Any matrix $A \in \mathbb{C}^{m \times n}$ has a singular value decomposition given by
$$A = U_{m \times r} \Sigma_{r \times r} V_{r \times n}^*$$ where $r$ is the rank of the matrix $A$. The matrices $U_{m \times r}$ and $V_{n \times r}$ are orthonormal matrices i.e. $U^*U = I_{r \times r}$ and $V^* V = I_{r \times r}$. The matrix $\Sigma_{r \times r}$ is a diagonal matrix with its diagonal entries $$\Sigma_{11} \geq \Sigma_{22} \geq \Sigma_{33} \geq \cdots \geq \Sigma_{rr} > 0$$ i.e.
$$A = \underbrace{\begin{bmatrix} \vec{u}_1 & \vec{u}_2 & \cdots & \vec{u}_r\end{bmatrix}}_{U} \underbrace{\begin{bmatrix} \Sigma_{11} & 0 & 0 & \cdots & 0\\0 & \Sigma_{22} & 0 & \cdots & 0\\ 0 & 0 & \Sigma_{33} & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & \Sigma_{rr}\end{bmatrix}}_{\Sigma} \underbrace{\begin{bmatrix} \vec{v}_1 & \vec{v}_2 & \cdots & \vec{v}_r\end{bmatrix}^*}_{V^*}$$
where $\vec{u}_k \in \mathbb{C}^{m \times 1}, \vec{v}_k \in \mathbb{C}^{n \times 1}$ and $\vec{u}_j \cdot \vec{u}_{k} = \vec{v}_j \cdot \vec{v}_{k} = \delta_{jk}$.
Note that the above decomposition can also be written as shown below.
$$A = \Sigma_{11} \vec{u}_1 \vec{v}_1^* + \Sigma_{22} \vec{u}_2 \vec{v}_2^* + \Sigma_{33} \vec{u}_3 \vec{v}_3^* + \cdots + \Sigma_{rr} \vec{u}_r \vec{v}_r^*$$
If we have that $$\Sigma_{11} \geq \Sigma_{22} \geq \Sigma_{33} \geq \cdots \geq \Sigma_{\ell \ell} > \tau \geq \Sigma_{\ell+1,\ell+1} \geq \cdots \geq \Sigma_{rr} > 0$$ then we can write $A$ as
$$A = \underbrace{\Sigma_{11} \vec{u}_1 \vec{v}_1^* + \Sigma_{22} \vec{u}_2 \vec{v}_2^* + \cdots + \Sigma_{\ell, \ell} \vec{u}_{\ell} \vec{v}_{\ell}^*}_{A_{\ell}} + \underbrace{\Sigma_{\ell+1, \ell+1} \vec{u}_{\ell+1} \vec{v}_{\ell+1}^* + \cdots + \Sigma_{rr} \vec{u}_r \vec{v}_r^*}_{A-A_{\ell} = \tilde{A}_{\ell}}$$
Now note that $A_r$ can be written as
$$A_{\ell} = \underbrace{\begin{bmatrix} \vec{u}_1 & \vec{u}_2 & \cdots & \vec{u}_{\ell}\end{bmatrix}}_{U^{(\ell)}} \underbrace{\begin{bmatrix} \Sigma_{11} & 0 & 0 & \cdots & 0\\0 & \Sigma_{22} & 0 & \cdots & 0\\ 0 & 0 & \Sigma_{33} & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & \Sigma_{{\ell},{\ell}}\end{bmatrix}}_{\Sigma^{(\ell)}} \underbrace{\begin{bmatrix} \vec{v}_1 & \vec{v}_2 & \cdots & \vec{v}_{\ell}\end{bmatrix}^*}_{V^{(\ell)^*}}$$
and
$$\tilde{A}_{\ell} = \underbrace{\begin{bmatrix} \vec{u}_{\ell+1} & \vec{u}_{\ell+2} & \cdots & \vec{u}_{r}\end{bmatrix}}_{\tilde{U}^{(\ell)}} \underbrace{\begin{bmatrix} \Sigma_{{\ell+1},{\ell+1}} & 0 & \cdots & 0\\0 & \Sigma_{{\ell+2},{\ell+2}} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \Sigma_{r,r}\end{bmatrix}}_{\tilde{\Sigma}^{(\ell)}} \underbrace{\begin{bmatrix} \vec{v}_{\ell+1} & \vec{v}_{\ell+2} & \cdots & \vec{v}_{r}\end{bmatrix}^*}_{\tilde{V}^{(\ell)^*}}$$
where $\vec{u}_k \in \mathbb{C}^{m \times 1}, \vec{v}_k \in \mathbb{C}^{n \times 1}$ and $\vec{u}_j \cdot \vec{u}_{k} = \vec{v}_j \cdot \vec{v}_{k} = \delta_{jk}$.
Hence, you have the decomposition $$A = \underbrace{U^{(\ell)} \Sigma^{(\ell)} V^{(\ell)^*}}_{A_{\ell}} + \underbrace{\tilde{U}^{(\ell)} \tilde{\Sigma}^{(\ell)} \tilde{V}^{(\ell)^*}}_{\tilde{A}_{\ell}}$$ where the singular values of $\tilde{A}_{\ell}$ are not greater than $\tau$.