{†} 
MAF layers ()  3, 4^{*}, 5^{†$} 
MAF hidden features  64, 128^{*†$} 
MAF hidden blocks  1^{*†}, 2^{$} 
Fixed learning rate  0.01^{*}, 0.005^{†}, 0.001^{$} 
# of mixture components  20^{†$}, 50, 100, 200, 300^{*} 
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