Given a corrupted input image, Con\textit{ffusion}, repurposes a pretrained diffusion model to generate lower and upper bounds around each reconstructed pixel. The true pixel value is guaranteed to fall within these bounds with probability $p$.

**InfoGAN** is a type of generative adversarial network that modifies the [GAN](https://paperswithcode.com/method/gan) objective to
encourage it to learn interpretable and meaningful representations. This is done by maximizing the
mutual information between a fixed small subset of the GAN’s noise variables and the observations.

Formally, InfoGAN is defined as a minimax game with a variational regularization of mutual information and the hyperparameter $\lambda$:

$$ \min\_{G, Q}\max\_{D}V\_{INFOGAN}\left(D, G, Q\right) = V\left(D, G\right) - \lambda{L}\_{I}\left(G, Q\right) $$

Where $Q$ is an auxiliary distribution that approximates the posterior $P\left(c\mid{x}\right)$ - the probability of the latent code $c$ given the data $x$ - and $L\_{I}$ is the variational lower bound of the mutual information between the latent code and the observations.

In the practical implementation, there is another fully-connected layer to output parameters for the conditional distribution $Q$ (negligible computation ontop of regular GAN structures). Q is represented with a [softmax](https://paperswithcode.com/method/softmax) non-linearity for a categorical latent code. For a continuous latent code, the authors assume a factored Gaussian.

InfoGAN

InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets

Conffusion

Conffusion: Confidence Intervals for Diffusion Models

**Temporal Distribution Matching**, or **TDM**,  is a module used in the [AdaRNN](https://paperswithcode.com/method/adarnn) architecture to match the distributions of the discovered periods to build a time series prediction model $\mathcal{M}$ Given the learned time periods, the TDM module is designed to learn the common knowledge shared by different periods via matching their distributions. Thus, the learned model $\mathcal{M}$ is expected to generalize well on unseen test data compared with the methods which only rely on local or statistical information.

Within the context of AdaRNN, Temporal Distribution Matching aims to adaptively match the distributions between the [RNN](https://paperswithcode.com/methods/category/recurrent-neural-networks) cells of two periods while capturing the temporal dependencies. TDM introduces the importance vector $\mathbf{\alpha} \in \mathbb{R}^{\hat{V}}$ to learn the relative importance of $V$ hidden states inside the RNN, where all the hidden states are weighted with a normalized $\alpha$. Note that for each pair of periods, there is an $\mathbf{\alpha}$, and we omit the subscript if there is no confusion. In this way, we can dynamically reduce the distribution divergence of cross-periods.

Given a period-pair $\left(\mathcal{D}\_{i}, \mathcal{D}\_{j}\right)$, the loss of temporal distribution matching is formulated as:

$$
\mathcal{L}\_{t d m}\left(\mathcal{D}\_{i}, \mathcal{D}\_{j} ; \theta\right)=\sum_{t=1}^{V} \alpha\_{i, j}^{t} d\left(\mathbf{h}\_{i}^{t}, \mathbf{h}\_{j}^{t} ; \theta\right)
$$

where $\alpha\_{i, j}^{t}$ denotes the distribution importance between the periods $\mathcal{D}\_{i}$ and $\mathcal{D}\_{j}$ at state $t$.

All the hidden states of the RNN can be easily computed by following the standard RNN computation. Denote by $\delta(\cdot)$ the computation of a next hidden state based on a previous state. The state computation can be formulated as

$$
\mathbf{h}\_{i}^{t}=\delta\left(\mathbf{x}\_{i}^{t}, \mathbf{h}\_{i}^{t-1}\right)
$$

The final objective of temporal distribution matching (one RNN layer) is:

$$
\mathcal{L}(\theta, \mathbf{\alpha})=\mathcal{L}\_{\text {pred }}(\theta)+\lambda \frac{2}{K(K-1)} \sum\_{i, j}^{i \neq j} \mathcal{L}\_{t d m}\left(\mathcal{D}\_{i}, \mathcal{D}\_{j} ; \theta, \mathbf{\alpha}\right)
$$

where $\lambda$ is a trade-off hyper-parameter. Note that in the second term, we compute the average of the distribution distances of all pairwise periods. For computation, we take a mini-batch of $\mathcal{D}_{i}$ and $\mathcal{D}\_{j}$ to perform forward operation in RNN layers and concatenate all hidden features. Then, we can perform TDM using the above equation.

Source	Conffusion: Confidence Intervals for Diffusion Models
Year	2000
Data Source	CC BY-SA - https://paperswithcode.com