$$
\begin{array}{l} \mathcal {L} (\boldsymbol {\theta}, \sigma) = \mathbb {E} _ {\boldsymbol {x}, \boldsymbol {y}} \left\| \sum_ {k = 1} ^ {K} \boldsymbol {a} _ {k} [ \boldsymbol {w} _ {k} ^ {\top} \boldsymbol {y} ] _ {+} - \boldsymbol {x} \right\| ^ {2} \\ = \mathbb {E} _ {\boldsymbol {x}, \boldsymbol {y}} \left[ \sum_ {i = 1} ^ {K} \sum_ {j = 1} ^ {K} \boldsymbol {a} _ {i} ^ {\top} \boldsymbol {a} _ {j} [ \boldsymbol {w} _ {i} ^ {\top} \boldsymbol {y} ] _ {+} [ \boldsymbol {w} _ {j} ^ {\top} \boldsymbol {y} ] _ {+} - 2 \sum_ {i = 1} ^ {K} \boldsymbol {x} ^ {\top} \boldsymbol {a} _ {i} [ \boldsymbol {w} _ {i} ^ {\top} \boldsymbol {y} ] _ {+} + \| \boldsymbol {x} \| ^ {2} \right] \\ = \sum_ {i = 1} ^ {K} \sum_ {j = 1} ^ {K} \boldsymbol {a} _ {i} ^ {\top} \boldsymbol {a} _ {j} \mathbb {E} _ {\boldsymbol {x}, \boldsymbol {y}} \left[ \left[ \boldsymbol {w} _ {i} ^ {\top} \boldsymbol {y} \right] _ {+} \left[ \boldsymbol {w} _ {j} ^ {\top} \boldsymbol {y} \right] _ {+} \right] - 2 \sum_ {i = 1} ^ {K} \mathbb {E} _ {\boldsymbol {x}, \boldsymbol {y}} \left[ \boldsymbol {x} ^ {\top} \boldsymbol {a} _ {i} \left[ \boldsymbol {w} _ {i} ^ {\top} \boldsymbol {y} \right] _ {+} \right] + \mathbb {E} \| \boldsymbol {x} \| ^ {2} \\ = \sum_ {i = 1} ^ {K} \sum_ {j = 1} ^ {K} \boldsymbol {a} _ {i} ^ {\top} \boldsymbol {a} _ {j} \mathbb {E} _ {\boldsymbol {x}} \left[ \mathbb {E} _ {\boldsymbol {y} | \boldsymbol {x}} \left[ \left[ \boldsymbol {w} _ {i} ^ {\top} \boldsymbol {y} \right] _ {+} \left[ \boldsymbol {w} _ {j} ^ {\top} \boldsymbol {y} \right] _ {+} \right] \right] - 2 \sum_ {i = 1} ^ {K} \mathbb {E} _ {\boldsymbol {x}} \left[ \mathbb {E} _ {\boldsymbol {y} | \boldsymbol {x}} \left[ \boldsymbol {x} ^ {\top} \boldsymbol {a} _ {i} \left[ \boldsymbol {w} _ {i} ^ {\top} \boldsymbol {y} \right] _ {+} \right] \right] + \mathbb {E} \| \boldsymbol {x} \| ^ {2} \\ = \sum_ {i = 1} ^ {K} \sum_ {j = 1} ^ {K} \boldsymbol {a} _ {i} ^ {\top} \boldsymbol {a} _ {j} \mathbb {E} _ {\boldsymbol {x}} \left[ \mathbb {E} _ {\boldsymbol {y} | \boldsymbol {x}} \left[ \left[ \boldsymbol {w} _ {i} ^ {\top} \boldsymbol {y} \right] _ {+} \right] \mathbb {E} _ {\boldsymbol {y} | \boldsymbol {x}} \left[ \left[ \boldsymbol {w} _ {j} ^ {\top} \boldsymbol {y} \right] _ {+} \right] \right] - 2 \sum_ {i = 1} ^ {K} \mathbb {E} _ {\boldsymbol {x}} \left[ \mathbb {E} _ {\boldsymbol {y} | \boldsymbol {x}} \left[ \boldsymbol {x} ^ {\top} \boldsymbol {a} _ {i} \left[ \boldsymbol {w} _ {i} ^ {\top} \boldsymbol {y} \right] _ {+} \right] \right] + \mathbb {E} \| \boldsymbol {x} \| ^ {2} \\ + \sum_ {i = 1} ^ {K} \sum_ {j = 1} ^ {K} \boldsymbol {a} _ {i} ^ {\top} \boldsymbol {a} _ {j} \mathbb {E} _ {\boldsymbol {x}} \left[ \mathbb {E} _ {\boldsymbol {y} | \boldsymbol {x}} \left[ \left[ \boldsymbol {w} _ {i} ^ {\top} \boldsymbol {y} \right] _ {+} \left[ \boldsymbol {w} _ {j} ^ {\top} \boldsymbol {y} \right] _ {+} \right] \right] - \sum_ {i = 1} ^ {K} \sum_ {j = 1} ^ {K} \boldsymbol {a} _ {i} ^ {\top} \boldsymbol {a} _ {j} \mathbb {E} _ {\boldsymbol {x}} \left[ \mathbb {E} _ {\boldsymbol {y} | \boldsymbol {x}} \left[ \left[ \boldsymbol {w} _ {\boldsymbol {i}} ^ {\top} \boldsymbol {y} \right] _ {+} \right] \right. \mathbb {E} _ {\boldsymbol {y} | \boldsymbol {x}} \left[ \left[ \boldsymbol {w} _ {\boldsymbol {j}} ^ {\top} \boldsymbol {y} \right] _ {+} \right] \\ = \mathbb {E} _ {\left(\boldsymbol {x}\right)} \left\| \sum_ {i = 1} ^ {K} \boldsymbol {a} _ {i} \tilde {\phi} \left(\boldsymbol {w} _ {i} ^ {\top} \boldsymbol {x}\right) - \boldsymbol {x} \right\| ^ {2} + \sum_ {i = 1} ^ {K} \sum_ {j = 1} ^ {K} \boldsymbol {a} _ {i} ^ {\top} \boldsymbol {a} _ {j} \mathbb {E} _ {\boldsymbol {x}} \left[ \mathbb {E} _ {\boldsymbol {y} | \boldsymbol {x}} \left[ \left[ \boldsymbol {w} _ {i} ^ {\top} \boldsymbol {y} \right] _ {+} \left[ \boldsymbol {w} _ {j} ^ {\top} \boldsymbol {y} \right] _ {+} \right] \right] \\ - \sum_ {i = 1} ^ {K} \sum_ {j = 1} ^ {K} \boldsymbol {a} _ {i} ^ {\top} \boldsymbol {a} _ {j} \mathbb {E} _ {\boldsymbol {x}} \left[ \mathbb {E} _ {\boldsymbol {y} | \boldsymbol {x}} \left[ \left[ \boldsymbol {w} _ {i} ^ {\top} \boldsymbol {y} \right] _ {+} \right] \mathbb {E} _ {\boldsymbol {y} | \boldsymbol {x}} \left[ \left[ \boldsymbol {w} _ {j} ^ {\top} \boldsymbol {y} \right] _ {+} \right] \right]. \\ \end{array}
$$