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Suppose I have I have vectors $x_1,\ldots,x_m$ in $\mathbb{R}^d$, with $\|x_i\|=1$ for all $i$

Is there a name or statistical/geometric interpretation of the following quantity?

$$\sum_{ij} \langle x_i, x_j \rangle ^2 $$

This seems to measure how far the examples are from being mutually orthogonal, evaluating to $m$ for perfectly orthogonal set of $x$'s. Wondering if this comes up in literature.

Motivation: reciprocal of this quantity is proportional to drop in expected error norm squared after applying one step of gradient descent on least squares objective on dataset with examples $x$

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  • $\begingroup$ What does $\|x_i\|$ means? L2 norm? If so, maybe the sum represents volume of a sphere or some driven quantity of it. $\endgroup$ Commented Aug 5, 2022 at 21:30
  • $\begingroup$ It is the adjacency matrix for cosine similarity. $\endgroup$ Commented Aug 6, 2022 at 6:00

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