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“…Proof. Note that according to the structure of the Euclidean projection over S n (see for instance Lemma 6 in [12]), when the condition in the if statement holds on some iteration t, then indeed the projection of Y t+1 onto S n is given by the rank one matrix u 1 u ⊤ 1 , and thus X t+1 is equivalent to the standard projected gradient update step:…”

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“…Proof. Note that according to the structure of the Euclidean projection over S n (see for instance Lemma 6 in [12]), when the condition in the if statement holds on some iteration t, then indeed the projection of Y t+1 onto S n is given by the rank one matrix u 1 u ⊤ 1 , and thus X t+1 is equivalent to the standard projected gradient update step:…”

“…Thus, starting from iteration T 1 and onwards, by invoking Theorem 7 in [12], it directly follows that the projection of Y t+1 onto S n is indeed always rank-one, and thus from this point on, only projected gradient steps are used.…”