Discrete Distribution Networks

A Novel Generative Model with Simple Principles and Unique Properties

Lei Yang

 

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Left: Illustrates the process of image reconstruction and latent acquisition in DDN. Each layer of DDN outputs $K$ distinct images, here $K=3$, to approximate the distribution $P(X)$. The sampler then selects the image most similar to the target from these and feeds it into the next DDN layer. As the number of layers increases, the generated images become increasingly similar to the target. For generation tasks, the sampler is simply replaced with a random choice operation.
Right: Shows the tree-structured representation space of DDN's latent variables. Each sample can be mapped to a leaf node on this tree.

Abstract

DDN enables more general zero-shot conditional generation. DDN supports zero-shot conditional generation across non-pixel domains, and notably, without relying on gradient, such as text-to-image generation using a black-box CLIP model. Images enclosed in yellow borders serve as the ground truth. The abbreviations in the table header correspond to their respective tasks as follows: “SR” stands for Super-Resolution, with the following digit indicating the resolution of the condition. “ST” denotes Style Transfer, which computes Perceptual Losses with the condition.

Overview of Discrete Distribution Networks

(a) The data flow during the training phase of DDN is shown at the top. As the network depth increases, the generated images become increasingly similar to the training images. Within each Discrete Distribution Layer (DDL), $K$ samples are generated, and the one closest to the training sample is selected as the generated image for loss computation. These $K$ output nodes are optimized using Adam with the Split-and-Prune method. The right two figures demonstrate the two model paradigms supported by DDN. (b) Single Shot Generator Paradigm: Each neural network layer and DDL has independent weights. (c) Recurrence Iteration Paradigm: All neural network layers and DDLs share weights. For inference, replacing the Guided Sampler in the DDL with a random choice enables the generation of new images.

Objective function

The DDN model consists of $L$ layers of Discrete Distribution Layers (DDL). For a given layer $l$, denoted as $f_l$, the input is the selected sample from the previous layer, ${\mathbf{x}}_{l-1}^{\ast }$. The layer generates $K$ new samples, $f_l({\mathbf{x}}_{l-1}^{\ast })$, from which we select the sample ${\mathbf{x}}_l^{\ast }$ that is closest to the current training sample $\mathbf{x}$, along with its corresponding index $k_l^{\ast }$. The loss $J_l$ for this layer is then computed only on the selected sample ${\mathbf{x}}_l^{\ast }$.

$$k_l^{\ast }=\underset{k\in \{1,\dots ,K\}}{\mathrm{argmin}}{\Vert f_l({\mathbf{x}}_{l-1}^{\ast })[k]-\mathbf{x}\Vert }^2$$ $${\mathbf{x}}_l^{\ast }=f_l({\mathbf{x}}_{l-1}^{\ast })[k_l^{\ast }]$$ $$J_l={\Vert {\mathbf{x}}_l^{\ast }-\mathbf{x}\Vert }^2$$

Here, ${\mathbf{x}}_0^{\ast }=\mathbf{0}$ represents the initial input to the first layer. For simplicity, we omit the details of input/output feature, neural network blocks and transformation operations in the equations.

By recursively unfolding the above equations, we can derive the latent variable ${\mathbf{k}}_{1:L}^{\ast }$ and the global objective function $J$.

$${\mathbf{k}}_{1:L}^{\ast }=[k_1^{\ast },k_2^{\ast },\dots ,k_L^{\ast }]={\left[\underset{k\in \{1,\dots ,K\}}{\mathrm{argmin}}{\Vert \mathcal{F}([{\mathbf{k}}_{1:l-1}^{\ast },k])-\mathbf{x}\Vert }^2\right]}_{l=1}^L$$

$$J=\frac{1}{L}\sum _{l=1}^L{\Vert \mathcal{F}({\mathbf{k}}_{1:l}^{\ast })-\mathbf{x}\Vert }^2$$

Here, $\mathcal{F}$ represents the composite function formed from $f_l$, defined as: $\mathcal{F}({\mathbf{k}}_{1:l})=f_l(f_{l-1}(\dots f_1({\mathbf{x}}_0)[k_1]\dots )[k_{l-1}])[k_l]$. Finally, we average the L2 loss across all layers to obtain the final loss for the entire network.

Toy examples for two-dimensional data generation

The numerical values at the bottom of each figure represent the Kullback-Leibler (KL) divergence. Due to phenomena such as “dead nodes” and “density shift”, the application of Gradient Descent alone fails to properly fit the Ground Truth (GT) density. However, by employing the Split-and-Prune strategy, the KL divergence is reduced to even lower than that of the Real Samples.

Random samples from DDN trained on face image

Zero-Shot Conditional Generation guided by CLIP

The text at the top is the guide text for that column.

Zero-Shot Conditional Generation with Multiple Conditions

The DDN balances the steering forces of CLIP and Inpainting according to their associated weights.

Conditional DDN performing coloring and edge-to-RGB tasks

Columns 4 and 5 display the generated results under the guidance of other images, where the produced image strives to adhere to the style of the guided image as closely as possible while ensuring compliance with the condition. The resolution of the generated images is 256x256.

Hierarchical Generation Visualization of DDN

We trained a DDN with output level $L=3$ and output nodes $K=8$ per level on MNIST dataset, its latent hierarchical structure is visualized as recursive grids. Each sample with a colored border represents an intermediate generation product. The samples within the surrounding grid of each colored-bordered sample are refined versions generated conditionally based on it (enclosed by the same color frontier). The small samples without colored borders are the final generated images. The larger the image, the earlier it is in the generation process, implying a coarse version. The large image in the middle is the average of all the generated images.