# How to find upper and lower bound

Let $$\Sigma \in S_{++}^n$$ be a symmetric positive definite matrix with all diagonal entries equal to one. Let $$U \in \mathbb{R}^{n \times k_1}$$, $$W \in \mathbb{R}^{n \times k_2}$$, $$\Lambda \in \mathbb{R}^{k_1 \times k_1}$$ and $$T \in \mathbb{R}^{k_2 \times k_2}$$, where $$\Lambda$$ and $$T$$ are both diagonal matrices with positive elements, and $$n > k_2 > k_1$$. We also know $$\text{trace}(\mathbf{\Lambda}) = \mu \times \text{trace}(\mathbf{T})$$, and the sum of the absolute values of all the elements of $$U$$ is less than $$W$$. Then how can I find upper and lower bounds on

\begin{align*} \frac{\|\Sigma - UTU^\top\|_F^2}{\|\Sigma - W\Lambda W^\top\|_F^2} \end{align*}

in terms of $$\mu$$, $$W$$, $$\Lambda$$ and $$\Sigma$$?

We first find an upper bound on $$\|UTU^\top \|_F^2$$ in terms of $$W$$ and $$\Lambda$$-
\begin{align*} \|UTU^\top \|_F^2 &\leq \| U\|_F^4 \|T \|_F^2 \\ & \leq \frac{1}{\mu} \| W\|_F^4 \|\Lambda \|_F^2 \\ \end{align*}
Now, assuming that $$\|\Sigma \|_F^2 \geq \| UTU^\top \|_F^2$$ and $$\|\Sigma \|_F^2 \geq \| W\Lambda W^\top \|_F^2$$
\begin{align*} \|\Sigma\|_F^2 - \| UTU^\top \|_F^2 \leq \|\Sigma - UTU^\top \|_F^2 \leq \|\Sigma \|_F^2+\| UTU^\top \|_F^2 \\ \|\Sigma\|_F^2 - \frac{1}{\mu} \| W\|_F^4 \|\Lambda \|_F^2 \leq \|\Sigma - UTU^\top \|_F^2 \leq \|\Sigma \|_F^2+\frac{1}{\mu} \| W\|_F^4 \|\Lambda \|_F^2 \\ \Rightarrow \frac{\|\Sigma\|_F^2 - \frac{1}{\mu} \| W\|_F^4 \|\Lambda \|_F^2}{\|\Sigma - W\Lambda W^\top \|_F^2} \leq \frac{\|\Sigma - UTU^\top \|_F^2}{{\|\Sigma - W\Lambda W^\top \|_F^2}} \leq \frac{\|\Sigma \|_F^2+\frac{1}{\mu} \| W\|_F^4 \|\Lambda \|_F^2}{{\|\Sigma - W\Lambda W^\top \|_F^2}} \\ \Rightarrow \frac{\|\Sigma\|_F^2 - \frac{1}{\mu} \| W\|_F^4 \|\Lambda \|_F^2}{\|\Sigma\|_F^2 + \|W\Lambda W^\top \|_F^2} \leq \frac{\|\Sigma - UTU^\top \|_F^2}{{\|\Sigma - W\Lambda W^\top \|_F^2}} \leq \frac{\|\Sigma \|_F^2+\frac{1}{\mu} \| W\|_F^4 \|\Lambda \|_F^2}{{\|\Sigma\|_F^2 - \|W\Lambda W^\top \|_F^2}} \\ \end{align*} Assume that $$c\|\Sigma \|_F^2 = \| W\Lambda W^\top \|_F^2$$ where $$0 \leq c\leq 1$$, we get
\begin{align*} \frac{\|\Sigma\|_F^2 - \frac{1}{\mu} \| W\|_F^4 \|\Lambda \|_F^2}{(1+c)\|\Sigma\|_F^2 } \leq \frac{\|\Sigma - UTU^\top \|_F^2}{{\|\Sigma - W\Lambda W^\top \|_F^2}} \leq \frac{\|\Sigma \|_F^2+\frac{1}{\mu} \| W\|_F^4 \|\Lambda \|_F^2}{{(1-c)\|\Sigma\|_F^2 }} \\ \end{align*} As $$W$$, $$\Lambda$$ and $$\Sigma$$ are fixed, let $$\frac{\|W \|_F^4 \| \Lambda\|_F^2}{\|\Sigma \|_F^2} = t$$, then we have \begin{align*} \frac{1 - \frac{t}{\mu}}{(1+c) } \leq \frac{\|\Sigma - UTU^\top \|_F^2}{{\|\Sigma - W\Lambda W^\top \|_F^2}} \leq \frac{1 + \frac{t}{\mu}}{(1-c) } \\ \end{align*}