**Stand-Alone Self Attention** (SASA) replaces all instances of spatial [convolution](https://paperswithcode.com/method/convolution) with a form of self-attention applied to [ResNet](https://paperswithcode.com/method/resnet) producing a fully, stand-alone self-attentional model.

**Linear Regression** is a method for modelling a relationship between a dependent variable and independent variables. These models can be fit with numerous approaches. The most common is *least squares*, where we minimize the mean square error between the predicted values $\hat{y} = \textbf{X}\hat{\beta}$ and actual values $y$: $\left(y-\textbf{X}\beta\right)^{2}$.

We can also define the problem in probabilistic terms as a generalized linear model (GLM) where the pdf is a Gaussian distribution, and then perform maximum likelihood estimation to estimate $\hat{\beta}$.

Image Source: [Wikipedia](https://en.wikipedia.org/wiki/Linear_regression)

Linear Regression

SASA

Stand-Alone Self-Attention in Vision Models

Semantic Cross Attention (SCA) is based on cross attention, which we restrict with respect to a semantic mask.

The goal of SCA is two-fold depending on what is the query and what is the key. Either it allows to give the feature map information from a semantically restricted set of latents or, respectively, it allows a set of latents to retrieve information in a semantically restricted region of the feature map. 

SCA is defined as:  

\begin{equation}
    \text{SCA}(I_{1}, I_{2}, I_{3}) = \sigma\left(\frac{QK^T\odot I_{3} +\tau \left(1-I_{3}\right)}{\sqrt{d_{in}}}\right)V \quad ,
\end{equation}

where $I_{1},I_{2},I_{3}$ the inputs, with $I_{1}$ attending $I_{2}$, and $I_{3}$ the mask that forces tokens from $I_1$ to attend only specific tokens from $I_2$. The attention values requiring masking are filled with $-\infty$ before the softmax. (In practice $\tau{=}-10^9$),  $Q {=} W_QI_{1}$, $K {=} W_KI_{2}$ and $V {=} W_VI_{2}$ the queries, keys and values, and $d_{in}$ the internal attention dimension. $\sigma(.)$ is the softmax operation.

Let $X\in\mathbb{R}^{n\times C}$ be the feature map with n the number of pixels, and C the number of channels. Let $Z\in\mathbb{R}^{m\times d}$ be a set of $m$ latents of dimension $d$ and $s$ the number of semantic labels. Each semantic label is attributed $k$ latents, such that $m=k\times s$. Each semantic label mask is assigned $k$ copies in $S{\in}\{0;1\}^{n \times m}$. 

We can differentiate 3 types of SCA:

(a) SCA with pixels $X$ attending latents $Z$: $\text{SCA}(X, Z, S)$, where $W_{Q} {\in} \mathbb{R}^{n\times d_{in}}$ and $W_{K}, W_{V} {\in} \mathbb{R}^{m\times d_{in}}$.
The idea is to force the pixels from a semantic region to attend latents that are associated with the same label. 

(b) SCA with latents $Z$ attending pixels $X$: $\text{SCA}(Z, X, S)$, where $W_{Q}{\in} \mathbb{R}^{m\times d_{in}}$, $W_{K}, W_{V} {\in} \mathbb{R}^{n\times d_{in}}$. 
The idea is to semantically mask attention values to enforce latents to attend semantically corresponding pixels.

(c) SCA with latents $Z$ attending themselves: $\text{SCA}(Z, Z, M)$, where $W_{Q}, W_{K}, W_{V} {\in} \mathbb{R}^{n\times d_{in}}$. We denote $M\in\mathbb{N}^{m\times m}$ this mask, with $M_{\text{latents}}(i,j) {=} 1$ if the semantic label of latent $i$ is the same as the one of latent $j$; $0$ otherwise.
The idea is to let the latents only attend latents that share the same semantic label.

Source	Stand-Alone Self-Attention in Vision Models
Year	2000
Data Source	CC BY-SA - https://paperswithcode.com