SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\tilde {\mathrm {X}} _ {i} ^ {\prime} = \tilde {\mathrm {X}} _ {i} - \operatorname {p r o j} \left(\tilde {\mathrm {X}} _ {i}, \mathcal {Z}\right) = \mathrm {X} _ {i} - \left(\boldsymbol {\Sigma} _ {\mathrm {X Z}}\right) _ {i} \boldsymbol {\Sigma} _ {\mathrm {Z Z}} ^ {+} (\mathrm {Z} - \mathbb {E} [ \mathrm {Z} ]) - \mathbb {E} [ \mathrm {X} _ {i} ]. \tag {9}
$$
Putting it all together. Plugging Eq. 9 into  $\mathrm{X}_i' = \tilde{\mathrm{X}}_i' + \mathbb{E}[\mathrm{X}_i]$  and combining all components into vector form yields
$$
\mathrm {X} _ {\text {L E A C E}} ^ {\prime} = \mathrm {X} - \Sigma_ {\mathrm {X Z}} \Sigma_ {\mathrm {Z Z}} ^ {+} (\mathrm {Z} - \mathbb {E} [ \mathrm {Z} ]), \tag {10}
$$
J Notation Key
$\mathcal{Z}$  The space of one-hot labels  $\{(z_1,\ldots z_k)\in \{0,1\} ^k\mid \sum_{j = 1}^k z_j = 1\}\}$  (treated interchangeably with the integers  $\{1,\dots ,k\}$  when convenient
X, Z Integrable (i.e. finite first moment) random vectors taking values in  $\mathbb{R}^d$  and  $\mathbb{R}^k$  respectively (or their realized values inside an expectation, e.g. in  $\mathbb{E}[f(\mathrm{X})]$ ). Z is sometimes restricted to the one-hot labels  $\mathcal{Z}$ , in which case we assume each  $\mathbb{P}(\mathrm{Z} = j) > 0$ .
$\mathrm{X}_i, \mathrm{Z}_j$  The  $i^{\text{th}}$  and  $j^{\text{th}}$  components thereof, themselves scalar random variables (or their realized values inside an expectation).
$\xi, \zeta$  Scalar random variables taking values in  $\mathbb{R}$ .
$\eta$  A predictor function  $\mathbb{R}^d\to \mathcal{Z}$  (or its value  $\eta (\mathrm{X})$  when inside an expectation).
$\mathcal{V}$  A space of predictor functions  $\{\eta (\cdot ;\pmb {\theta}):\mathbb{R}^d\to \mathbb{R}^k\mid \pmb {\theta}\in \Theta \}$ , parameterized by  $\pmb{\theta}$  and containing all constant functions.
A space of loss functions  $\{\mathcal{L}:\mathbb{R}^k\times \mathcal{Z}\to [0,\infty)\}$
$r$  An erasure function  $\mathbb{R}^d\to \mathbb{R}^d$ , hopefully making a minimal edit to  $X$  that eliminates the ability to predict labels  $Z$  with predictors in  $\nu$ .
A A matrix with entries in  $\mathbb{R}$
$A_{ij}$  The entry thereof at the  $i^{\mathrm{th}}$  row and  $j^{\mathrm{th}}$  column.
$\mathbf{A}^{+}$  The Moore-Penrose pseudoinverse of  $\mathbf{A}$ .
A column vector with entries in  $\mathbb{R}$
$v_{i}$  The  $i^{\mathrm{th}}$  component thereof.