Vision
Image Tensor Basics
\[\mathbf{X} \in \mathbb{R}^{H \times W \times C}\]
\[\mathbf{X} \in \mathbb{R}^{C \times H \times W}\]
\[\mathrm{vec}(\mathbf{X}) \in \mathbb{R}^{HWC}\]
Convolution
\[Y_{i,j} = \sum_{u=0}^{k_h-1} \sum_{v=0}^{k_w-1} K_{u,v} X_{i+u, j+v}\]
\[Y_{o,i,j} = \sum_{c=1}^{C_{\mathrm{in}}} \sum_{u=0}^{k_h-1} \sum_{v=0}^{k_w-1}
K_{o,c,u,v} X_{c, i+u, j+v}\]
\[H_{\mathrm{out}} = \left\lfloor \frac{H + 2p_h - k_h}{s_h} \right\rfloor + 1\]
\[W_{\mathrm{out}} = \left\lfloor \frac{W + 2p_w - k_w}{s_w} \right\rfloor + 1\]
Pooling
\[Y_{i,j}^{\mathrm{max}} = \max_{0 \leq u < k_h,\ 0 \leq v < k_w} X_{i+u, j+v}\]
\[Y_{i,j}^{\mathrm{avg}} = \frac{1}{k_h k_w} \sum_{u=0}^{k_h-1} \sum_{v=0}^{k_w-1} X_{i+u, j+v}\]
\[H_{\mathrm{out}} = \left\lfloor \frac{H + 2p_h - k_h}{s_h} \right\rfloor + 1, \quad
W_{\mathrm{out}} = \left\lfloor \frac{W + 2p_w - k_w}{s_w} \right\rfloor + 1\]
Vision Losses
\[\mathcal{L}_{\mathrm{CE}} = - \sum_{i=1}^{K} y_i \log \hat{y}_i\]
\[\mathcal{L}_{\mathrm{BCE}} = - \sum_{i=1}^{K} \left( y_i \log \hat{y}_i + (1-y_i)\log(1-\hat{y}_i) \right)\]
\[\mathcal{L}_{\mathrm{MSE}} = \frac{1}{N} \sum_{i=1}^{N} (y_i - \hat{y}_i)^2\]
Patch Embedding
\[N = \frac{H W}{P^2}\]
\[\mathbf{x}_p^{(n)} \in \mathbb{R}^{P^2 C}, \quad n = 1, \dots, N\]
\[\mathbf{z}_0^{(n)} = \mathbf{x}_p^{(n)} \mathbf{E}\]
\[\mathbf{Z}_0 = [\mathbf{x}_{\mathrm{cls}}; \mathbf{z}_0^{(1)}; \dots; \mathbf{z}_0^{(N)}] + \mathbf{E}_{\mathrm{pos}}\]
\[\mathbf{Z}'_\ell = \mathbf{Z}_{\ell-1} + \mathrm{MHA}(\mathrm{LayerNorm}(\mathbf{Z}_{\ell-1}))\]
\[\mathbf{Z}_\ell = \mathbf{Z}'_\ell + \mathrm{FFN}(\mathrm{LayerNorm}(\mathbf{Z}'_\ell))\]
\[\hat{\mathbf{y}} = \mathrm{softmax}(\mathbf{W}\mathbf{z}_{L}^{\mathrm{cls}})\]
