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\begin{array}{l} = \mathbb {E} _ {\mathcal {D}} \mathbb {E} _ {\boldsymbol {x} \sim \mathcal {P}} \mathbb {E} _ {\boldsymbol {\xi} \sim \mathcal {N} (0, \mathbb {I} _ {d})} \left\| \boldsymbol {x} - \left\langle t (\mu / \sqrt {d}, \sqrt {1 - \Delta} \boldsymbol {x} + \sqrt {\Delta} \boldsymbol {\xi}) \right\rangle_ {\boldsymbol {\mu} \sim \mathbb {P} _ {b}} \right\| ^ {2} \\ = \operatorname {m s e} _ {\circ} - \mathbb {E} _ {\mathcal {D}} \sum_ {s = \pm 1} \mathbb {E} _ {\boldsymbol {\xi}, \boldsymbol {\eta} \sim \mathcal {N} (0, \mathbb {I} _ {d})} \frac {\Delta}{\sigma^ {2} (1 - \Delta) + \Delta} \left\langle \left(\frac {(1 - b \sqrt {1 - \Delta}) s \boldsymbol {\mu} ^ {\top} \boldsymbol {\mu} ^ {\star}}{d} + \frac {\boldsymbol {\mu} ^ {\top} ((1 - b \sqrt {1 - \Delta}) \sigma \boldsymbol {\eta} - b \sqrt {\Delta} \boldsymbol {\xi})}{\sqrt {d}}\right) \right. \\ \tanh \left(\frac {\sqrt {1 - \Delta} s \frac {\mu^ {\top}}{2} \boldsymbol {\mu} ^ {\star} + \sqrt {1 - \Delta} \sigma \frac {\mu^ {\top}}{\sqrt {d}} \boldsymbol {\eta} + \sqrt {\Delta} \frac {\boldsymbol {\mu} ^ {\top} \boldsymbol {\xi}}{\sqrt {d}}}{\sigma^ {2} (1 - \Delta) + \Delta}\right) \Bigg \rangle_ {\boldsymbol {\mu} \sim \mathbb {P} _ {b}} \\ + \mathbb {E} _ {\mathcal {D}} \frac {1}{2} \sum_ {s = \pm 1} \mathbb {E} _ {\boldsymbol {\xi}, \boldsymbol {\eta} \sim \mathcal {N} (0, \mathbb {I} _ {d})} \left\langle \left(\frac {\Delta}{\sigma^ {2} (1 - \Delta) + \Delta}\right) ^ {2} \frac {\boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {\mu} _ {2}}{d} \tanh  \left(\frac {\sqrt {1 - \Delta} ^ {s} \frac {\boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {\mu} ^ {\star}}{d} + \sqrt {1 - \Delta} \sigma \frac {\boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {\eta}}{\sqrt {d}} + \sqrt {\Delta} \frac {\boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {\xi}}{\sqrt {d}}}{\sigma^ {2} (1 - \Delta) + \Delta}\right) \right. \\ \times \tanh  \left(\frac {\sqrt {1 - \Delta} ^ {s} \frac {\mu_ {2} ^ {\top} \boldsymbol {\mu} ^ {\star}}{d} + \sqrt {1 - \Delta} \sigma \frac {\boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {\eta}}{\sqrt {d}} + \sqrt {\Delta} \frac {\boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {\xi}}{\sqrt {d}}}{\sigma^ {2} (1 - \Delta) + \Delta}\right) \Bigg \rangle_ {\boldsymbol {\mu} _ {1}, \boldsymbol {\mu} _ {2} \sim \mathbb {P} _ {b}} \\ = \operatorname {m s e} _ {\circ} - \mathbb {E} _ {\mathcal {D}} \sum_ {s = \pm 1} \mathbb {E} _ {z \sim \mathcal {N} (0, 1)} \frac {\Delta}{\sigma^ {2} (1 - \Delta) + \Delta} \left\langle \left(\frac {(1 - b \sqrt {1 - \Delta}) s \boldsymbol {\mu} ^ {\top} \boldsymbol {\mu} ^ {\star}}{d}\right) \tanh  \left(\frac {\sqrt {1 - \Delta} s \boldsymbol {\mu} ^ {\top} \boldsymbol {\mu} ^ {\star}}{d} + \sqrt {(1 - \Delta) \sigma^ {2} + \Delta} \frac {\| \boldsymbol {\mu} \|}{\sqrt {d}} z}{\sigma^ {2} (1 - \Delta) + \Delta}\right) \right\rangle_ {\boldsymbol {\mu} \sim \mathbb {P} _ {b}} \\ + \mathbb {E} _ {\mathcal {D}} \frac {1}{2} \sum_ {s = \pm 1} \mathbb {E} _ {u, v \sim \mathcal {N} (0, \Omega)} \left\langle \left(\frac {\Delta}{\sigma^ {2} (1 - \Delta) + \Delta}\right) ^ {2} \frac {\boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {\mu} _ {2}}{d} \tanh  \left(\frac {\sqrt {1 - \Delta} ^ {\frac {s \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {\mu} ^ {\star}}{d}} + \sqrt {(1 - \Delta) \sigma^ {2} + \Delta} u}{\sigma^ {2} (1 - \Delta) + \Delta}\right) \right. \\ \times \tanh  \left(\frac {\sqrt {1 - \Delta} \frac {s \mu_ {2} ^ {\top} \mu^ {*}}{d} + \sqrt {(1 - \Delta) \sigma^ {2} + \Delta v}}{\sigma^ {2} (1 - \Delta) + \Delta}\right) \Bigg \rangle_ {\boldsymbol {\mu} _ {1}, \boldsymbol {\mu} _ {2} \sim \mathbb {P} _ {b}}. \tag {88} \\ \end{array}
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