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GADE: A Generative Adversarial Approach to Density Estimation and its Applications

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Abstract

Density estimation is a challenging unsupervised learning problem. Current maximum likelihood approaches for density estimation are either restrictive or incapable of producing high-quality samples. On the other hand, likelihood-free models such as generative adversarial networks, produce sharp samples without a density model. The lack of a density estimate limits the applications to which the sampled data can be put, however. We propose a generative adversarial density estimator (GADE), a density estimation approach that bridges the gap between the two. Allowing for a prior on the parameters of the model, we extend our density estimator to a Bayesian model where we can leverage the predictive variance to measure our confidence in the likelihood. Our experiments on challenging applications such as visual dialog or autonomous driving where the density and the confidence in predictions are crucial shows the effectiveness of our approach.

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Notes

  1. This is merely for theoretical analysis. In practice, we don’t need to explicitly define an invertible function.

  2. A randomized version of this lemma is proposed in Boutsidis et al. (2017).

  3. We observed in practice that if the generator function \(g_{z}\) shares weights with that of \(\phi \) (i.e. second layer of \(g_{z}\) with last layer of \(\phi \) and so forth) the approach does not perform well. In addition, even if we constrain each layer of the \(g_{z}\) network to match in distribution to layers of \(\phi \) the quality of the generated samples deteriorate.

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Authors and Affiliations

  1. Australian Institute for Machine Learning, The University of Adelaide, Adelaide, Australia

    M. Ehsan Abbasnejad, Javen Shi, Anton van den Hengel & Lingqiao Liu

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  1. M. Ehsan Abbasnejad
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Correspondence to M. Ehsan Abbasnejad.

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Communicated by Jun-Yan Zhu, Hongsheng Li, Eli Shechtman, Ming-Yu Liu, Jan Kautz, Antonio Torralba.

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Proof of Lemma 2

Proof of Lemma 2

Proof

Since B is a positive definite matrix with \(\Vert B\Vert _{2}<1\), it follows that,

$$\begin{aligned} \log \det {A}= & {} \log (\alpha ^{d}\det (A/\alpha ))\\= & {} d\log (\alpha )+\log \left( \prod _{i=1}^{d}\lambda _{i}(B)\right) \\= & {} d\log (\alpha )+\sum _{i=1}^{d}\log (\lambda _{i}(B))=d\log (\alpha )+\text {tr}\left( {\log \left( {B}\right) }\right) . \end{aligned}$$

where \(\lambda _{i}(B)\) is the ith eigenvalue of the matrix B. We know for the second line that

$$\begin{aligned} \text {tr}\left( {\log \left( {B}\right) }\right) =\sum _{i=1}^{d}\lambda _{i}(\log \left( {B}\right) ) =\sum _{i=1}^{d}\log (\lambda _{i}(B)). \end{aligned}$$

Then we have,

$$\begin{aligned} \text {tr}\left( {\log \left( {B}\right) }\right)= & {} \text {tr}\left( {\log \left( {(B-\mathbf {I}_{d})+\mathbf {I}_{d}}\right) }\right) \end{aligned}$$
(10)
$$\begin{aligned}= & {} \text {tr}\left( {\sum _{k=1}^{\infty } \frac{(-1)^{k+1}\left( B-\mathbf {I}_{d}\right) ^{k}}{k}}\right) . \end{aligned}$$
(11)

For the second equality we use Taylor expansion of the \(\log \) since all the eigenvalues of \(B-\mathbf {I}_{d}\) are in (0, 1) and the last equality follows by the linearity of the trace operator. \(\square \)

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Abbasnejad, M.E., Shi, J., van den Hengel, A. et al. GADE: A Generative Adversarial Approach to Density Estimation and its Applications. Int J Comput Vis 128, 2731–2743 (2020). https://doi.org/10.1007/s11263-020-01360-9

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