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Summary by Hugo Larochelle 3 years ago
This paper presents a novel neural network approach (though see [here](https://www.facebook.com/hugo.larochelle.35/posts/172841743130126?pnref=story) for a discussion on prior work) to density estimation, with a focus on image modeling. At its core, it exploits the following property on the densities of random variables. Let $x$ and $z$ be two random variables of equal dimensionality such that $x = g(z)$, where $g$ is some bijective and deterministic function (we'll note its inverse as $f = g^{-1}$). Then the change of variable formula gives us this relationship between the densities of $x$ and $z$:
$p_X(x) = p_Z(z) \left|{\rm det}\left(\frac{\partial g(z)}{\partial z}\right)\right|^{-1}$
Moreover, since the determinant of the Jacobian matrix of the inverse $f$ of a function $g$ is simply the inverse of the Jacobian of the function $g$, we can also write:
$p_X(x) = p_Z(f(x)) \left|{\rm det}\left(\frac{\partial f(x)}{\partial x}\right)\right|$
where we've replaced $z$ by its deterministically inferred value $f(x)$ from $x$.
So, the core of the proposed model is in proposing a design for bijective functions $g$ (actually, they design its inverse $f$, from which $g$ can be derived by inversion), that have the properties of being easily invertible and having an easy-to-compute determinant of Jacobian. Specifically, the authors propose to construct $f$ from various modules that all preserve these properties and allows to construct highly non-linear $f$ functions. Then, assuming a simple choice for the density $p_Z$ (they use a multidimensional Gaussian), it becomes possible to both compute $p_X(x)$ tractably and to sample from that density, by first samples $z\sim p_Z$ and then computing $x=g(z)$.
The building blocks for constructing $f$ are the following:
**Coupling layers**: This is perhaps the most important piece. It simply computes as its output $b\odot x + (1-b) \odot (x \odot \exp(l(b\odot x)) + m(b\odot x))$, where $b$ is a binary mask (with half of its values set to 0 and the others to 1) over the input of the layer $x$, while $l$ and $m$ are arbitrarily complex neural networks with input and output layers of equal dimensionality.
In brief, for dimensions for which $b_i = 1$ it simply copies the input value into the output. As for the other dimensions (for which $b_i = 0$) it linearly transforms them as $x_i * \exp(l(b\odot x)_i) + m(b\odot x)_i$. Crucially, the bias ($m(b\odot x)_i$) and coefficient ($\exp(l(b\odot x)_i)$) of the linear transformation are non-linear transformations (i.e. the output of neural networks) that only have access to the masked input (i.e. the non-transformed dimensions). While this layer might seem odd, it has the important property that it is invertible and the determinant of its Jacobian is simply $\exp(\sum_i (1-b_i) l(b\odot x)_i)$. See Section 3.3 for more details on that.
**Alternating masks**: One important property of coupling layers is that they can be stacked (i.e. composed), and the resulting composition is still a bijection and is invertible (since each layer is individually a bijection) and has a tractable determinant for its Jacobian (since the Jacobian of the composition of functions is simply the multiplication of each function's Jacobian matrix, and the determinant of the product of square matrices is the product of the determinant of each matrix). This is also true, even if the mask $b$ of each layer is different. Thus, the authors propose using masks that alternate across layer, by masking a different subset of (half of) the dimensions. For images, they propose using masks with a checkerboard pattern (see Figure 3). Intuitively, alternating masks are better because then after at least 2 layers, all dimensions have been transformed at least once.
**Squeezing operations**: Squeezing operations corresponds to a reorganization of a 2D spatial layout of dimensions into 4 sets of features maps with spatial resolutions reduced by half (see Figure 3). This allows to expose multiple scales of resolutions to the model. Moreover, after a squeezing operation, instead of using a checkerboard pattern for masking, the authors propose to use a per channel masking pattern, so that "the resulting partitioning is not redundant with the previous checkerboard masking". See Figure 3 for an illustration.
Overall, the models used in the experiments usually stack a few of the following "chunks" of layers: 1) a few coupling layers with alternating checkboard masks, 2) followed by squeezing, 3) followed by a few coupling layers with alternating channel-wise masks. Since the output of each layers-chunk must technically be of the same size as the input image, this could become expensive in terms of computations and space when using a lot of layers. Thus, the authors propose to explicitly pass on (copy) to the very last layer ($z$) half of the dimensions after each layers-chunk, adding another chunk of layers only on the other half. This is illustrated in Figure 4b.
Experiments on CIFAR-10, and 32x32 and 64x64 versions of ImageNet show that the proposed model (coined the real-valued non-volume preserving or Real NVP) has competitive performance (in bits per dimension), though slightly worse than the Pixel RNN.
**My Two Cents**
The proposed approach is quite unique and thought provoking. Most interestingly, it is the only powerful generative model I know that combines A) a tractable likelihood, B) an efficient / one-pass sampling procedure and C) the explicit learning of a latent representation. While achieving this required a model definition that is somewhat unintuitive, it is nonetheless mathematically really beautiful!
I wonder to what extent Real NVP is penalized in its results by the fact that it models pixels as real-valued observations. First, it implies that its estimate of bits/dimensions is an upper bound on what it could be if the uniform sub-pixel noise was integrated out (see Equations 3-4-5 of [this paper](http://arxiv.org/pdf/1511.01844v3.pdf)). Moreover, the authors had to apply a non-linear transformation (${\rm logit}(\alpha + (1-\alpha)\odot x)$) to the pixels, to spread the $[0,255]$ interval further over the reals. Since the Pixel RNN models pixels as discrete observations directly, the Real NVP might be at a disadvantage.
I'm also curious to know how easy it would be to do conditional inference with the Real NVP. One could imagine doing approximate MAP conditional inference, by clamping the observed dimensions and doing gradient descent on the log-likelihood with respect to the value of remaining dimensions. This could be interesting for image completion, or for structured output prediction with real-valued outputs in general. I also wonder how expensive that would be.
In all cases, I'm looking forward to saying interesting applications and variations of this model in the future!

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