epsilon net restricted inner product bounds the operator norm

[[concept]]

Topics

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Lemma

If $\mathcal{X}$ is an epsilon net of ${\mathbb{S}}^{d-1}$ or $B^d$, then $||\Delta |{|}_{\mathcal{X}}\le ||\Delta ||\le \frac{1}{1-2ϵ}||\Delta |{|}_{\mathcal{X}}$ (if $ϵ\in \left(0,\frac{1}{2}\right)$ for the second inequality)

Proof

The first inequality is immediate from the definition of the restricted inner product. So we prove only the second inequality:

Let $x\in {\mathbb{S}}^{d-1}$ be such that $|x^T\Delta x|=||\Delta ||$ and let $x_i\in \mathcal{X}$ such that $||x-x_i||\le ϵ$. Then

\lvert \lvert \Delta \rvert \rvert _{{\cal X}} &\geq \lvert x_{i}^T\Delta x_{i} \rvert \\ &= \lvert (x + (x_{i}-x))^T \Delta (x+(x_{i} - x)) \rvert \\ &= \underbrace{\lvert x^T\Delta x \rvert}_{\lvert \lvert \Delta \rvert \rvert } - \underbrace{\lvert x^T\Delta(x_{i}-x) \rvert}_{\leq \lvert \lvert x \rvert \rvert \cdot \lvert \lvert \Delta \rvert \rvert \cdot \lvert \lvert x_{i}-x \rvert \rvert \leq \epsilon \lvert \lvert \Delta \rvert \rvert } -\underbrace{\lvert (x_{i}-x)^T \Delta x_{i} \rvert }_{\leq \epsilon \lvert \lvert \Delta \rvert \rvert } \\ &\geq (1-2\epsilon) \lvert \lvert \Delta \rvert \rvert \end{align}$$ $$\tag*{$\blacksquare$}$$

NOTE

++++ interpretation of special $x$

References

References

See Also

• restricted inner product

TODO

• Finish the note and also update Random Matrix Lecture 04 clean

Mentions

Mentions

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