\subsection{General Approach}\label{sec:general-approach}
Although the problem \eqref{eqn:original_target} -- \eqref{eqn:original_rate_constraint} is not a standard GP, we can transform it to a Reversed GP by introducing an auxiliary variable ${t_0}$ \cite{Chiang2005}

\begin{eqnarray}
  {\mathop {\min }\limits_{{{\mathbf{S}}_P},{{\mathbf{S}}_I},\rho ,{t_0}} }&{1/{t_0}} \label{eqn:transformed_target} \\
  {{\text{ subject to }}}&{\frac{1}{2}\left[ {\left\| {{{\mathbf{S}}_I}} \right\|_F^2 + \left\| {{{\mathbf{S}}_P}} \right\|_F^2} \right] \leqslant P} \label{eqn:transformed_power_constraint} \\
  {}&{{t_0}/{z_{DC}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\rho } \right) \leqslant 1} \label{eqn:transformed_current_constraint} \\
  {}&{{2^{\bar R}}/\left[ {\prod\limits_{n = 0}^{N - 1} {\left( {1 + \frac{{(1 - \rho )}}{{\sigma _n^2}}{C_n}} \right)} } \right] \leqslant 1} \label{eqn:transformed_rate_constraint}
\end{eqnarray}

We cannot apply GP tools to the new problem yet, as $1/{z_{DC}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\rho } \right)$ and $1/\left[ {\prod\nolimits_{n = 0}^{N - 1} {\left( {1 + \frac{{(1 - \rho )}}{{\sigma _n^2}}{C_n}} \right)} } \right]$ are not posynomials. To solve this, \cite{Clerckx2018} suggested a conservative approach to approximate the terms with posynomials in the denominator by new posynomials, based on the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

Consider constraint \eqref{eqn:transformed_current_constraint} first. The posynomial at the denominator can be decomposed as the sum of monomials

\begin{equation}\label{eqn:transformed_current_posynomial_decomposition}
  {z_{DC}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\rho } \right) = \sum\limits_{k = 1}^K {{g_k}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\rho } \right)}
\end{equation}

Since monomial $\left\{ {{g_k}} \right\}$ is nonnegative for all $k$, the AM-GM inequality suggests a posynomial upper bound for the previous non-posynomial term

\begin{equation}\label{eqn:transformed_current_am_gm}
  \frac{1}{{\sum\limits_{k = 1}^K {{g_k}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\rho } \right)} }} \leqslant \prod\limits_{k = 1}^K {{{\left( {\frac{{{g_k}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\rho } \right)}}{{{\gamma _k}}}} \right)}^{ - {\gamma _k}}}}
\end{equation}

The nonnegative coefficients $\left\{ {{\gamma _k}} \right\}$ are chosen to satisfy $\sum\nolimits_{k = 1}^K {{\gamma _k}}  = 1$. Similarly, define $\bar \rho  = 1 - \rho $ and let $\left\{ {{g_{nk}}\left( {{{\mathbf{S}}_I},\bar \rho } \right)} \right\}$ be the monomials of the posynomial $1 + \frac{{\bar \rho }}{{\sigma _n^2}}{C_n}$, we have

\begin{equation}\label{eqn:transformed_rate_posynomial_decomposition}
  1 + \frac{{\bar \rho }}{{\sigma _n^2}}{C_n} = \sum\limits_{k = 1}^{{K_n}} {{g_{nk}}} \left( {{{\mathbf{S}}_I},\bar \rho } \right)
\end{equation}

Apply the AM-GM inequality to \eqref{eqn:transformed_rate_posynomial_decomposition}, we have

\begin{equation}\label{eqn:transformed_rate_am_gm}
  \frac{1}{{1 + \frac{{\bar \rho }}{{\sigma _n^2}}{C_n}}} \leqslant \prod\limits_{k = 1}^{{K_n}} {{{\left( {\frac{{{g_{nk}}\left( {{{\mathbf{S}}_I},\bar \rho } \right)}}{{{\gamma _{nk}}}}} \right)}^{ - {\gamma _{nk}}}}}
\end{equation}

with ${\gamma _{nk}} \geqslant 0$ and $\sum\nolimits_{k = 1}^{{K_n}} {{\gamma _{nk}}}  = 1$. In this way, we transformed the problem into a standard GP

\begin{eqnarray}
  {\mathop {\min }\limits_{{{\mathbf{S}}_P},{{\mathbf{S}}_I},\rho ,\bar \rho ,{t_0}} }&{1/{t_0}} \label{eqn:general_target} \\
  {{\text{ subject to }}}&{\frac{1}{2}\left[ {\left\| {{{\mathbf{S}}_I}} \right\|_F^2 + \left\| {{{\mathbf{S}}_P}} \right\|_F^2} \right] \leqslant P} \label{eqn:general_power_constraint} \\
  {}&{{t_0}\prod\limits_{k = 1}^K {{{\left( {\frac{{{g_k}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\rho } \right)}}{{{\gamma _k}}}} \right)}^{ - {\gamma _k}}}}  \leqslant 1} \label{eqn:general_current_constraint} \\
  {}&{2^{\bar R}}\prod\limits_{n = 0}^{N - 1} {\prod\limits_{k = 1}^{{K_n}} {{{\left( {\frac{{{g_{nk}}\left( {{{\mathbf{S}}_I},\bar \rho } \right)}}{{{\gamma _{nk}}}}} \right)}^{ - {\gamma _{nk}}}}} }  \leqslant 1 \label{eqn:general_rate_constraint} \\
  {}&{\rho  + \bar \rho  \leqslant 1} \label{eqn:general_ratio_constraint}
\end{eqnarray}

It is worth noting that the tightness of the AM-GM inequality depends on the choice of $\left\{ {{\gamma _k},{\gamma _{nk}}} \right\}$. In this paper, we employ the iterative method proposed in \cite{Clerckx2018} that updates the coefficient sets at iteration $i$ with the previous solution ${{\mathbf{S}}_P^{(i - 1)},{\mathbf{S}}_I^{(i - 1)},{\rho ^{(i - 1)}}}$ by

\begin{eqnarray}
  {{\gamma _k} = \frac{{{g_k}\left( {{\mathbf{S}}_P^{(i - 1)},{\mathbf{S}}_I^{(i - 1)},{\rho ^{(i - 1)}}} \right)}}{{{z_{DC}}\left( {{\mathbf{S}}_P^{(i - 1)},{\mathbf{S}}_I^{(i - 1)},{\rho ^{(i - 1)}}} \right)}},}&{k = 1, \ldots ,K} \\
  {{\gamma _{nk}} = \frac{{{g_{nk}}\left( {{\mathbf{S}}_I^{(i - 1)},{{\bar \rho }^{(i - 1)}}} \right)}}{{1 + \frac{{{{\bar \rho }^{(i - 1)}}}}{{\sigma _n^2}}{C_n}\left( {{\mathbf{S}}_I^{(i - 1)}} \right)}},}&\begin{gathered}
  n = 0, \ldots ,N - 1 \hfill \\
  k = 1, \ldots ,{K_n} \hfill \\
\end{gathered}
\end{eqnarray}

Once $\left\{ {{\gamma _k},{\gamma _{nk}}} \right\}$ are obtained, we solve \eqref{eqn:general_target} -- \eqref{eqn:general_ratio_constraint} to obtain ${\mathbf{S}}_P^{(i)},{\mathbf{S}}_I^{(i)},{\rho ^{(i)}}$. The iteration is repeated until it converges. Algorithm \ref{alg:general} summarizes the procedures involved in the optimization. The successive approximation approach is also known as inner approximation method \cite{Marks1978}, which cannot guarantee a global optimal solution but the result satisfies the Karush–Kuhn–Tucker (KKT) conditions.

\begin{algorithm}
  \caption{General Waveform Design}
  \label{alg:general}
  \begin{algorithmic}[1]
    \State \textbf{Initialize:} $i \leftarrow 0$, ${\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star $ in \eqref{eqn:optimal_phases}, ${{{\mathbf{S}}_P},{{\mathbf{S}}_I}}$ in \eqref{eqn:initial_amplitude}, $\rho ,\bar \rho  = 1 - \rho ,\bar R,z_{DC}^{(0)} = 0$
    \Repeat
      \State $i \leftarrow i + 1,\mathop {{{\mathbf{S}}_P}}\limits^{..}  \leftarrow {{\mathbf{S}}_P},\mathop {{{\mathbf{S}}_I}}\limits^{..}  \leftarrow {{\mathbf{S}}_I},\ddot \rho  \leftarrow \rho ,\ddot{\bar{\rho}}  \leftarrow \bar \rho $
      \State ${\gamma _k} \leftarrow {g_k}\left( {\mathop {{{\mathbf{S}}_P}}\limits^{..} ,\mathop {{{\mathbf{S}}_I}}\limits^{..} ,{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\ddot \rho } \right)/{z_{DC}}\left( {\mathop {{{\mathbf{S}}_P}}\limits^{..} ,\mathop {{{\mathbf{S}}_I}}\limits^{..} ,{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\ddot \rho } \right),k = 1, \ldots ,K$
      \State ${\gamma _{nk}} \leftarrow {g_{nk}}\left( {\mathop {{{\mathbf{S}}_I}}\limits^{..} ,\ddot{\bar{\rho}} } \right)/\left( {1 + \frac{{\ddot{\bar{\rho}}}}{{\sigma _n^2}}{C_n}\left( {\mathop {{{\mathbf{S}}_I}}\limits^{..} } \right)} \right),n = 0, \ldots ,N - 1,k = 1, \ldots ,{K_n}$
      \State ${{\mathbf{S}}_P},{{\mathbf{S}}_I},\rho ,\bar \rho  \leftarrow \arg \min $ \eqref{eqn:general_target} -- \eqref{eqn:general_ratio_constraint}
      \State $z_{DC}^{(i)} \leftarrow {z_{DC}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\rho } \right)$
    \Until{$\left| {z_{DC}^{(i)} - z_{DC}^{(i - 1)}} \right| < $ or $i = {i_{\max }}$}
  \end{algorithmic}
\end{algorithm}

\subsection{Decoupled Design}\label{sec:decoupled-design}
For the transmitter with multiple antennas ($M > 1$), the previous method is involved with weight design across space and frequency. In this part, we will investigate an approach proposed in \cite{Clerckx2018} that decouples the optimization in spatial and frequency domains without impacting performance. As suggested by \eqref{eqn:mutual_information} and \eqref{eqn:power_waveform_second_order} -- \eqref{eqn:waveform_end}, the optimum weight vectors ${{\mathbf{w}}_{P,n}}$ and ${{\mathbf{w}}_{I,n}}$ that maximize the rate and energy correspond to the MRT beamformers, which are given by

\begin{align}\label{eqn:mrt_weights}
  {{\mathbf{w}}_{P,n}} &= {s_{P,n}}{\mathbf{h}}_n^H/\left\| {{{\mathbf{h}}_n}} \right\| \\
  {{\mathbf{w}}_{I,n}} &= {s_{I,n}}{\mathbf{h}}_n^H/\left\| {{{\mathbf{h}}_n}} \right\|
\end{align}

Therefore, the received power and information signals \eqref{eqn:received_signal} rewrites as

\begin{align}\label{eqn:mrt_received_components}
  {y_P}(t) &= \sum\limits_{n = 0}^{N - 1} {\left\| {{{\mathbf{h}}_n}} \right\|} {s_{P,n}}\cos \left( {{w_n}t} \right) = \Re \left\{ {\sum\limits_{n = 0}^{N - 1} {\left\| {{{\mathbf{h}}_n}} \right\|} {s_{P,n}}{e^{j{w_n}t}}} \right\} \\
  {y_I}(t) &= \sum\limits_{n = 0}^{N - 1} {\left\| {{{\mathbf{h}}_n}} \right\|} {s_{I,n}}{{\tilde x}_n}\cos \left( {{w_n}t} \right) = \Re \left\{ {\sum\limits_{n = 0}^{N - 1} {\left\| {{{\mathbf{h}}_n}} \right\|} {s_{I,n}}{{\tilde x}_n}{e^{j{w_n}t}}} \right\}
\end{align}

In this way, the weight optimization on multiple transmit antennas is converted into an equivalent problem on a single antenna. For the $n$-th subband, the equivalent channel gain is $\left\| {{{\mathbf{h}}_n}} \right\|$ with the power allocated to multisine and modulated waveform denoted by $s_{P,n}^2$ and $s_{I,n}^2$ ($\frac{1}{2}\sum\nolimits_{n = 0}^{N - 1} {\left( {s_{P,n}^2 + s_{I,n}^2} \right)}  \leqslant P$). The problem can be solved using Algorithm \ref{alg:general}, with the second and fourth order terms reduced to

\begin{align}\label{eqn:decoupled_terms}
  \mathbb{E}\left[ {{y_P}{{(t)}^2}} \right] &= \frac{1}{2}\sum\limits_{n = 0}^{N - 1} {{{\left\| {{{\mathbf{h}}_n}} \right\|}^2}} s_{P,n}^2 \\
  \mathbb{E}\left[ {{y_P}{{(t)}^4}} \right] &= \frac{3}{8}\sum\limits_{\substack{ {n_0},{n_1},{n_2},{n_3} \\ {n_0} + {n_1} = {n_2} + {n_3} }}  {\left[ {\prod\limits_{j = 0}^3 {{s_{P,{n_j}}}} \left\| {{{\mathbf{h}}_{{n_j}}}} \right\|} \right]}  \\
  \mathbb{E}\left[ {{y_I}{{(t)}^2}} \right] &= \frac{1}{2}\sum\limits_{n = 0}^{N - 1} {{{\left\| {{{\mathbf{h}}_n}} \right\|}^2}} s_{I,n}^2 \\
  \mathbb{E}\left[ {{y_I}{{(t)}^4}} \right] &= \frac{6}{8}{\left[ {\sum\limits_{n = 0}^{N - 1} {{{\left\| {{{\mathbf{h}}_n}} \right\|}^2}} s_{I,n}^2} \right]^2}
\end{align}

Hence, the target function ${z_{DC}}$ is only a function of two $N$-dimensional vectors ${{\mathbf{s}}_{P/I}} = \left[ {{s_{P/I,0}}, \ldots ,{s_{P/I,N - 1}}} \right]$, and the mutual information $I$ can be simplified as

\begin{equation}\label{eqn:decoupled_mutual_information}
  I\left( {{{\mathbf{s}}_I},\rho } \right) = {\log _2}\left( {\prod\limits_{n = 0}^{N - 1} {\left( {1 + \frac{{(1 - \rho )}}{{\sigma _n^2}}s_{I,n}^2{{\left\| {{{\mathbf{h}}_n}} \right\|}^2}} \right)} } \right)
\end{equation}

Similarly, we decompose the posynomials ${z_{DC}}\left( {{{\mathbf{s}}_P},{{\mathbf{s}}_I},\rho } \right) = \sum\limits_{k = 1}^K {{g_k}} \left( {{{\mathbf{s}}_P},{{\mathbf{s}}_I},\rho } \right)$ and $1 + \frac{{\bar \rho }}{{\sigma _n^2}}{C_n} = \sum\limits_{k = 1}^{{K_n}} {{g_{nk}}} \left( {{{\mathbf{s}}_I},\bar \rho } \right)$ with ${C_n} = s_{I,n}^2{\left\| {{{\mathbf{h}}_n}} \right\|^2}$, then apply the AM-GM inequality to the constraints with posynomials in the denominator. The equivalent GP problem write as

\begin{eqnarray}
  {\mathop {\min }\limits_{{{\mathbf{s}}_P},{{\mathbf{s}}_I},\rho ,\bar \rho ,{t_0}} }&{1/{t_0}} \label{eqn:decoupled_target} \\
  {{\text{ subject to }}}&{\frac{1}{2}\left[ {\left\| {{{\mathbf{s}}_I}} \right\|^2 + \left\| {{{\mathbf{s}}_P}} \right\|^2} \right] \leqslant P} \label{eqn:decoupled_power_constraint} \\
  {}&{{t_0}\prod\limits_{k = 1}^K {{{\left( {\frac{{{g_k}\left( {{{\mathbf{s}}_P},{{\mathbf{s}}_I},\rho } \right)}}{{{\gamma _k}}}} \right)}^{ - {\gamma _k}}}}  \leqslant 1} \label{eqn:decoupled_current_constraint} \\
  {}&{2^{\bar R}}\prod\limits_{n = 0}^{N - 1} {\prod\limits_{k = 1}^{{K_n}} {{{\left( {\frac{{{g_{nk}}\left( {{{\mathbf{s}}_I},\bar \rho } \right)}}{{{\gamma _{nk}}}}} \right)}^{ - {\gamma _{nk}}}}} }  \leqslant 1 \label{eqn:decoupled_rate_constraint} \\
  {}&{\rho  + \bar \rho  \leqslant 1} \label{eqn:decoupled_ratio_constraint}
\end{eqnarray}

Following \eqref{eqn:initial_amplitude}, the amplitudes of power and information waveform can be initialized to

\begin{equation}\label{eqn:initial_amplitude_decoupled}
  {s_{P,n}} = {s_{I,n}} = c\left\| {{{\mathbf{h}}_n}} \right\|
\end{equation}

Compared with the general approach, the decoupled design guarantees the same performance by a joint space-frequency design with a lower computational complexity, which converts the original $N \times M$ matrices ${{{\mathbf{S}}_P},{{\mathbf{S}}_I}}$ to $N$-dimensional vectors ${{{\mathbf{s}}_P},{{\mathbf{s}}_I}}$ via MRT beamformers. Algorithm \ref{alg:decoupled} summarizes the optimization process of the decoupling strategy.

\begin{algorithm}
  \caption{Decoupled Waveform Design}
  \label{alg:decoupled}
  \begin{algorithmic}[1]
    \State \textbf{Initialize:} $i \leftarrow 0$, ${\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star $ in \eqref{eqn:optimal_phases}, ${{{\mathbf{s}}_P},{{\mathbf{s}}_I}}$ in \eqref{eqn:initial_amplitude_decoupled}, $\rho ,\bar \rho  = 1 - \rho ,\bar R,z_{DC}^{(0)} = 0$
    \Repeat
      \State $i \leftarrow i + 1,\mathop {{{\mathbf{s}}_P}}\limits^{..}  \leftarrow {{\mathbf{s}}_P},\mathop {{{\mathbf{s}}_P}}\limits^{..}  \leftarrow {{\mathbf{s}}_I},\ddot \rho  \leftarrow \rho ,\ddot{\bar{\rho}}  \leftarrow \bar \rho $
      \State ${\gamma _k} \leftarrow {g_k}\left( {{{\mathop {\mathbf{s}}\limits^{..} }_P},{{\mathop {\mathbf{s}}\limits^{..} }_I},{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\ddot \rho } \right)/{z_{DC}}\left( {{{\mathop {\mathbf{s}}\limits^{..} }_P},{{\mathop {\mathbf{s}}\limits^{..} }_I},{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\ddot \rho } \right),k = 1, \ldots ,K$
      \State ${\gamma _{nk}} \leftarrow {g_{nk}}\left( {{{\mathop {\mathbf{s}}\limits^{..} }_I},\ddot{\bar{\rho}} } \right)/\left( {1 + \frac{{\ddot{\bar{\rho}} }}{{\sigma _n^2}}{C_n}\left( {{{\mathop {\mathbf{s}}\limits^{..} }_I}} \right)} \right),n = 0, \ldots ,N - 1,k = 1, \ldots ,{K_n}$
      \State ${{\mathbf{s}}_P},{{\mathbf{s}}_I},\rho ,\bar \rho  \leftarrow \arg \min $ \eqref{eqn:decoupled_target} -- \eqref{eqn:decoupled_ratio_constraint}
      \State $z_{DC}^{(i)} \leftarrow {z_{DC}}\left( {{{\mathbf{s}}_P},{{\mathbf{s}}_I},{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\rho } \right)$
    \Until{$\left| {z_{DC}^{(i)} - z_{DC}^{(i - 1)}} \right| < $ or $i = {i_{\max }}$}
  \end{algorithmic}
\end{algorithm}

\subsection{Lower Bound}\label{sec:lower-bound}
The deterministic multisine waveform not only boosts the harvested energy but also avoids interference to the modulated waveform. To highlight its benefit on rate and energy, we compare the performance of superposed waveform to two baselines. In the first case, there is no multisine component. Only modulated waveform is used for WIPT (i.e. ${{\mathbf{S}}_P} = 0,\frac{1}{2}\left\| {{{\mathbf{S}}_I}} \right\|_F^2 = P$) and the twofold benefit disappears. In the second baseline, it is assumed that the power waveform behaves as a deterministic multisine from WPT perspective but as CSCG distributed from WIT perspective. Therefore, the energy benefit of the multisine is maintained but the rate benefit is lost. The power waveform creates an interference term $\sqrt {1 - \rho } {{\mathbf{h}}_n}{{\mathbf{w}}_{P,n}}$ to the information waveform, and the lower bound of the mutual information writes as

\begin{equation}\label{eqn:mutual_information_lower_bound}
  {I_{LB}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{{\mathbf{\Phi }}_P},{{\mathbf{\Phi }}_I},\rho } \right) = \sum\limits_{n = 0}^{N - 1} {{{\log }_2}} \left( {1 + \frac{{(1 - \rho ){{\left| {{{\mathbf{h}}_n}{{\mathbf{w}}_{I,n}}} \right|}^2}}}{{\sigma _n^2 + (1 - \rho ){{\left| {{{\mathbf{h}}_n}{{\mathbf{w}}_{P,n}}} \right|}^2}}}} \right)
\end{equation}

It leads to a smaller rate-energy region than the ideal case. Also, the MRT beamformers ${{\mathbf{w}}_{P,n}},{{\mathbf{w}}_{I,n}}$ in \eqref{eqn:mrt_weights} are suboptimal due to the interference, and the corresponding phases ${\mathbf{\Phi }}_P^ \star, {\mathbf{\Phi }}_I^ \star $ in \eqref{eqn:optimal_phases} are not the best solution for $M > 1$. Minimum Mean Squared Error (MMSE) combiner can be further exploited for a better joint design over space and frequency domains.

Consider the suboptimal phases ${\mathbf{\Phi }}_P^ \star $ and ${\mathbf{\Phi }}_I^ \star $ in \eqref{eqn:optimal_phases} for simplicity. In such cases, the target function is still as \eqref{eqn:target_function_truncated} while the lower bound of the achievable rate is now

\begin{equation}\label{mutual_information_lower_bound_rewritten}
  {I_{LB}}\left( {{{\mathbf{S}}_I},{\mathbf{\Phi }}_I^ \star ,\rho } \right) = {\log _2}\left( {\prod\limits_{n = 0}^{N - 1} {\left( {1 + \frac{{(1 - \rho ){C_n}}}{{\sigma _n^2 + (1 - \rho ){D_n}}}} \right)} } \right)
\end{equation}

with ${C_n} = \sum\nolimits_{{m_0},{m_1}} {\prod\nolimits_{j = 0}^1 {{s_{I,n,{m_j}}}{A_{n,{m_j}}}} } $ and ${D_n} = \sum\nolimits_{{m_0},{m_1}} {\prod\nolimits_{j = 0}^1 {{s_{P,n,{m_j}}}{A_{n,{m_j}}}} } $. Thus, the previous rate constraint \eqref{eqn:transformed_rate_constraint} is replaced by

\begin{equation}\label{eqn:transformed_rate_constraint_lower_bound}
  {2^{\bar R}}\frac{{\prod\limits_{n = 0}^{N - 1} {\left( {1 + \frac{{\bar \rho }}{{\sigma _n^2}}{D_n}} \right)} }}{{\prod\limits_{n = 0}^{N - 1} {\left( {1 + \frac{{\bar \rho }}{{\sigma _n^2}}\left( {{D_n} + {C_n}} \right)} \right)} }} \leqslant 1
\end{equation}

Decompose the posynomials in the denominators as

\begin{equation}\label{eqn:posynomial_lower_bound}
  1 + \frac{{\bar \rho }}{{\sigma _n^2}}\left( {{D_n} + {C_n}} \right) = \sum\limits_{j = 1}^{{J_n}} {{f_{nj}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},\rho } \right)}
\end{equation}

where $\left\{ {{f_{nj}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},\rho } \right)} \right\}$ is the monomial terms. With a proper choice of nonnegative $\left\{ {{\gamma _{nj}}} \right\}$ satisfying $\sum\nolimits_{j = 1}^{{J_n}} {{\gamma _{nj}}}  = 1$, the standard GP can be written as

\begin{eqnarray}
  {\mathop {\min }\limits_{{{\mathbf{S}}_P},{{\mathbf{S}}_I},\rho ,\bar \rho ,{t_0}} }&{1/{t_0}} \label{eqn:lower_bound_target} \\
  {{\text{subject to}}}&{\frac{1}{2}\left[ {\left\| {{{\mathbf{S}}_I}} \right\|_F^2 + \left\| {{{\mathbf{S}}_P}} \right\|_F^2} \right] \leqslant P} \label{eqn:lower_bound_power_constraint} \\
  {}&{{t_0}\prod\limits_{k = 1}^K {{{\left( {\frac{{{g_k}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\rho } \right)}}{{{\gamma _k}}}} \right)}^{ - {\gamma _k}}}}  \leqslant 1} \label{eqn:lower_bound_current_constraint} \\
  {}&{{2^{\bar R}}\prod\limits_{n = 0}^{N - 1} {\left( {1 + \frac{{\bar \rho }}{{\sigma _n^2}}{D_n}\left( {{{\mathbf{S}}_P}} \right)} \right)} \prod\limits_{j = 1}^{{J_n}} {{{\left( {\frac{{{f_{nj}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},\rho } \right)}}{{{\gamma _{nj}}}}} \right)}^{ - {\gamma _{nj}}}}}  \leqslant 1} \label{eqn:lower_bound_rate_constraint} \\
  {}&{\rho  + \bar \rho  \leqslant 1 \label{eqn:lower_bound_ratio_constraint}}
\end{eqnarray}

Algorithm \ref{alg:lower-bound} shows the basic idea to obtain the lower-bound of R-E region. It boils down to Algorithm \ref{alg:general} when the interference posynomial ${D_n} = 0$ for all $n$.

\begin{algorithm}
  \caption{Lower-Bound of R-E Region}
  \label{alg:lower-bound}
  \begin{algorithmic}[1]
    \State \textbf{Initialize:} $i \leftarrow 0$, ${{\mathbf{w}}_{P,n}},{{\mathbf{w}}_{I,n}}$ in \eqref{eqn:mrt_weights}, ${{{\mathbf{S}}_P},{{\mathbf{S}}_I}}$ in \eqref{eqn:initial_amplitude}, $\rho ,\bar \rho  = 1 - \rho ,\bar R,z_{DC}^{(0)} = 0$
    \Repeat
      \State $i \leftarrow i + 1,\mathop {{{\mathbf{S}}_P}}\limits^{..}  \leftarrow {{\mathbf{S}}_P},\mathop {{{\mathbf{S}}_I}}\limits^{..}  \leftarrow {{\mathbf{S}}_I},\ddot \rho  \leftarrow \rho ,\ddot{\bar{\rho}}  \leftarrow \bar \rho $
      \State ${\gamma _k} \leftarrow {g_k}\left( {\mathop {{{\mathbf{S}}_P}}\limits^{..} ,\mathop {{{\mathbf{S}}_I}}\limits^{..} ,\ddot \rho } \right)/{z_{DC}}\left( {\mathop {{{\mathbf{S}}_P}}\limits^{..} ,\mathop {{{\mathbf{S}}_I}}\limits^{..} ,\ddot \rho } \right),k = 1, \ldots ,K$
      \State ${\gamma _{nj}} \leftarrow {f_{nj}}\left( {\mathop {{{\mathbf{S}}_P}}\limits^{..} ,\mathop {{{\mathbf{S}}_I}}\limits^{..} ,\ddot \rho } \right)/\left( {1 + \frac{{\ddot{\bar{\rho}}}}{{\sigma _n^2}}\left( {{D_n}\left( {\mathop {{{\mathbf{S}}_P}}\limits^{..} } \right) + {C_n}\left( {\mathop {{{\mathbf{S}}_I}}\limits^{..} } \right)} \right)} \right)$
      \State ${{\mathbf{S}}_P},{{\mathbf{S}}_I},\rho ,\bar \rho  \leftarrow \arg \min $ \eqref{eqn:lower_bound_target} -- \eqref{eqn:lower_bound_ratio_constraint}
      \State $z_{DC}^{(i)} \leftarrow {z_{DC}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},\rho } \right)$
    \Until{$\left| {z_{DC}^{(i)} - z_{DC}^{(i - 1)}} \right| < $ or $i = {i_{\max }}$}
  \end{algorithmic}
\end{algorithm}

\subsection{PAPR Constraints}\label{sec:papr-constraints}
Another practical constraint at the transmitter is PAPR. We assume the modulated information waveform is with unit PAPR (by PSK or FSK) so that the limitation only influence the design of multisine power waveform. Following \eqref{eqn:power_waveform}, the PAPR constraint on antenna $m$ writes as

\begin{equation}\label{eqn:papr_power_waveform}
  {\text{PAPR}_m} = \frac{{\mathop {\max }\limits_t {{\left| {{x_{P,m}}(t)} \right|}^2}}}{{\mathbb{E}\left[ {{{\left| {{x_{P,m}}(t)} \right|}^2}} \right]}} = \frac{{\mathop {\max }\limits_t {{\left| {{x_{P,m}}(t)} \right|}^2}}}{{\frac{1}{2}{{\left\| {{{\mathbf{s}}_{P,m}}} \right\|}^2}}} \leqslant \eta
\end{equation}

We assume the optimum phases ${\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star $ in \eqref{eqn:optimal_phases} are used in the optimization of ${{\mathbf{S}}_I},{{\mathbf{S}}_P}$. To handle the PAPR constraint \eqref{eqn:papr_power_waveform}, we introduce an oversampling factor ${O_s}$ to sample the power waveform at ${t_q} = qT/N{O_s}$ for $q = 0, \ldots ,N{O_s} - 1$ with $T = 1/\Delta f$. For a sufficiently large ${O_s}$, the PAPR constraint can be expressed as

\begin{equation}\label{eqn:papr_sample}
  {\left| {{x_{P,m}}\left( {{t_q}} \right)} \right|^2} \leqslant \frac{1}{2}\eta {\left\| {{{\mathbf{s}}_{P,m}}} \right\|^2}
\end{equation}

where the l.h.s. obtained from equation \eqref{eqn:power_waveform} is

\begin{equation}\label{eqn:papr_average_sample}
  {\left| {{x_{P,m}}\left( {{t_q}} \right)} \right|^2} = \sum\limits_{{n_0},{n_1}} {{s_{P,{n_0},m}}{s_{P,{n_1},m}}\cos \left( {{w_{{n_0}}}{t_q} + \phi _{P,{n_0},m}^ \star } \right)\cos \left( {{w_{{n_1}}}{t_q} + \phi _{P,{n_1},m}^ \star } \right)}
\end{equation}

However, ${\left| {{x_{P,m}}\left( {{t_q}} \right)} \right|^2}$ is no longer a posynomial as some coefficients can be negative with time-varying arguments. It is named signomial \cite{Boyd2007} and can be decomposed either as the sum of monomials or as the difference of two posynomials

\begin{equation}\label{eqn:papr_signomial}
  {\left| {{x_{P,m}}\left( {{t_q}} \right)} \right|^2} = {f_{mq}}\left( {{{\mathbf{S}}_P},{\mathbf{\Phi }}_P^ \star } \right) = {f_{mq1}}\left( {{{\mathbf{S}}_P},{\mathbf{\Phi }}_P^ \star } \right) - {f_{mq2}}\left( {{{\mathbf{S}}_P},{\mathbf{\Phi }}_P^ \star } \right)
\end{equation}

Therefore, the PAPR constraint rewrites as

\begin{equation}\label{eqn:papr_standard}
  \frac{{{f_{mq1}}\left( {{{\mathbf{S}}_P},{\mathbf{\Phi }}_P^ \star } \right)}}{{\frac{1}{2}\eta {{\left\| {{{\mathbf{s}}_{P,m}}} \right\|}^2} + {f_{mq2}}\left( {{{\mathbf{S}}_P},{\mathbf{\Phi }}_P^ \star } \right)}} \leqslant 1
\end{equation}

Similarly, denote the posynomial at the denominator as

\begin{equation}\label{eqn:papr_denominator}
  \frac{1}{2}\eta {\left\| {{{\mathbf{s}}_{P,m}}} \right\|^2} + {f_{mq2}}\left( {{{\mathbf{S}}_P},{\mathbf{\Phi }}_P^ \star } \right) = \sum\limits_{k = 1}^{{K_{mq2}}} {{g_{mq2k}}} \left( {{{\mathbf{S}}_P},{\mathbf{\Phi }}_P^ \star } \right)
\end{equation}

With a proper choice of nonnegative $\left\{ {{\gamma _{mq2k}}} \right\}$ satisfying $\sum\nolimits_{k = 1}^{{K_{mq2}}} {{\gamma _{mq2k}}}  = 1$, we apply AM-GM inequality to \eqref{eqn:papr_standard} and obtain the new constraint

\begin{equation}\label{eqn:papr_equivalent_inequality}
  {f_{mq1}}\left( {{{\mathbf{S}}_P},{\mathbf{\Phi }}_P^ \star } \right)\prod\limits_{k = 1}^{{K_{mq2}}} {{{\left( {\frac{{{g_{mq2k}}\left( {{{\mathbf{S}}_P},{\mathbf{\Phi }}_P^ \star } \right)}}{{{\gamma _{mq2k}}}}} \right)}^{ - {\gamma _{mq2k}}}}}  \leqslant 1
\end{equation}

In this way, the optimization problem \eqref{eqn:general_target} -- \eqref{eqn:general_ratio_constraint} with an extra PAPR constraint \eqref{eqn:papr_power_waveform} is replaced by a standard GP

\begin{eqnarray}
  {\mathop {\min }\limits_{{{\mathbf{S}}_P},{{\mathbf{S}}_I},\rho ,\bar \rho ,{t_0}} }&{1/{t_0}} \label{eqn:papr_target} \\
  {{\text{ subject to }}}&{\frac{1}{2}\left[ {\left\| {{{\mathbf{S}}_I}} \right\|_F^2 + \left\| {{{\mathbf{S}}_P}} \right\|_F^2} \right] \leqslant P} \label{eqn:papr_power_constraint} \\
  {}&{{t_0}\prod\limits_{k = 1}^K {{{\left( {\frac{{{g_k}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\rho } \right)}}{{{\gamma _k}}}} \right)}^{ - {\gamma _k}}}}  \leqslant 1} \label{eqn:papr_current_constraint} \\
  {}&{{2^{\bar R}}\prod\limits_{n = 0}^{N - 1} {\prod\limits_{k = 1}^{{K_n}} {{{\left( {\frac{{{g_{nk}}\left( {{{\mathbf{S}}_I},\bar \rho } \right)}}{{{\gamma _{nk}}}}} \right)}^{ - {\gamma _{nk}}}}} }  \leqslant 1} \label{eqn:papr_rate_constraint} \\
  {}&{{f_{mq1}}\left( {{{\mathbf{S}}_P},{\mathbf{\Phi }}_P^ \star } \right)\prod\limits_{k = 1}^{{K_{mq2}}} {{{\left( {\frac{{{g_{mq2k}}\left( {{{\mathbf{S}}_P},{\mathbf{\Phi }}_P^ \star } \right)}}{{{\gamma _{mq2k}}}}} \right)}^{ - {\gamma _{mq2k}}}}}  \leqslant 1} \label{eqn:papr_papr_constraint} \\
  {}&{\rho  + \bar \rho  \leqslant 1} \label{eqn:papr_ratio_constraint}
\end{eqnarray}

Algorithm \ref{alg:papr} shows the gist of the optimization procedure. For the system with multiple transmit antenna and valid PAPR constraints, the decoupling strategy is suboptimal since the arguments of cosines are indeed frequency-dependent. Also, the multisine waveform is oversampled to satisfy the PAPR constraint, which further increases the overall computational complexity.

\begin{algorithm}
  \caption{Waveform Design with PAPR Constraints}
  \label{alg:papr}
  \begin{algorithmic}[1]
    \State \textbf{Initialize:} $i \leftarrow 0$, ${\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star $ in \eqref{eqn:optimal_phases}, ${{{\mathbf{S}}_P},{{\mathbf{S}}_I}}$ in \eqref{eqn:initial_amplitude}, $\rho ,\bar \rho  = 1 - \rho ,\bar R,z_{DC}^{(0)} = 0$
    \Repeat
      \State $i \leftarrow i + 1,\mathop {{{\mathbf{S}}_P}}\limits^{..}  \leftarrow {{\mathbf{S}}_P},\mathop {{{\mathbf{S}}_I}}\limits^{..}  \leftarrow {{\mathbf{S}}_I},\ddot \rho  \leftarrow \rho ,\ddot{\bar{\rho}}  \leftarrow \bar \rho $
      \State ${\gamma _k} \leftarrow {g_k}\left( {\mathop {{{\mathbf{S}}_P}}\limits^{..} ,\mathop {{{\mathbf{S}}_I}}\limits^{..} ,{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\ddot \rho } \right)/{z_{DC}}\left( {\mathop {{{\mathbf{S}}_P}}\limits^{..} ,\mathop {{{\mathbf{S}}_I}}\limits^{..} ,{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\ddot \rho } \right),k = 1, \ldots ,K$
      \State ${\gamma _{nk}} \leftarrow {g_{nk}}\left( {\mathop {{{\mathbf{S}}_I}}\limits^{..} ,\ddot{\bar{\rho}}} \right)/\left( {1 + \frac{{\ddot{\bar{\rho}}}}{{\sigma _n^2}}{C_n}\left( {\mathop {{{\mathbf{S}}_I}}\limits^{..} } \right)} \right),n = 0, \ldots ,N - 1,k = 1, \ldots ,{K_n}$
      \State ${\gamma _{mq2k}} \leftarrow {g_{mq2k}}\left( {\mathop {{{\mathbf{S}}_P}}\limits^{..} ,{\mathbf{\Phi }}_P^ \star } \right)/\left( {\frac{1}{2}\eta {{\left\| {\mathop {{{\mathbf{s}}_{P,m}}}\limits^{..} } \right\|}^2} + {f_{mq2}}\left( {\mathop {{{\mathbf{S}}_P}}\limits^{..} ,{\mathbf{\Phi }}_P^ \star } \right)} \right),$
      \Statex $m = 1, \ldots ,M,q = 0, \ldots ,N{O_s} - 1,k = 1, \ldots ,{K_{mq2}}$
      \State ${{\mathbf{S}}_P},{{\mathbf{S}}_I},\rho ,\bar \rho  \leftarrow \arg \min $ \eqref{eqn:papr_target} -- \eqref{eqn:papr_ratio_constraint}
      \State $z_{DC}^{(i)} \leftarrow {z_{DC}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^ \star ,{\mathbf{\Phi }}_I^ \star ,\rho } \right)$
    \Until{$\left| {z_{DC}^{(i)} - z_{DC}^{(i - 1)}} \right| < $ or $i = {i_{\max }}$}
  \end{algorithmic}
\end{algorithm}

\subsection{Multiple Rectennas}\label{sec:multiple-rectennas}
The R-E region is expected to be enlarged by using multiple rectennas. In this part, we extend the general MISO strategy in \cite{Clerckx2018} to $U$ rectennas, which can either serve a single user in a point-to-point MIMO or spread across multiple users in an MU-MISO. Note that there exists a tradeoff between the energy harvested in different rectennas since they have different preference on the transmitted waveform. The fairness issue can be solved by introducing weight ${v_u}$ for rectenna $u = 1, \ldots ,U$ and considering the weighted sum of DC components as a new target function

\begin{equation}\label{eqn:weighted_target}
  {Z_{DC}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{{\mathbf{\Phi }}_P},{{\mathbf{\Phi }}_I},\rho } \right) = \sum\limits_{u = 1}^U {{v_u}{z_{DC,u}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{{\mathbf{\Phi }}_P},{{\mathbf{\Phi }}_I},\rho } \right)}
\end{equation}

With multiple rectennas, the frequency response is extended to

\begin{equation}\label{eqn:mo_channel}
  {h_{n,m,u}} = {A_{n,m,u}}{e^{j{{\bar \psi }_{n,m,u}}}}
\end{equation}

Therefore, the phase of the received signal on rectenna $u$ in subband $n$ transmitted by antenna $m$ equals

\begin{equation}\label{eqn:received_phase}
  {\psi _{n,m,u}} = {\phi _{n,m}} + {{\bar \psi }_{n,m,u}}
\end{equation}

where ${\phi _{n,m}}$ is the beamforming phase. In such cases, it is impossible to ensure ${\psi _{n,m,u}} = 0$ for all rectennas as there are three constraints $n,m,u$ but only two variables $n,m$. With a specific phase design, the arguments of cosines in \eqref{eqn:power_waveform_second_order} -- \eqref{eqn:waveform_end} are not guaranteed to be zero such that the target function ${Z_{DC}}$ is indeed a signomial.

Denoting ${\widetilde {\mathbf{h}}_{n,u}} = \sqrt {{k_2}{v_u}} {{\mathbf{h}}_{n,u}}$, the channel matrix for subband $n$ can be constructed as

\begin{equation}\label{eqn:mo_channel_matrox}
  {\widetilde {\mathbf{H}}_n} = {\left[ {\widetilde {\mathbf{h}}_{n,1}^T \ldots \widetilde {\mathbf{h}}_{n,U}^T} \right]^T}
\end{equation}

It is mentioned in \cite{Clerckx2016} that a possible phase choice ${\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime $ is to set the $\left( {n,m} \right)$ entries as

\begin{equation}\label{eqn:mo_phases}
  \phi _{P,n,m}^\prime  = \phi _{I,n,m}^\prime  = \angle {v_{{\text{max}},n,m}}
\end{equation}

where ${v_{{\text{max}},n,m}}$ is the $m$-th term of the dominant right singular vector ${{\mathbf{v}}_{{\text{max}},n}}$ that can be obtained by singular value decomposition of ${\widetilde {\mathbf{H}}_n}$. Also, the subband amplitudes can be initialized using the maximum eigenvalue $\sigma _n^\prime $

\begin{equation}\label{eqn:mo_initial_amplitude}
  {s_{P,n,m}} = {s_{I,n,m}} = c\sigma _n^\prime
\end{equation}

where $c$ is the coefficient to guarantee the transmit power constraint. Note the initialization is irrelevant to $m$.

To convert the problem into a standard GP, we introduce an auxiliary variable ${t_0}$ and rewrite the signomial as the difference of two posynomials

\begin{equation}\label{eqn:mo_current_signomial}
  {Z_{DC}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right) = {f_1}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right) - {f_2}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right) \geqslant {t_0}
\end{equation}

Furthermore, we can decompose the first posynomial as

\begin{equation}\label{eqn:mo_current_posynomial}
  {f_1}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right) = \sum\limits_{k = 1}^{{K_1}} {{g_{1k}}} \left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)
\end{equation}

With a proper choice of nonnegative $\left\{ {{\gamma _{1k}}} \right\}$ satisfying $\sum\nolimits_{k = 1}^{{K_1}} {{\gamma _{1k}}}  = 1$, \eqref{eqn:mo_current_signomial} rewrites as

\begin{align}\label{eqn:mo_current_signomial_rewritten}
  \frac{{{t_0} + {f_2}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)}}{{{f_1}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)}} &= \left( {{t_0} + {f_2}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)} \right) \nonumber \\
  &\quad \prod\limits_{k = 1}^{{K_1}} {{{\left( {\frac{{{g_{1k}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)}}{{{\gamma _{1k}}}}} \right)}^{ - {\gamma _{1k}}}}} \\
  &\leqslant 1
\end{align}

Similarly, the denominator of rate constraint \eqref{eqn:transformed_rate_constraint} is indeed a product of signomials, with each factor expressed as

\begin{equation}\label{eqn:mo_rate_signomial}
  1 + \frac{{(1 - \rho )}}{{\sigma _n^2}}{C_n} = {f_{1nk}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right) - {f_{2nk}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)
\end{equation}

Therefore, we can rewrite the rate constraint \eqref{eqn:transformed_rate_constraint} as

\begin{align}\label{eqn:mo_rate_constraint_original}
  \prod\limits_{n = 0}^{N - 1} {\left( {1 + \frac{{(1 - \rho )}}{{\sigma _n^2}}{C_n}} \right)}  &= \prod\limits_{n = 0}^{N - 1} {\left( {{f_{1nk}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right) - {f_{2nk}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)} \right)}  \\
  &\geqslant {2^{\bar R}} \label{eqn:mo_rate_constraint_original_end}
\end{align}

One possible approach is to unwrap the result of signomial multiplication as a new signomial. In this way, \eqref{eqn:mo_rate_constraint_original_end} is reduced to

\begin{equation}\label{eqn:mo_rate_constraint_unwrapped}
  f_1^\prime \left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right) - f_2^\prime \left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right) \geqslant {2^{\bar R}}
\end{equation}

which is equivalent to

\begin{equation}\label{eqn:mo_rate_constraint_rewritten}
  \frac{{{2^{\bar R}} + f_2^\prime \left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)}}{{f_1^\prime \left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)}} \leqslant 1
\end{equation}

By decomposing $f_1^\prime \left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right) = \sum\nolimits_{j = 1}^{{J_1}} {g_{1j}^\prime \left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)} $ and introducing another nonnegative coefficient set $\left\{ {\gamma _{1j}^\prime } \right\}$ with $\sum\nolimits_{j = 1}^{{J_1}} {\gamma _{1j}^\prime }  = 1$ for AM-GM inequality, it can be converted to

\begin{equation}\label{eqn:mo_rate_constraint_standard}
  \left( {{2^{\bar R}} + f_2^\prime \left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)} \right)\prod\limits_{j = 1}^{{J_1}} {{{\left( {\frac{{g_{1j}^\prime \left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)}}{{\gamma _{1j}^\prime }}} \right)}^{ - \gamma _{1j}^\prime }}}  \leqslant 1
\end{equation}

Hence, the problem is transformed into a standard GP

\begin{eqnarray}
  {\mathop {\min }\limits_{{{\mathbf{S}}_P},{{\mathbf{S}}_I},\rho ,\bar \rho ,{t_0}} }&{1/{t_0}} \label{eqn:mo_target} \\
  {{\text{ subject to }}}&{\frac{1}{2}\left[ {\left\| {{{\mathbf{S}}_I}} \right\|_F^2 + \left\| {{{\mathbf{S}}_P}} \right\|_F^2} \right] \leqslant P} \label{eqn:mo_power_constraint} \\
  {}&{\left( {{t_0} + {f_2}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)} \right)\prod\limits_{k = 1}^{{K_1}} {{{\left( {\frac{{{g_{1k}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)}}{{{\gamma _{1k}}}}} \right)}^{ - {\gamma _{1k}}}}}  \leqslant 1} \label{eqn:mo_current_constraint} \\
  {}&{\left( {{2^{\bar R}} + f_2^\prime \left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)} \right)\prod\limits_{j = 1}^{{J_1}} {{{\left( {\frac{{g_{1j}^\prime \left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)}}{{\gamma _{1j}^\prime }}} \right)}^{ - \gamma _{1j}^\prime }}}  \leqslant 1} \label{eqn:mo_rate_constraint} \\
  {}&{\rho  + \bar \rho  \leqslant 1 \label{eqn:mo_ratio_constraint}}
\end{eqnarray}

It is worth noting that the GP method is not the best optimization approach due to the predetermined suboptimal beamforming phases ${\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime $. Also, the unwrap process from \eqref{eqn:mo_rate_constraint_original} to \eqref{eqn:mo_rate_constraint_unwrapped} significantly increases the computational complexity and is more suitable for small $n$ and $m$. The procedure is concluded in Algorithm \ref{alg:mo}.

\begin{algorithm}
  \caption{Waveform Design for Multiple Rectennas}
  \label{alg:mo}
  \begin{algorithmic}[1]
    \State \textbf{Initialize:} $i \leftarrow 0$, ${{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime }$ in \eqref{eqn:mo_phases}, ${{{\mathbf{S}}_P},{{\mathbf{S}}_I}}$ in \eqref{eqn:mo_initial_amplitude}, $\rho ,\bar \rho  = 1 - \rho ,\bar R,Z_{DC}^{(0)} = 0$
    \Repeat
      \State $i \leftarrow i + 1,\mathop {{{\mathbf{S}}_P}}\limits^{..}  \leftarrow {{\mathbf{S}}_P},\mathop {{{\mathbf{S}}_I}}\limits^{..}  \leftarrow {{\mathbf{S}}_I},\ddot \rho  \leftarrow \rho ,\ddot{\bar{\rho}}  \leftarrow \bar \rho $
      \State ${\gamma _{1k}} \leftarrow {g_{1k}}\left( {\mathop {{{\mathbf{S}}_P}}\limits^{..} ,\mathop {{{\mathbf{S}}_I}}\limits^{..} ,{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\ddot \rho } \right)/{f_1}\left( {\mathop {{{\mathbf{S}}_P}}\limits^{..} ,\mathop {{{\mathbf{S}}_I}}\limits^{..} ,{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\ddot \rho } \right),k = 1, \ldots ,{K_1}$
      \State $\gamma _{1j}^\prime  \leftarrow g_{1j}^\prime \left( {\mathop {{{\mathbf{S}}_P}}\limits^{..} ,\mathop {{{\mathbf{S}}_I}}\limits^{..} ,{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\ddot \rho } \right)/f_1^\prime \left( {\mathop {{{\mathbf{S}}_P}}\limits^{..} ,\mathop {{{\mathbf{S}}_I}}\limits^{..} ,{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\ddot \rho } \right),j = 1, \ldots ,{J_1}$
      \State ${{\mathbf{S}}_P},{{\mathbf{S}}_I},\rho ,\bar \rho  \leftarrow \arg \min $ \eqref{eqn:mo_target} -- \eqref{eqn:mo_ratio_constraint}
      \State $Z_{DC}^{(i)} \leftarrow {Z_{DC}}\left( {{{\mathbf{S}}_P},{{\mathbf{S}}_I},{\mathbf{\Phi }}_P^\prime ,{\mathbf{\Phi }}_I^\prime ,\rho } \right)$
    \Until{$\left| {Z_{DC}^{(i)} - Z_{DC}^{(i - 1)}} \right| < $ or $i = {i_{\max }}$}
  \end{algorithmic}
\end{algorithm}


\section{Renormalization Group}
Block spin trnasformation for matrice slacing $a$ to $la$.
The parttion function is
\begin{align}
    Z &=
    \sum_{ \left\{ \sigma \right\}} e^{K\left[ \sigma \right]}\\
    &=
    \sum_{ \left\{ \sigma' \right\}} e^{K' \left[ \sigma' \right]}
\end{align}
where
\begin{align}
    K'
    &=
    K_0'
    +
    K'_2
    \sum_{n,m} \sigma'_{i} \sigma'_{j}
    +
    K'_3
    \sum_{n-m,n} \sigma'_i \sigma'_j
    +
    K_4' \sum_{\square}
    \sigma'_i \sigma'_j \sigma'_k \sigma'_l
    + \cdots
\end{align}
At fixed points
\begin{align}
    K^* &= R\left( K^* \right)
\end{align}
which we can expand near fixed point
\begin{align}
    K'_{\alpha}
    -
    K_{\alpha}^*
    &=
    R\left( K \right)
    -
    K^*\\
    &\approx
    R\left( K^* \right)
    +
    \left( K - K^* \right)_{\beta}
    \left.
    \frac{\partial R}{\partial K_{\beta}}
    \right|_{K=K^*}
    +
    \cdots
\end{align}
For simplicity assume the matrix $\frac{\partial R_\alpha}{\partial K_\beta}$
is symmetric.
Then if $\lambda > 1$ we say it is a relevant direciton,
but if $\lambda < 1$ it is irrelevant.
And if it's equal to zero then it's marginal.
THey are relevant for areason.

Big diagram here.
\begin{figure}[h]
    \begin{center}
        \includesvg{rgflow}
    \end{center}
    \caption{RG flow}%
    \label{fig:rgflow}
\end{figure}
\begin{align}
    K &=
    \beta J\\
    &=
    \frac{J}{k_B \left( T - T_c + T_c \right)}\\
    &\approx
    \frac{J}{k_B T_c}
    \left(
    1
    +
    \underbrace{\frac{T - T_c}{T_c}}_{t}
    + \cdots
    \right)
\end{align}

Critical exponent $\nu$.
\begin{align}
    t' &= \lambda t\\
    t^{(n)} &= \lambda^n t\\
\end{align}
so the correlation length scales like
\begin{align}
    \xi\left( t' \right) &= \frac{1}{l} \xi\left( t \right)\\
    \xi\left( t^{(n)} \right)
    &=
    \frac{\xi\left( t \right)}{l^n}
\end{align}
with
\begin{align}
    n
    &=
    \frac{\ln \left( b/t \right)}{\ln \lambda}
\end{align}
Note
\begin{align}
    \lambda^n t
    &=
    e^{n \ln \lambda} t
    =
    e^{\frac{\ln (b/t)}{\ln \lambda} \ln \lambda} t\\
    &=
    \frac{b}{t} t\\
    &= b
\end{align}
And
\begin{align}
    l^n &=
    e^{n\ln l}\\
    &=
    e^{\frac{\ln(b/t)}{\ln \lambda} \ln \lambda}\\
    &=
    \left( \frac{b}{t} \right)^{\frac{\ln l}{\ln \lambda}}
\end{align}
so then
\begin{align}
    \xi\left( t \right)
    &=
    l^n \xi\left( t' \right)
    =
    \underbrace{l^n}_{t/b}
    \xi\left( \underbrace{\lambda^n t}_{b} \right)\\
    &\sim
    t^{- \frac{\ln l}{\ln \lambda}} = t^{-\nu}
\end{align}
Hence
\begin{align}
    \nu &=
    \frac{\ln l}{\ln \lambda}
\end{align}

Now let us calculate the $\alpha$.
Consider $f(t)$ to be the free energy per site.
\begin{align}
    l^d f(t)
    &-
    f\left( t' \right)
\end{align}
which implies
\begin{align}
    f(t)
    &=
    \underbrace{
    \frac{1}{l^{dn}}
    }_{
    \left( \frac{t}{b} \right)^{-d \underbrace{
    \frac{\ln l}{\ln \lambda}}_{
    \nu
    }}
    }
    f\left( \lambda^n t \right)\\
    &\sim t^{d\nu}
\end{align}
Recalling that
\begin{align}
    n &=
    \frac{\ln b/t}{\ln \lambda}
\end{align}
And the specific heat is proportional to
\begin{align}
    C &\sim
    -\frac{\partial^2 f}{\partial t^2}
    \sim
    t^{\underbrace{d\nu - 2}_{\alpha}}
\end{align}
hence
\begin{align}
    \alpha &= d\nu - 2
\end{align}
which is the Josephson relation.


\documentclass{article}

\usepackage{style/preamble}
\usepackage{parskip}
\usepackage[normalem]{ulem}

\begin{document}
  \title{\sout{Problem Set} 5 - Affine charts}
  \date{}
  \maketitle

\section{Singularities}
\textbf{Notation:} Set of solutions of a bunch of polynomial equations is called a \emph{variety}. Varities in $\bbc^n$ are called \emph{affine varities} and varities in $\bbp^n$ are called \emph{projective varieties}.

We'll start with two examples of singularities, one in $\bbc^2$ and one in $\bbp^1$.

\begin{ex}
  The affine variety given by $p(z,w) = z^2 - w^2$ is an intersection of two lines, and hence has a singularity at $(0,0)$.
  In general, if the lowest degree terms in $p(z,w)$ are of degree $\ge 2$ then the corresponding variety has a singularity at $(0,0)$ and hence is not a Riemann surface.
  As far as varieties in $\bbc^2$ are concerned, this is the only way singularities arise but in higher dimensions the singularities are much more complicated and hence interesting.
\end{ex}

\begin{ex}[Hyperelliptic curve]
  Consider $p(z,w) = z^2 - q(w)$ with $q(w) = w^4 - w$.
  The homogenization of $p$ is $\conj{p}(z,w,t) = z^2 t^2 - w^4 - w t^3$.
  The points at $\infty$ are given by plugging in $t = 0$,
  \begin{align*}
    \conj{p}(z,w,t) &= 0 \\
    w^4 &= 0
  \end{align*}
  So there is exactly one point at $\infty$, namely $[1:0:0]$.

  The projection map $\conj{S_p} \rightarrow \bbp^1$ sending $[z:w:1] \mapsto w$ and $[1:0:0] \mapsto \infty$ is a degree 2 map which ramifies at 5 points: the roots of $q(z)$ and $\infty$.

  Plugging in Riemann-Hurwitz we get:
  \begin{align*}
    \chi(\conj{S_p})
    &= 2 \cdot \chi(\bbp^1) - \sum_{5}\mathrm{index} - 1 \\
    &= 4 - 5 \\
    &= -1
  \end{align*}
  But there is no surface with Euler characteristic -1.
  The reason we see this is that $\conj{S_p}$ is not a Riemann surface and has a singularity at $\infty$.
\end{ex}

\section{Affine charts on $\bbp^2$}
There is a natural cover of $\bbp^2$ given by three open sets
\begin{align*}
  \bbp^2
  &= \set{[z : w : t] \mid \mbox{ not all } z,w,t \mbox{ zero}} \\
  &= \set{[z : w : 1]} \cup \set{[z : 1 : t]} \cup \set{[1 : w : t]} \\
  &=: U_t \cup U_w \cup U_z
\end{align*}
We can define charts on these as
\begin{align*}
  \varphi_z: U_z &\longrightarrow \bbc^2 \\
  [1 : w : t] &\longmapsto (w,t)\\\\
  \varphi_z: U_w &\longrightarrow \bbc^2 \\
  [z : 1 : t] &\longmapsto (z,t)\\\\
  \varphi_z: U_t &\longrightarrow \bbc^2 \\
  [z : w : 1] &\longmapsto (z,w)
\end{align*}
A projective variety $\conj{S_p}$ cut out by the polynomial $\conj{p}(z,w,t)$ is a Riemann surface if and only if the affine varities $\conj{S_p} \cap U_t$, $\conj{S_p} \cap U_w$, and $\conj{S_p} \cap U_z$ are Riemann surfaces.
These are called \emph{affine charts} on $\conj{S_p}$.

The varieties
$\conj{S_p} \cap U_t$, $\conj{S_p} \cap U_w$, $\conj{S_p} \cap U_z$
can be described as
\begin{align*}
  \conj{S_p} \cap U_t
  &= \set{(z,w) \mid \conj{p}(z,w,1) = 0} \\\\
  \conj{S_p} \cap U_w
  &= \set{(z,t) \mid \conj{p}(z,1,t) = 0} \\\\
  \conj{S_p} \cap U_z
  &= \set{(w,t) \mid \conj{p}(1,w,t) = 0}
\end{align*}
In order to check that a projective variety is a Riemann surface, we first break the variety into affine charts, and then check that each of the charts is a Riemann surface using the Jacobian.

\section{Jacobian}
\begin{theorem}
  The affine variety $S = \set{(z,w) : p(z,w) = 0} \subseteq \bbc^2$ is a Riemann surface if the Jacobian, defined as
  \begin{align*}
    J(z,w) := \begin{bmatrix} \dfrac{\partial p}{\partial z} & \dfrac{\partial p}{\partial w} \end{bmatrix}
  \end{align*}
  does not vanish, at all points $(z,w)$ in $S$.
\end{theorem}
\begin{proof}[Proof]
  The reason is essentially that if at $(z,w)$, we have $\dfrac{\partial p}{\partial z} \neq 0$ then projection onto the $w$ coordinate locally defines a chart around $(z,w)$.
  Similarly, if at $(z,w)$, we have $\dfrac{\partial p}{\partial w} \neq 0$ then projection onto the $z$ coordinate locally defines a chart around $(z,w)$.
  If both are non-zero then both charts are valid and the deritvatives of transition functions are given by the rational functions
  \begin{align*}
    \left(\frac{\partial p}{\partial w}\right) \cdot \left(\frac{\partial p}{\partial z}\right)^{-1}
  \end{align*}
  which are complex differentiable as the denominator is non-zero.
\end{proof}

\begin{ex}
  The polynomial $z^2 - q(w)$ has Jacobian $\begin{bmatrix} 2z & -q'(w) \end{bmatrix}$. The Jacobian vanishes 0 precisely when $z = 0$ and $q'(w) = 0$. For this to be true for a point on the curve, $z=0 \implies q(w) = 0$. Both $q(w) = 0 $ and $q'(w) = 0$ implies that $w$ is a repeated root of $q$.
  Thus the corresponding variety is a Riemann surface if $q(w)$ has no repeated roots.
\end{ex}

\begin{ex}[Fermat]
  For the projective variety cut out by $\conj{p}(z,w,t) = z^p + w^p - t^p$, the three affine charts are given by
  \begin{align*}
    z^p + w^p - 1  &&& \begin{bmatrix} pz^{p-1} & pw^{p-1} \end{bmatrix}\\
    z^p + 1 - t^p &&& \begin{bmatrix} pz^{p-1} & pt^{p-1} \end{bmatrix}\\
    1 + w^p - t^p &&& \begin{bmatrix} pw^{p-1} & pt^{p-1} \end{bmatrix}
  \end{align*}
  The Jacobians vanish at $(0,0)$ but these points are not on the affine varities.
\end{ex}

\begin{ex}[Elliptic curves]
  The equation $z^2 = w^3 + w$ has homogenization $\conj{p}(z,w,t) = z^2 t - w^3 - w t^2$.
  In the three charts this polynomial and the Jacobians become
  \begin{align*}
    z^2  - w^3 - w  &&& \begin{bmatrix} 2z & -3w^2 - 1 \end{bmatrix}\\
    z^2 t - 1 - t^2 &&& \begin{bmatrix} 2zt & z^2 - 2t \end{bmatrix}\\
    t - w^3 - w t^2 &&& \begin{bmatrix} 1 - 2wt & -3w^2 - t^2 \end{bmatrix}
  \end{align*}
  It is easy to see that all the Jacobians do not vanish anywhere on the varieties.
\end{ex}

\begin{ex}[Hyperelliptic curves]
  The equation $z^2 = w^4 + w$ has homogenization $\conj{p}(z,w,t) = z^2 t^2 - w^3 - w t^3$.
  In the three charts this polynomial and the Jacobians become
  \begin{align*}
    z^2  - w^4 - w  &&& \begin{bmatrix} 2z & -4w^3 - 1 \end{bmatrix}\\
    z^2 t^2 - 1 - t^3 &&& \begin{bmatrix} 2zt^2 & 2z^2t - 3t^2 \end{bmatrix}\\
    t^2 - w^3 - w t^3 &&& \begin{bmatrix} 2t - 3wt^2 & -3w^2 - t^2 \end{bmatrix}
  \end{align*}
  In this case, the third Jacobian vanishes at the point $(0,0)$ which is on the curve, and hence our original projective variety is singular at the point at $\infty$.
\end{ex}

It is possible to remove singularities of hyperelliptic curves ($z^2 = q(w)$ with $\deg q > 3$) by putting charts at $\infty$ artificially.
The resulting Riemann surface has genus $\left \lfloor{\dfrac{\deg q - 1}{2}}\right \rfloor $. See \\ \url{https://en.wikipedia.org/wiki/Hyperelliptic_curve#Genus_of_the_curve}.

\end{document}


\subsection{Functions and brackets}

\subsubsection{Functions (or maps)}

Functions take other terms, and are themselves terms. For example if we wanted to know if someone can legally drive in a specific country, we could use:

\(P(you,age(UK))\)

A function may not be able to produce an output for all inputs. For examples \(age(green)\) has no interpretation.

Functions can also take different numbers of inputs. Constants, such as “you” and “UK” can be shown as functions with \(0\) inputs. As a result we could instead write:

\(P(you(),age(UK()))\)

We generally denote functions with a lower case letter, so would instead write:

\(P(y(),a(b()))\)

Functions are also called maps.


\problemname{Counting Greedily Increasing Supersequences}
Given a permutation $A = (a_1, a_2, \dots, a_N)$ of the integers $1, 2, \dots, N$, we define the \emph{greedily increasing subsequence} (GIS) in the following way.

Let $g_1 = a_1$. For every $i > 1$, let $g_i$ be the leftmost integer in $A$ that is strictly larger than $g_{i-1}$.
If for a given $i$ there is no such integer, we say that the GIS of the sequence is the sequence $(g_1, g_2, ..., g_{i - 1})$.

For example, consider the permutation $(2, 3, 1, 5, 4, 7, 6)$.
First, we have $g_1 = 2$.
The leftmost integer larger than $2$ is $3$, so $g_2 = 3$.
The leftmost integer larger than $3$ is $5$ ($1$ is too small), so $g_3 = 5$.
Finally, $g_4 = 7$.
Thus, the GIS of $(2, 3, 1, 5, 4, 7, 6)$ is $(2, 3, 5, 7)$.

Given a sequence $G = (g_1, g_2, \dots, g_L)$, how many permutations $A$ of the integers $1, 2, \dots, N$ have $G$ as its GIS?

\section*{Input}
The first line of input contains the integers $1 \le N \le 10^6$, the number of elements of the permutation $A$,
and $1 \le L \le 10^6$, the length of the sequence $G$.

The next line contains $L$ positive integers between $1$ and $N$, the elements $g_1, \dots, g_L$ of the sequence $G$.

\section*{Output}
Output a single integer: the number of $N$-element permutations having the given sequence as its GIS.
Since this number may be large, output it modulo the prime number $10^9 + 7$.


\chapter{Combining Models}
\section{Introduction}
\begin{description}
	\item[\textbf{committees}] Train $L$ different models and then make predictions using the average of the predictions made by each model.
	\item[\textbf{boosting}] Train multiple models in sequence in which the error function used to train a particular model depends on the performance of the previous models.
	\item[\textbf{decision trees}] Different models are responsible for making predictions in different regions of input space.
	\item[\textbf{mixtures of experts}] Models are viewed as mixture distributions in which the component densities,as well as the mixing coefficients,are conditioned on the input variables.
	\begin{align}
	p(t|\vec{x})=\sum\limits_{k=1}^{K}\pi_k(\vec{x})p(t|\vec{x},k)
	\end{align}
	in which $\pi_k(\vec{x})=p(k|\vec{x})$ represents the input-dependent mixing coefficients,and $k$ indexes the model.
\end{description}

\section{Bayesian Model Averaging}
In Bayesian model averaging the whole data set is generated by a single model.By contrast,when we combine multiple models,we see that different data points within the data set can potentially be generated from different values of the latent variable $\vec{z}$ and hence by different components.

\section{Committees}
The simplest way to construct a committee is to average the predictions of a set of individual models,to cancel the contribution arising from variance and bias.

\textbf{Bootstrap} data to introduce variability between the different models within the committee.Suppose we generate $M$ bootstrap data sets
\begin{align}
y_{COM}(\vec{x})=\dfrac{1}{M}\sum\limits_{m=1}^{M}y_m(\vec{x})
\end{align}
where $m=1,...,M$.This procedure is known as \textbf{bootstrap aggregation} or \textbf{bagging}.

\section{Boosting}
Here we describe the most widely used form of boosting algorithm:\textbf{AdaBoost},short for 'adaptive boosting'.The base classifiers are known as \textbf{weak learners} and are trained in \textbf{sequence} using a \textbf{weighted form of the data set} in which the weighting coefficient associated with each data point depends on the performance of the previous classifiers.

Consider a two-class classification problem,in which the training data comprises input vectors $\vec{x}_1,...,\vec{x}_N$ along with corresponding binary target variables $t_1,...,t_N$ where $t_n\in \{-1,1\}$.Each data point is given an associated weighting parameter $w_n$,initially set $1/N$ for all.A base classifier function $y(\vec{x})\in \{-1,1\}$
\begin{SCfigure*}
	\caption{Shematic illustration of boosting framework}
	\includegraphics{prml/Figure14.1}
\end{SCfigure*}

\begin{algorithm}[H]
	\caption{\color{red}{AdaBoost}}
	\label{algo:AdaBoost}
	\DontPrintSemicolon % Some LaTeX compilers require you to use \dontprintsemicolon instead
	\KwIn{A set $\vec{X} = \{\vec{x}_1, \vec{x}_2, \ldots, \vec{x}_N\}$,$\{t_1,...,t_N\}$}
	\KwOut{$y(\vec{x})$,.}
	1. Initialize the data weighting coefficients $\{w_n\}$ by setting $w_n^{1}=1/N$ for $n=1,...,N$. \;
	2. \For{$m=1,...,N$}{
		(a) Fit a classifier $y_m(\vec{x})$ to the training data by minimizing the weighted error function
		\begin{align}
		J_m=\sum\limits_{n=1}^{N}w_n^{(m)}I(y_m(\vec{x}_n)\neq t_n)
		\end{align}
		where $I(y_m(\vec{x}_n)\neq t_n)$ is the indicator function and equals $1$ when $y_m(\vec{x}_n)= t_n$ and $0$ otherwise. \;
		(b) Evaluate  the quantities
		\begin{align}
		\epsilon_m=\dfrac{\sum_{n=1}^{N}w_n^{(m)}I(y_m(\vec{x}_n)\neq t_n)}{\sum_{n=1}^{N}w_n^{(m)}}
		\end{align}
		and then use these to evaluate
		\begin{align}
		\alpha_m=\ln\{\dfrac{1-\epsilon_m}{\epsilon_m}\}.
		\end{align}
		\;
		(c) Update the data weighting coefficients
		\begin{align}
		w_n^{(m+1)}=w_n^{(m)}\exp\{\alpha_m I(y_m(\vec{x}_m)\neq t_n)\}
		\end{align}
	}
	3. Make predictions using the final model,which is given by
	\begin{align}
	Y_M(\vec{x})=sign(\sum_{m=1}^{M}\alpha_m y_m(\vec{x}))
	\end{align}

\end{algorithm}

\subsection{Minimizing exponential error}
Consider the exponential error function defined by
\begin{align}
E=\sum_{n=1}^{N}\exp\{-t_n f_m(\vec{x}_n)\}
\end{align}
where $f_m(\vec{x})$ is a classifier defined in terms of a linear combination of base classifiers $y_l(\vec{x})$ of the form
\begin{align}
f_m(\vec{x})=\dfrac{1}{2}\sum_{l=1}^{m}\alpha_l y_l(\vec{x})
\end{align}
and $t_n\in \{-1,1\}$ are the training set target values.Our goal is to minimize $E$ with respect to the weighting coefficients $\alpha_l$ and parameters of the base classifiers $y_l(\vec{x})$.

Separating off the contribution from base classifier $y_m(\vec{x})$,
\begin{align}
E&=\sum_{n=1}^{N}\exp\{-t_n f_m(\vec{x}_n)\} \\
&=\sum_{n=1}^{N}\exp\{-t_n \dfrac{1}{2}\sum_{l=1}^{m}\alpha_l y_l(\vec{x}) \} \\
&=\sum_{n=1}^{N}\exp\{-t_n f_{m-1}(\vec{x}_n)-\dfrac{1}{2}t_n \alpha_m y_m(\vec{x}_n)\} \\
&=\sum_{n=1}^{N}w_n^{(m)}\exp\{-\dfrac{1}{2}t_n \alpha_m y_m(\vec{x}_n)\}
\end{align}
where the coefficients $w_n^{(m)}=\exp\{-t_n f_{m-1}(\vec{x}_n)\}$ can be viewed as constants because we are optimizing only $\alpha_m$ and $y_m(\vec{x})$.Denote by $\mathcal{T}_m$ the set of data points correctly classified by $y_m(\vec{x})$ and misclassified points by $\mathcal{M}_m$,then we in turn rewrite the error function
\begin{align}
E &=e^{-\alpha_m/2}\sum_{n\in\mathcal{T}_m}w_n^{(m)}+e^{\alpha_m/2}\sum_{n\in\mathcal{M}_m}w_n^{(m)} \\
&=(e^{\alpha_m/2}-e^{-\alpha_m/2})\sum_{n=1}^{N}w_n^{(m)}I(y_m(\vec{x}_n)\neq t_n)+e^{-\alpha_m/2}\sum_{n=1}^{N}w_n^{(m)}
\end{align}
Then we can minimize this with respect to $y_m(\vec{x}_n)$ and $\alpha_m$.
\begin{align}
\because w_n^{(m)}=\exp\{-t_n f_{m-1}(\vec{x}_n)\} \\
\therefore w_n^{(m+1)}=\exp\{-t_n f_{m}(\vec{x}_n)\} \\
\therefore w_n^{(m+1)}=w_n^{(m)}\exp\{-\dfrac{1}{2}t_n\alpha_m y_m(\vec{x}_n)\}
\end{align}
Making use of the fact that
\begin{align}
t_n y_m(\vec{x})=1-2I(y_m(\vec{x}_n)\neq t_n)
\end{align}
we see updates at the next iteration
\begin{align}
w_n^{(m+1)}=w_n^{(m)}\exp(-\alpha_m/2)\exp\{\alpha_m I(y_m(\vec{x}_n)\neq t_n)\}
\end{align}
Because the term $\exp(-\alpha_m/2)$ is independent of $n$,so can be discarded.

\subsection{Error functions for boosting}

\section{Tree-based Models}

\section{Conditional Mixture Models}


% !TEX root = ../main.tex
% chktex-file 21
% chktex-file 46
\section{Effects of Coarsening}%
\label{sec:cons}

We have just seen how to compute a coarsened graph $G_c$ via the REC algorithm.
What remains to be answered is how coarsening affects the performance of graph algorithms when $G_c$ is used as a proxy for the original graph $G$.
In the first step we will present a graph similarity measure.
Using this measure we will then put bounds on the dissimilarity of $G$ and $G_c$.
Finally we will use those bounds to analyze the influence of coarsening on the performance of the \textit{spectral clustering algorithm}.
The results described in this section were found by \citet{Loukas2018}.

\subsection{Restricted Spectral Similarity}%
\label{sec:cons:rss}

To compare a graph $G$ with its coarsened version $G_c$ we will use the notion of \textit{spectral similarity}.
As we have seen before, $G$ can be viewed as an operator that transforms an input signal $x \in \mathbb{R}^N$.
We described this transform in terms of the graph's Laplacian $L$,
more specifically in terms of the Laplacian's eigenbasis ${\{ u_k \}}_{k = 1}^{N}$ and spectrum ${\{ \lambda_k \}}_{k = 1}^{N}$.
Similarly the coarsened graph $G_c$ can be described in terms of its Laplacian $L_c$.
Since $L \in \mathbb{R}^{N \times N}$ and $L_c \in \mathbb{R}^{n \times n}$ act on signal spaces of different dimensionality, they cannot be compared directly however.
Instead we compare $L$ with $\widetilde{L} = C^{\top} L_c C$, the upsampled version of $L_c$.
We say $\widetilde{L}$ is an \textit{$\varepsilon$-approximation} of $L$ iff.\  $\widetilde{L}$ scales the eigenvectors $u_k$ of $L$ by a factor of roughly $\lambda_k$ in the direction of $u_k$:
\begin{wrapfigure}{r}{0.25\textwidth}
	\centering
	\includegraphics[width=\linewidth]{gfx/cons/rss.pdf}
	\caption{See \cref{eq:cons:rss}.}\label{fig:cons:rss}
\end{wrapfigure}
\begin{align}
	\forall k \leq K:\ (1 - \varepsilon) \underbrace{u_k^{\top} L u_k}_{\lambda_k} \leq u_k^{\top} \widetilde{L} u_k \leq (1 + \varepsilon) \underbrace{u_k^{\top} L u_k}_{\lambda_k} \text{ with } \varepsilon \geq 0\label{eq:cons:rss}
\end{align}
The reason \cref{eq:cons:rss} is only quantified for the first $K$ eigenvectors of $L$ is, that $\text{rank}(L) = N - c > n - c = \text{rank}(\widetilde{L})$;
i.e.\ $\widetilde{L}$ has a higher dimensional null space\footnote{%
	Here $c$ denotes the number of connected components of $G$.
	They reduce the rank since $\lambda_1 = \cdots = \lambda_c = 0$.
}.
Thus the inequality in \cref{eq:cons:rss} cannot hold for a signal $x$ that is in the null space of $\widetilde{L}$ but not in that of $L$.
Therefore the similarity condition is restricted to the first $K$ eigenvectors, as they represent the most important ``low-frequency'' components of $G$.
This restricted condition is called \textit{restricted spectral similarity} (RSS).

The choice of $K$ in RSS depends on the level of detail that should be considered when comparing $L$ and $\widetilde{L}$.
If $G$ has $c'$ clusters, i.e.\ connected subgraphs with relatively few or even no edges going out of it, a choice of $K = c'$ is reasonable.
That way RSS checks whether the clusters of $G$ are preserved in the coarsened graph $G_c$ and how much the connectedness between the clusters changes.
Details like the connections within clusters on the other hand do not have a strong influence on the RSS similarity if $K = c'$, since they are described by the ``high-frequency'' eigenvectors $u_\ell$ of $L$ where $\ell > K$.

\subsection{Bounding REC via RSS}%
\label{sec:cons:bound}

Now we will use RSS to bound the eigenvalue distortion caused by a single application of the REC coarsening algorithm.
It can be shown that the probability of satisfying the RSS condition for a single eigenvector $u_k$ with a sufficiently small eigenvalue $\lambda_k$ is lower-bounded by
\begin{align}
	\forall \lambda_k \leq \frac{\nu_\text{min}}{4}:\ P\mkern-2mu\left((1 - \varepsilon) \lambda_k \leq u_k^{\top} \widetilde{L} u_k \leq (1 + \varepsilon) \lambda_k\right)
	\geq 1 - \frac{r \nu_{\text{max}}}{2 \varepsilon d_{\text{avg}}} \left(1 + \frac{3 - 4 \lambda_k}{\nu_{\text{min}}}\right)\label{eq:cons:bound}
\end{align}
where $r = \frac{N - n}{N}$ is the graph reduction ratio, $d_{\text{avg}} = \frac{1}{N} \sum_{v_i \in \mathcal{V}} d_i$ is the average weighted vertex degree and $\nu_{\text{min}}$, $\nu_{\text{max}}$ are the minimum and maximum of the neighborhood weights ${\{ d_i + d_j - w_{i j}  \}}_{e_{i j} \in \mathcal{E}}$.
For a formal proof of this bound we refer to \citet[Suppl.~2]{Loukas2018}.
Here we will instead give an intuition for its key statements:
\begin{enumerate}[label=\textbf{\arabic*.}]
	\item \textbf{$\varepsilon$-term:}
		Unsurprisingly the RSS bound is inversely proportional to $\varepsilon$.
		Relaxing the RSS bound makes it is more likely to be satisfied.
	\item \textbf{$r$-term:}
		The RSS bound is linearly dependent on the reduction ratio $r$.
		The more the graph size is reduced, the smaller the probability of satisfying the bound.
	\item \textbf{$\lambda_k / \nu_{\text{min}}$-term:}
		The bound is also linearly dependent on the eigenvalue $\lambda_k$.
		Additionally, since $\lambda_k$ is proportional to the arbitrarily large weights $w_{i j}$, it is normalized via $\nu_{\text{min}}$.
		To get an intuition for the influence of the eigenvalue on the RSS bound, two opposing effects have to be considered:
		\begin{enumerate}[label=(\roman*)] % chktex 36
			\item \textbf{Negative effect:}\label{itm:cons:bound:neg}
				The eigenvector $u_k$ for a large $\lambda_k$ represents a ``high-frequency'' component of $G$ that is more easily distorted by coarsening than a smooth ``low-frequency'' eigenvector $u_\ell$ with eigenvalue $\lambda_{\ell \ll k}$ (see~\cref{fig:sgt:graphFourier}).
				Formally this means that the approximated eigenvector $\widetilde{u}_k := \Pi u_k$ has a lower overlap with the original $u_k$ than $\widetilde{u}_\ell$ has with its original $u_\ell$, i.e.\ $\langle u_k, \widetilde{u}_k \rangle < \langle u_\ell, \widetilde{u}_\ell \rangle$.
				$\pmb{\Rightarrow}$ This effect causes the RSS probability bound to \textit{decrease} with increasing, more easily distorted, eigenvalues.
			\item \textbf{Positive effect:}\label{itm:cons:bound:pos}
				Recall that $u_k^\top \widetilde{L} u_k = \widetilde{u}_k^\top L \widetilde{u}_k$ and $\lambda_k = u_k^\top L u_k$.
				RSS checks whether $L$ transforms $u_k$ similarly to its approximate $\widetilde{u}_k$.
				When writing the latter transform in terms of $L$'s eigenbasis we get $\widetilde{u}_k^\top L \widetilde{u}_k = \sum_{m = 1}^N \lambda_m \langle u_m, \widetilde{u}_k \rangle^2$.
				For large eigenvalues $\lambda_k$ the self-overlap $\langle u_k, \widetilde{u}_k \rangle^2$ is weighted more strongly than the overlap with low-valued eigenvectors $u_{\ell \ll k}$.
				$\pmb{\Rightarrow}$ This effect causes the RSS probability bound to \textit{increase} with increasing, more strongly weighted, eigenvalues.
		\end{enumerate}
		As long as the eigenvector distortion described in~\ref{itm:cons:bound:neg} is not too large ($\lambda_k \leq \frac{\nu_\text{min}}{4}$), the positive effect described in~\ref{itm:cons:bound:pos} dominates, i.e.\ $\lambda_k \langle u_k, \widetilde{u}_k \rangle^2 > \lambda_\ell \langle u_\ell, \widetilde{u}_\ell \rangle^2$ for $k \gg \ell$ despite $\langle u_k, \widetilde{u}_k \rangle < \langle u_\ell, \widetilde{u}_\ell \rangle$.
		This is why the RSS probability bound increases with increasing $\lambda_k$'s, up to the point where they become too large and the bound becomes undefined.
		This can be seen in \cref{fig:cons:example:regular}.
		It shows the RSS bound for a $20$-regular graph, which is defined up to $\lambda_{20}$, after that the eigenvector distortion becomes too large.
	\item \textbf{$\nu_{\text{max}}/d_{\text{avg}}$-term:}
		Finally the bound also depends on the quotient between the maximum weight $\nu_{\text{max}}$ within an edge neighborhood $\mathcal{N}_{i j}$ and the average weighted vertex degree $d_{\text{avg}}$.
		This quotient can be interpreted as a measure of how much the weight distribution within $G$ varies when comparing local clusters with the global average.
		Regular graphs minimize this quotient, as all their local clusters have the same weight.
		The key implication is that the distortion of the graph Laplacian caused by coarsening is proportional to the regularity of the coarsened graph, i.e.\ coarsening distorts regular graphs less than highly irregular graphs.
		This can be seen in \crefrange{fig:cons:example:regular}{fig:cons:example:bunny} for three increasingly irregular example graphs.
\end{enumerate}
\begin{figure}[ht]
	\centering
	\begin{subfigure}{0.33\textwidth}
		\includegraphics[width=\linewidth]{gfx/cons/example/regular.png}
		\caption{Regular graph ($d = 20$)}\label{fig:cons:example:regular}
	\end{subfigure}%
	\begin{subfigure}{0.33\textwidth}
		\includegraphics[width=\linewidth]{gfx/cons/example/yeast.png}
		\caption{Yeast (protein network)}\label{fig:cons:example:yeast}
	\end{subfigure}%
	\begin{subfigure}{0.33\textwidth}
		\includegraphics[width=\linewidth]{gfx/cons/example/bunny.png}
		\caption{Bunny (point cloud)}\label{fig:cons:example:bunny}
	\end{subfigure}
	\caption{%
		Comparison of the RSS-bound and the actual error constants $\varepsilon$ for the eigenvectors of multiple coarsened graphs ($r = 0.4$).
		Confidence bounds for $p_s \in \{0.5, 0.7\}$ are shown.
		\source{Loukas2018}
	}\label{fig:cons:example}
\end{figure}

\vspace{-1em}%
\subsection{Implications for Spectral Clustering}%
\label{sec:cons:sc}

Lastly we will evaluate the effect coarsening has on the performance of the \textit{spectral clustering algorithm}~\cite{Luxburg2007}.
As we have seen in \cref{sec:sgt:spectrum}, the positive and negative vertex signal strengths of the Fiedler vector $u_2$ describe a $2$-clustering of the graph $G$ where the cluster boundary can be interpreted as a heat-flow bottleneck.
This idea can be extended to the $K$-clustering scenario by considering the first $K$ eigenvectors $U_K = \{ u_2, \dots, u_K \}$ instead of only looking at $u_2$.
To obtain a vertex clustering, the vertex basis vectors ${\{ b_i \}}_{v_i \in \mathcal{V}}$ are projected onto the spectral basis $U_K$, i.e.\ the Fourier transform $\widehat{b}_i = {(\langle u_2, b_i \rangle, \dots, \langle u_K, b_i \rangle)}^\top$ is computed for each vertex.
Afterwards regular $K$-means clustering is performed on the Fourier-transformed vertex basis ${\{\,\widehat{b}_i \}}_{v_i \in \mathcal{V}}$ to get the clustering $S = (S_1, \dots, S_K)$ with $S_k \subseteq \mathcal{V}$.

For large graphs $G$, computing the eigenbasis $U_K$ of $L \in \mathbb{R}^{N \times N}$ is expensive.
To speed this up, one can compute the lower dimensional eigenbasis $\widetilde{U}_K = \{ C^\top u'_2, \dots, C^\top u'_K \}$ instead, where $u'_k$ are the eigenvectors of the coarsened Laplacian $L_c \in \mathbb{R}^{n \times n}$.
Since we want a clustering on the original vertex set $\mathcal{V}$, not the coarsened vertices $\mathcal{V}_c$, the coarse eigenvectors $u'_k \in \mathbb{R}^n$ need to be upsampled to $C^\top u'_k \in \mathbb{R}^N$.
In total this is often cheaper than computing $U_K$ directly.

The question now is how much worse the spectral clustering results become when the vertex basis vectors ${\{ b_i \}}_{v_i \in \mathcal{V}}$ are projected onto $\widetilde{U}_K$ instead of $U_K$.
To answer this question, it was shown by \citet[Coroll.~5.1]{Loukas2018} that the RSS bound also implies a bound on the absolute spectral clustering error;
therefore the intuitions we discussed in \cref{sec:cons:bound} also hold for spectral clustering.
Most notably this implies that the absolute clustering error introduced by coarsening depends on the regularity of the graph, i.e.\ regular graphs are best suited to speed up spectral clustering via coarsening.
\Cref{fig:cons:mnist} shows the relative performance decrease caused by coarsening for an exemplary regular graph.
\begin{figure}[ht]
	\centering
	\begin{tikzpicture}
		\begin{axis}[
			width=0.5\linewidth,
			height=0.3\linewidth,
			xmin=8, xmax=42,
			ymin=0, ymax=0.04,
			xtick={10,20,30,40},
			xticklabels={10\%,20\%,30\%,40\%},
			xtick pos=bottom,
			ytick={0,0.01,0.02,0.03,0.04},
			yticklabels={0\%,1\%,2\%,3\%,4\%},
			ytick pos=left,
			xlabel={reduction ratio $r$},
			ylabel={relative error},
			ymajorgrids=true,
			label style={font=\tiny},
			tick label style={font=\tiny},
			tick align=outside,
			y tick label style={/pgf/number format/.cd, fixed, scaled ticks=false, precision=2, /tikz/.cd}
		]
			\addplot [mark=*, color=blau] table [x=r, y=ref10, col sep=comma] {data/cons/mnist.csv};
		\end{axis}
	\end{tikzpicture}%
	\quad\includegraphics[width=0.29\linewidth]{gfx/cons/mnistGraph.png}
	\caption{%
		(Left)~Relative spectral clustering error on a coarsened $12$-nearest neighbor similarity graph for $N=1000$ images from the MNIST dataset.
		Only images of the digits 0 to 4 were sampled, thus $K = 5$.
		(Right)~The full MNIST similarity graph $G$ with vertex colors indicating the clustering obtained via $\widetilde{U}_K$. Edges that are contracted in $G_c$ are shown in red.
		\source{Loukas2018}
	}\label{fig:cons:mnist}
\end{figure}
% \vspace{-0.7cm}