After all the regularization defined, optimization of the Eq. (
2) is one of the key to solve the tracking. The Eq. (
2) can be minimized using ADMM [
6] to achieve the optimal solution benefitting from its convexity. Specifically, we introduce the auxiliary variable
\({\mathbf {g}}={\mathbf {f}}\) and the step size parameter
\(\gamma \) to construct the following augmented Lagrange function,
$$\begin{aligned} {\mathcal {L}}&= \frac{1}{2}\left\| \sum _{c=1}^{C}{\mathbf {x}}^c_t * {\mathbf {f}}^c_t - {\mathbf {y}}\right\| ^2_F + \frac{\lambda }{2}\sum _{c=1}^{C}\left\| {\mathbf {w}}_t \odot {\mathbf {g}}^c_t\right\| ^2_F \nonumber \\&\quad + \frac{\mu }{2}\sum _{c=1}^{C}\left\| {\mathbf {f}}^c_t - {\mathbf {f}}^c_s\right\| ^2_F+ \frac{\eta }{2}\left\| \sum _{c=1}^{C}{\mathbf {x}}^c_t * {\mathbf {f}}^c_t - {\mathbf {r}}\right\| ^2_F \nonumber \\&\quad + \sum _{c=1}^{C}\left( {\mathbf {f}}^c_t - {\mathbf {g}}^c_t\right) ^\mathrm {T}{\mathbf {s}}^c_t+ \frac{\gamma }{2}\sum _{c=1}^{C}\left\| {\mathbf {f}}^c_t - {\mathbf {g}}^c_t\right\| ^2_F, \end{aligned}$$
(10)
where
\({\mathbf {r}}={\mathbf {R}}_{t-1}[\psi \bigtriangleup ]\), and
\({\mathbf {s}}\) refers to the Lagrange multiplier. By introducing
\({\mathbf {h}}=\frac{1}{\gamma }{\mathbf {s}}\), Eq. (
10) can be reformulated as,
$$\begin{aligned} \begin{aligned} {\mathcal {L}}&= \frac{1}{2}\left\| \sum _{c=1}^{C}{\mathbf {x}}^c_t * {\mathbf {f}}^c_t - {\mathbf {y}}\right\| ^2_F + \frac{\lambda }{2}\sum _{c=1}^{C}\left\| {\mathbf {w}}_t \odot {\mathbf {g}}^c_t\right\| ^2_F\\&\quad + \frac{\mu }{2}\sum _{c=1}^{C}\left\| {\mathbf {f}}^c_t - {\mathbf {f}}^c_s\right\| ^2_F+ \frac{\eta }{2}\left\| \sum _{c=1}^{C}{\mathbf {x}}^c_t * {\mathbf {f}}^c_t - {\mathbf {r}}\right\| ^2_F\\&\quad + \frac{\gamma }{2}\sum _{c=1}^{C}\left\| {\mathbf {f}}^c_t - {\mathbf {g}}^c_t + {\mathbf {h}}^c_t\right\| ^2_F. \end{aligned} \end{aligned}$$
(11)
Then, the following subproblems are alternately optimized via ADMM formulation.
$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} {\mathbf {f}}^{i+1} = \underset{{\mathbf {f}}}{\arg \min }\frac{1}{2}\left\| \sum _{c=1}^{C}{\mathbf {x}}^c_t * {\mathbf {f}}^c_t - {\mathbf {y}}\right\| ^2_F + \frac{\mu }{2}\sum _{c=1}^{C}\left\| {\mathbf {f}}^c_t - {\mathbf {f}}^c_s\right\| ^2_F\\ \qquad +\frac{\eta }{2}\left\| \sum _{c=1}^{C}{\mathbf {x}}^c_t * {\mathbf {f}}^c_t - {\mathbf {r}}\right\| ^2_F + \frac{\gamma }{2}\sum _{c=1}^{C}\left\| {\mathbf {f}}^c_t - {\mathbf {g}}^c_t + {\mathbf {h}}^c_t\right\| ^2_F\\ \\ {\mathbf {g}}^{i+1} = \underset{{\mathbf {g}}}{\arg \min }\frac{\lambda }{2}\sum _{c=1}^{C}\left\| {\mathbf {w}}_t \odot {\mathbf {g}}^c_t\right\| ^2_F + \frac{\gamma }{2}\sum _{c=1}^{C}\left\| {\mathbf {f}}^c_t - {\mathbf {g}}^c_t + {\mathbf {h}}^c_t\right\| ^2_F\\ \\ {\mathbf {h}}^{i+1} = {\mathbf {h}}^i+{\mathbf {f}}^{i+1}-{\mathbf {g}}^{i+1} \end{array}\right. }. \end{aligned}\nonumber \\ \end{aligned}$$
(12)
Subproblem \({\mathbf {f}}\): For the first subproblem of Eq. (
12), it can be transformed into the frequency domain using Parseval’s formulation as,
$$\begin{aligned} \begin{aligned} \widehat{{\mathbf {f}}}^*&= \underset{\widehat{{\mathbf {f}}}}{\arg \min }\frac{1}{2}\left\| \sum _{c=1}^{C}\widehat{{\mathbf {x}}}^c_t \odot \widehat{{\mathbf {f}}}^c_t - \widehat{{\mathbf {y}}}\right\| ^2_F + \frac{\mu }{2}\sum _{c=1}^{C}\left\| \widehat{{\mathbf {f}}}^c_t - \widehat{{\mathbf {f}}}^c_s\right\| ^2_F\\&\quad + \frac{\eta }{2}\left\| \sum _{c=1}^{C}\widehat{{\mathbf {x}}}^c_t \odot \widehat{{\mathbf {f}}}^c_t - \widehat{{\mathbf {r}}}\right\| ^2_F + \frac{\gamma }{2}\sum _{c=1}^{C}\left\| \widehat{{\mathbf {f}}}^c_t - \widehat{{\mathbf {g}}}^c_t + \widehat{{\mathbf {h}}}^c_t\right\| ^2_F, \end{aligned} \end{aligned}$$
(13)
where
\(^{\hat{}}\) denotes the discrete Fourier transform (DFT). The
j-th element of the label
\(\widehat{{\mathbf {y}}}\) relies on the
j-th element of the sample
\(\widehat{{\mathbf {x}}}_t\) and the filter
\(\widehat{{\mathbf {f}}}_t\) across all
C channels.
\({\mathcal {V}}\left( {\mathbf {f}}\right) \in {\mathbb {R}}^C\) is the vector consisting of the
j-th element of
\({\mathbf {f}}\) along the channels. Equation (
13) can be further decomposed into
\(M \times N\) subproblems, where each subproblem is defined as,
$$\begin{aligned} \begin{aligned} {{\mathcal {V}}_j(\widehat{{\mathbf {f}}}^*)}&= \underset{{\mathcal {V}}_j(\widehat{{\mathbf {f}}})}{\arg \min }\frac{1}{2}\left\| {\mathcal {V}}_j(\widehat{{\mathbf {x}}}_t)^{\mathrm {T}}{\mathcal {V}}_j(\widehat{{\mathbf {f}}}_t) - \widehat{{\mathbf {y}}}_j\right\| ^2_F\\&\quad + \frac{\mu }{2}\left\| {\mathcal {V}}_j(\widehat{{\mathbf {f}}}_t) - {\mathcal {V}}_j(\widehat{{\mathbf {f}}}_s)\right\| ^2_F\\&\quad + \frac{\eta }{2}\left\| {\mathcal {V}}_j(\widehat{{\mathbf {x}}}_t)^{\mathrm {T}}{\mathcal {V}}_j(\widehat{{\mathbf {f}}}_t) - \widehat{{\mathbf {r}}}_j\right\| ^2_F + \frac{\gamma }{2}\left\| {\mathcal {V}}_j(\widehat{{\mathbf {f}}}_t) \right. \\&\quad \left. - {\mathcal {V}}_j(\widehat{{\mathbf {g}}}_t) + {\mathcal {V}}_j(\widehat{{\mathbf {h}}}_t)\right\| ^2_F, \end{aligned} \end{aligned}$$
(14)
where superscript
\(^\mathrm {T}\) on a complex vector or matrix indicates conjugate transpose operation. Taking the derivative of Eq. (
14) as zero, the closed-form solution of
\({\mathcal {V}}_j(\widehat{{\mathbf {f}}}^*)\) can be denoted as,
$$\begin{aligned} {\mathcal {V}}_j(\widehat{{\mathbf {f}}}^*) = \left[ \left( 1+\eta \right) {\mathcal {V}}_j(\widehat{{\mathbf {x}}}_t){\mathcal {V}}_j(\widehat{{\mathbf {x}}}_t)^\mathrm {T} + \left( \mu +\gamma \right) \right] ^{-1}{\mathbf {q}}, \end{aligned}$$
(15)
where the vector
\({\mathbf {q}} = {\mathcal {V}}_j(\widehat{{\mathbf {x}}}_t)\widehat{{\mathbf {y}}}_j + \eta {\mathcal {V}}_j(\widehat{{\mathbf {x}}}_t)\widehat{{\mathbf {r}}}_j + \gamma {\mathcal {V}}_j(\widehat{{\mathbf {g}}}_t) - \gamma {\mathcal {V}}_j(\widehat{{\mathbf {h}}}_t) + \mu {\mathcal {V}}_j(\widehat{{\mathbf {f}}}_s)\). Since
\({\mathcal {V}}_j(\widehat{{\mathbf {x}}}_t){\mathcal {V}}_j(\widehat{{\mathbf {x}}}_t)^\mathrm {T}\) is a rank-1 matrix, Eq. (
15) can be further rewritten via the Sherman–Morrsion formulation [
32] as,
$$\begin{aligned} {\mathcal {V}}_j(\widehat{{\mathbf {f}}})^*=\frac{1}{\mu +\gamma }\left[ {\mathbf {I}}-\frac{{\mathcal {V}}_{j}(\widehat{{\mathbf {x}}}){\mathcal {V}}_{j}(\widehat{{\mathbf {x}}})^\mathrm {T}}{\frac{\mu +\gamma }{1+\eta }+{\mathcal {V}}_{j}(\widehat{{\mathbf {x}}})^\mathrm {T} {\mathcal {V}}_{j}(\widehat{{\mathbf {x}}})}\right] {\mathbf {q}}. \end{aligned}$$
(16)
Note that Eq. (
16) only contains vector multiply–add operation, thus it can be computed efficiently.
\({\mathbf {f}}\) can be further obtained by the IDFT of
\(\widehat{{\mathbf {f}}}\).
Subproblem \({\mathbf {g}}\): For the second subproblem of Eq. (
12), each element of
\({\mathbf {g}}\) can be computed independently as,
$$\begin{aligned} \begin{aligned} {\mathbf {g}}^* = \frac{\gamma \left( {\mathbf {f}} + {\mathbf {h}}\right) }{\lambda \left( {\mathbf {w}} \odot {\mathbf {w}}\right) + \gamma {\mathbf {I}}}. \end{aligned} \end{aligned}$$
(17)
Lagrangian multiplier update: The Lagrange multiplier is updated as,
$$\begin{aligned} \begin{aligned} {\mathbf {h}}^{i+1} = {\mathbf {h}}^i + {\mathbf {f}}^{*(i+1)} - {\mathbf {g}}^{*(i+1)}, \end{aligned} \end{aligned}$$
(18)
where the subscript
i represents the
i-th iteration.
\({\mathbf {f}}^*\) and
\({\mathbf {g}}^*\) are the solution of subproblem
\({\mathbf {f}}\) and
\({\mathbf {g}}\), respectively.