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\section{Introduction to the Standard Model}\label{sm-basics}
The Standard Model (SM) of particle physics presents one of the key achievements of modern day particle physics and provides a variety of phenomenological predictions that are experimentally verifiable to high accuracy.
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The SM gauge group is given by
\begin{align}\label{gsm}
  \gsm&=\su(3)_C\times\su(2)_T\times\un(1)_Y,
\end{align}
where each group factor corresponds to a specific kind of interaction in the model.
The field content of the SM is given by the scalar doublet $\phi$, the gauge vectors $B^\mu$, $W_k^\mu$, $\mathcal{A}_a^\mu$, and three generations of left-handed\footnote{Right-handed fermions can be acommodated for by making use of charge conjugation.} Weyl fermions $\psi_f$~\cite{arthur, pdg}.

\noindent All these fields transform in their respective $\gsm$-representations as can be seen in tab.\,\ref{sm-fields}.
\begin{table}[H]\centering
\caption{Field content of the standard model. For the sake of simplicity generational indices are supressed.\label{sm-fields}}
\begin{tabular}{ccc}
field& spin& representation\\
$\phi$&$0$&$\rep{1}{2}{\frac12}$\\
$q_L$&$\sfrac12$&$\rep{3}{2}{\frac16}$\\
$u_R^c$&$\sfrac12$&$\rep{3}{1}{-\frac23}$\\
$d_R^c$&$\sfrac12$&$\rep{3}{1}{\frac13}$\\
$\ell_L$&$\sfrac12$&$\rep{1}{2}{-\frac12}$\\
$e_R^c$&$\sfrac12$&$\rep{1}{1}{1}$\\
$B^\mu$&$1$&$\rep{1}{1}{0}$\\
$W_k^\mu$&$1$&$\rep{1}{3}{0}$\\
$\mathcal{A}_a^\mu$&$1$&$\rep{8}{1}{0}$
\end{tabular}
\end{table}
The $\su(2)$-doublets can be identified with the familiar fields as:
\begin{align}
\phi=\vect{\phi_+\\\phi_0},\quad\quad q_L=\vect{u_L\\d_L},\quad\quad\ell_L=\vect{\nu_L\\e_L}.
\end{align}
After electroweak symmetry breaking, the electric charge is
\begin{align}
Q&=Y+T_3,
\end{align}
so that the well-known particle spectrum of the SM is recovered~\cite{arthur, pdg}.

\noindent From a theoretical perspective the gauge group in eq.\ \eqref{gsm} is rather unattractive because of its cumbersome structure and the appearance of several distinct and seemingly unrelated representations from which the fields emerge. Theories with simpler gauge groups and fields transforming in fewer representations, would be more appealing if their phenomenology could be broken down to the SM in some suitable low-energy limit. In the literature many such models are discussed, with the \textsc{Georgi-Glashow} model being pivotal in understanding how the SM-representations may originate. Here, $\gsm$ is embedded in the simple group $\su(5)$ and the SM-fields fit neatly into the representations:
\begin{align}
  \mathbf{5}&=\rep{3}{1}{-\frac13}\oplus\rep{1}{2}{\frac12}\\
  \mathbf{10}&=\rep{3}{2}{\frac16}\oplus\crep{3}{1}{-\frac23}\oplus\rep{1}{1}{1}\\
  \mathbf{24}&=\rep{8}{1}{0}\oplus\rep{1}{3}{0}\oplus\rep{1}{1}{0}\oplus\rep{3}{2}{-\frac53}\oplus\crep{3}{2}{\frac53}.
\end{align}
Nevertheless, there are some impediments damping this rather promising result, most strikingly the existance of other fields transforming in the residual representations, which could lead to proton decay and other unwanted processes~\cite{su5}.
From there, one can go higher and embed $\gsm$, using the $\su(5)$-unification scheme, in higher-dimensional symmetry groups. This could lead to a breaking chain \mbox{$E_8\!\to\!E_7\!\to\!E_6\!\to\!\operatorname{SO}(10)\!\to\!\su(5)\!\to\!\gsm$} motivated by symmetries in current candidates for a Theory of Everything like String Theory~\cite{arthur, pdg, ramond}.

\noindent As expected from gauge theories, the dynamical terms in the SM are constructed out of the field strength tensors, which in turn are obtained from the commutator of their covariant derivatives \mbox{$F^{(i)}_{\mu\nu}\propto[D^{(i)}_\mu, D^{(i)}_\nu]$}:
\begin{align}
  \lsm&\supset\frac1{2g_i^2}\tr\left[F^{(i)}_{\mu\nu}F^{(i)\mu\nu}\right].
\end{align}
Here, the trace is performed over gauge indices and normalised to $\sfrac12$ in the $\un(1)$-case. The gauge sector thus is determined by the gauge couplings $g_i$.\footnote{Usually, the couplings $g_1$, $g_2$, $g_3$ are called $g'$, $g$, and $g_\mathrm{s}$.}
 Furthermore, for $\su(3)$ there exist nontrivial gauge transformations related to the fundamental group structure which can not be removed by suitable redefinitions of the fields. These terms are quantified by a parameter $\theta_\mathrm{QCD}$ and contribute a term:
\begin{align}
\lsm&\supset\frac{g_\mathrm{s}^2\theta_\mathrm{QCD}}{16\pi^2}\epsilon^{\mu\nu\rho\sigma}\tr\left[F^{(3)}_{\mu\nu}F^{(3)}_{\rho\sigma}\right].
\end{align}
Generally, this term will lead to CP-violation in the strong sector, thus making it favourable to set $\theta_\mathrm{QCD}=0$. The effect of a nonzero $\theta_\mathrm{QCD}$ could be measured as an electric dipole moment for the neutron which is heavily supressed by observation constituting the \textit{strong CP-problem}~\cite{arthur, pdg}. Many solutions to this problem are proposed in the literature, a famous example being the \textsc{Peccei-Quinn} proposal promoting $\theta_\mathrm{QCD}$ to a dynamical field~\cite{pq, pq-2}.

\noindent The fermionic fields receive their usual kinetic terms constructed with the covariant derivative in the relevant representation.
\begin{align}
  \lsm&\supset\bar{\psi}_i\imag\slashed{D}\psi_i\\
  D_\mu&=\partial_\mu-\imag qA^k_\mu\mathcal{R}(T_k).
\end{align}
For reasons of notational brevity the internal index structure of the fields will be supressed in the following, with exception of the cases where certain caveats must be considered~\cite{arthur, pdg}.

\noindent The terms in the Higgs sector are given in the form of a scalar kinetic term, the quartic potential, and the Yukawa couplings to the fermions:
\begin{align}
  \lsm\supset-\left(D^\mu\phi\right)^\dagger\left(D_\mu\phi\right)+\mu\phi^\dagger\phi-\lambda\left(\phi^\dagger\phi\right)^2-\left(\lambda^\psi\left[\bar{\psi}\phi\psi\right]_\numb{1}+\hc\right).
\end{align}
The square brackets denote that the contractions of the gauge indices are performed in a gauge invariant way. This will imply that the left-handed fermionic weak isospin doublets are contracted with the Higgs doublet while potentially remaining colour indices are contracted with the right-handed fermion field such, that the whole term couples left- and right-handed fields together, leading to the behaviour expected from a mass term.
For our three types of massive fermions (up/down-type quarks and charged leptons) the singlets constructible with $\phi$ look like
\begin{align}
\lsm&\supset-\lambda^u\left[\bar{q}_L\tilde{\phi}u_R\right]_\numb{1}-\lambda^d\left[\bar{q}_L\phi d_R\right]_\numb{1}-\lambda^e\left[\bar{\ell}_L\phi e_R\right]_\numb{1}+\hc
\end{align}
Here, the conjugated field $\tilde{\phi}^\alpha=\epsilon^{\alpha\beta}\phi^*_\beta$ was used to generate masses for the up-type quarks. This will be one of the reasons why an additional Higgs doublet is needed in the MSSM \cite{arthur, pdg, peskin, higgs}.

\noindent In its potential the Higgs field acquires a vacuum ecpectation value (VEV) of
\begin{align}
\left\langle\phi_0\right\rangle&=v\equiv\sqrt{\frac\mu{2\lambda}}.
\end{align}
Perturbing with a real scalar $h$ around this VEV will lead to the emergence of a mass for the gauge bosons \mbox{$W^\pm_\mu\propto W^1_\mu\mp\imag W^2_\mu$} and $Z^0_\mu$ as well as a massless photon $A_\mu$ from the mixing of $B_\mu$ and $W^3_\mu$ via the process known as \textit{electroweak symmetry breaking}.
The masses generated are directly related to the respective couplings and the VEV:
\begin{align}
m_h&=2\sqrt{\lambda}v\\
m_f&=v\lambda^f\\
m_W&=\frac{gv}{\sqrt{2}}\\
m_Z&=\sqrt{\frac{g^2+{g'}^2}2}v.
\end{align}
To make matters more complicated, in the SM three families of fermions are present and the Yukawa couplings $\lambda^f$ are promoted to general complex \mbox{$3\!\times\!3$}- matrices $\lambda^f_{ij}$ which can be diagonalised with bi-unitary flavour transformations
\begin{align}
\lambda^f\to V_f^\dagger\lambda^fU_f&=v^{-1}\operatorname{diag}\left(m^{(1)}_f, m^{(2)}_f, m^{(3)}_f\right).
\end{align}
Two of these matrices, $V_u$ and $V_d$, will meet again in the gauge interaction part of the Dirac terms and form the \textsc{Cabibbo-Kobayashi-Maskawa} (CKM) matrix in the quark sector.
On the leptonic side the absence of neutrino masses allows to compensate for these unitary transformations completely~\cite{arthur, pdg}. If right-handed neutrinos would be included, the \textsc{Pontecorvo-Maki-Nakagawa-Sakata} (PMNS) matrix would describe similar effects, although it is – as of today – unknown how neutrinos gain their masses and if they are Dirac or Majorana fermions~\cite{pdg}.

\noindent Given in the form described above, the Standard Model possesses 60 parameters:
Three gauge couplings $g'$, $g$, $g_\mathrm{s}$, and the vacuum angle $\theta_\mathrm{QCD}$ in the pure gauge sector. The Higgs potential contributes the two parameters $\lambda$ and $v$ as well as 54 parameters from the three Yukawa matrices $\lambda^u$, $\lambda^d$, and $\lambda^e$.\\
Not all these parameters are physical and as already emphasised, flavour rotations can be used to remove redundant parameters. This would allow a $\un(3)$-transformation for each fermion field-type \mbox{$\psi\in\{q_L, u_R^c, d_R^c, \ell_L, e_R^c\}$}, but the accidental symmetries related to conservation of baryon number $B$ and the lepton numbers $L_e$, $L_\mu$, and $L_\tau$ can not be used to eliminate parameters.\footnote{At this point it should be noted, that the SM would possess a $\un(3)^5$ flavour symmetry in the absence of mass or Yukawa terms, but their existence breaks this symmetry. Because of this, the symmetry-breaking parameters, e.\ g.\ the Yukawa matrices, can be reduced by the flavour rotations comprising the now broken symmetry. Therefore, unbroken accidental symmetries, like baryon and lepton number phase rotations, can not be used to change parameters in the model.} Thus, only the symmetry $\un(3)^5/\un(1)^4$ remains to remove 41 parameters. Henceforth, the Standard Model without neutrino masses depends on 19 free parameters. Out of these, 14 correspond to real values like couplings and masses, while three describe mixing angles and two give CP-violating phases~\cite{arthur, pdg}.
If neutrino masses were included, together with the PMNS matrix the three neutrino masses would lead to four additional mixing angles and a CP-violating phase. Depending on them being Dirac or Majorana particles, two additional CP-violating phases in the latter case could be present and the SM with massive neutrinos would have 26 or 28 physical parameters~\cite{pdg}.

\section{The Minimal Supersymmetric Standard Model}\label{mssm-basics}
To obtain a minimal supersymmetric extension of the SM, all previous fields are embedded in corresponding chiral (real) superfields $\hat{\Phi}_i$ ($\hat{V}_i$).
Furthermore, it must be taken into account that the Higgsino accompanying the Higgs boson from before – now denoted $H_d$ – introduces a gauge anomaly in the model. To correct for this, a second Higgs doublet, called $H_u$, with opposite charge(s) is needed. In addition, it can be seen that the need for a holomorphic superpotential would forbid the Yukawa term giving masses to the up-type quarks and a new independent field $H_u$ must be included, which takes the position of the previously complex conjugated field $\tilde{\phi}$.
Implementing these considerations leads to the field content of the MSSM depicted in tab.\,\ref{mssm-fields}.
\begin{table}[H]\centering
\caption{Table of MSSM-superfields and their components, from~\cite{pdg}.\label{mssm-fields}}
\bgroup
\def\arraystretch{1.4}
\begin{tabular}{cccl}
  super field& bosonic field& fermionic field& representation\\\hline
  $\hat{V}_8$& $g$& $\tilde{g}$& $\quad\quad\rep{8}{1}{0}$\\
  $\hat{V}$& $W^0$, $W^\pm$& $\tilde{W}^0$, $\tilde{W}^\pm$& $\quad\quad\rep{1}{3}{0}$\\
  $\hat{V}'$& $B$& $\tilde{B}$& $\quad\quad\rep{1}{1}{0}$\\
  $\hat{L}$& $(\tilde{\nu}_L, \tilde{e}_L)$& $(\nu_L, e_L)$& $\quad\quad\rep{1}{2}{-\frac12}$\\
  $\hat{E}^c$& $\tilde{e}_R^c$& $e_R^c$& $\quad\quad\rep{1}{1}{1}$\\
  $\hat{Q}$& $(\tilde{u}_L, \tilde{d}_L)$& $(u_L, d_L)$& $\quad\quad\rep{3}{1}{\frac16}$\\
  $\hat{U}^c$& $\tilde{u}_R^c$& $u_R^c$& $\quad\quad\rep{3}{1}{-\frac23}$\\
  $\hat{D}^c$& $\tilde{d}_R^c$& $d_R^c$& $\quad\quad\rep{3}{1}{\frac13}$\\
  $\hat{H}_u$& $\left(H_u^+, H_u^0\right)$& $\left(\tilde{H}_u^+, \tilde{H}_u^0\right)$& $\quad\quad\rep{1}{2}{\frac12}$\\
  $\hat{H}_d$& $(H_d^0, H_d^-)$& $(\tilde{H}^0_d, \tilde{H}_d^-)$& $\quad\quad\rep{1}{2}{-\frac12}$
\end{tabular}
\egroup
\end{table}
\noindent The Lagrangian of the MSSM is constructed out of SUSY conserving terms and SUSY breaking terms where the latter introduce most of the new parameters~\cite{abc, primer, pdg, arthur, haber}.

\noindent The gauge terms are built straightforwardly, using the appropriately\footnote{For the abelian gauge field the definition could be simplified further to \mbox{$\mathcal{W}_\alpha=-\sfrac14\bar{D}^2D_\alpha\hat{V}$}.} defined field strength super fields
\begin{align}
\mathcal{W}_{i, \alpha}&\equiv-\frac14\bar{D}^2e^{-\hat{V}_i}D_\alpha e^{\hat{V}_i},
\end{align}
in an appropriate gauge.
The internal gauge dynamics are then given by
\begin{align}
  \lmssm\supset&\frac1{2g_i^2}\tr\left[\int\dd^2\theta\mathcal{W}_i^\alpha\mathcal{W}_{i, \alpha}+\hc\right].
\end{align}
The $\theta$-parameter can be included by complexifying\footnote{The key idea would be to go from a real coupling $\sfrac1{2g_i^2}$ to complex \mbox{$\tau\equiv\sfrac{1}{2g_\mathrm{s}^2}-\sfrac{\imag g_\mathrm{s}^2\theta_\mathrm{QCD}}{16\pi^2}$} and respecting $\tau$ in the hermitian conjugates.} the gauge coupling~\cite{lykken, primer, abc}.
The kinetic Kähler terms just become
\begin{align}
  \lmssm\supset&\int\dd^2\theta\dd^2\bar{\theta}\left[\hat{\Phi}^\dagger_ie^{2\hat{V}_i}\hat{\Phi}_i\right]_\numb{1}\\
  \hat{V}_i\equiv&\hat{V}_8^a\mathcal{R}_i(T_a)+\hat{V}^k\mathcal{R}_i(T_k)+Y_i\hat{V}'.
\end{align}
The $D$-terms in the Higgs sector of this potential will lead to the emergence of a quartic term in the effective Higgs potential for the two doublets. This has rather suprising consequences – at least at tree level – for the mass of the lightest Higgs field, but more on this in section~\ref{mssm-details}.
In the MSSM, a quartic coupling of Higgs fields can not be included in the superpotential interaction terms since it would be non-renormalisable~\cite{higgs, haber, pdg}.

\noindent When considering the contributions
\begin{align}
  \lmssm\supset&\int\dd^2\theta W(\hat{\Phi}_i)+\hc
\end{align}
from the superpotential $W$, naïvely all quadratic and cubic terms consisting of holomorphic gauge singlets are allowed leading to
\begin{align}\nonumber
  W=&\lambda_d\left[\hat{H}_d\hat{Q}\hat{D}\right]_\numb{1}+\lambda_e\left[\hat{H}_d\hat{L}\hat{E}\right]_\numb{1}-\lambda_u\left[\hat{H}_u\hat{Q}\hat{U}\right]_\numb{1}+\mu \left[\hat{H}_u\hat{H}_d\right]_\numb{1}\\
  &+a\left[\hat{L}\hat{H}_u\right]_\numb{1}+b\left[\hat{Q}\hat{L}\hat{D}\right]_\numb{1}+c\left[\hat{U}\hat{U}\hat{D}\right]_\numb{1}+d\left[\hat{L}\hat{L}\hat{E}\right]_\numb{1}.
\end{align}
The upper line contains the Yukawa couplings as well as the $\mu$-parameter, which will be the only parameter leading to masses for the Higgsinos.
Unfortunately the terms in the lower line introduce processes, violating baryon and lepton number conservation, and can not be excluded by renormalisability arguments as in the SM. As often, it would be attractive to rely on a suitable symmetry not obeyed by the unwanted terms. A promising candidate for such a symmetry would be the $\un(1)_R$ R-symmetry, transforming component fields differently, but using it continuously would also forbid the $\mu$-term and lead to massless Higgsinos and therefore striking contradiction with experiment. Breaking $\un(1)_R$ down to R-parity $\mathcal{Z}_2$ circumvents this problem and implies the assignment
\begin{align}
R=(-1)^{3(B-L)+2s},
\end{align}
for each component field.\footnote{An equivalent assignment makes use of matter parity \mbox{$P_m=(-1)^{3(B-L)}$}, defined on superfields, which can be seen more easily.}
Restricting to R-parity conservation has the importand consequence that superpartners of the ordinary SM particles can only be produced and annihilated in pairs, implying the existance of a lightest supersymmetric particle (LSP). Being weakly interacting (most probably) massive particles (`WIMPs'), such LSPs are rather attractive aspirants for dark matter particles~\cite{pdg, arthur, primer, abc}.

\noindent In the MSSM, SUSY is broken explicitly and the origin of the breaking terms is not considered. They could originate in a hidden sector, e.\ g.\ via gauge- or gravity-mediation, but in the end both will lead to the inclusion of \textit{soft}\footnote{Soft in this context meaning that the corresponding operators are relevant and therefore have mass dimension strictly below four.} SUSY breaking terms of the form:
\begin{align}\nonumber
  -\mathcal{L}^\mathrm{MSSM}_\mathrm{soft}\supset&\frac12M_i\tilde{\bar{\lambda}}_i\tilde{\lambda}_i+M^2_{\tilde{F}}\tilde{f}^\dagger\tilde{f}\\
  &+m_1^2H_d^\dagger H_d+m_2^2H_u^\dagger H_u+m_{12}^2\left(H_u\cdot H_d+\hc\right)\\\nonumber
  &+T_UH_u\tilde{Q}\tilde{U}+T_DH_d\tilde{Q}\tilde{D}+T_EH_d\tilde{L}\tilde{E}+\hc
\end{align}
The terms in the first line lead to masses and mass differences for the gauginos and sfermions, while the second line contributes to the Higgs potential. Lastly, the third line introduces additional trilinear couplings between the sfermion scalars and the Higgs fields.
Often a parametrisation\footnote{Here, it should be noted that one has to be careful regarding matrix indices, since it is not clear – at least ad hoc – , how the indices of $A_F$, $\lambda_f$, and $T_F$ are related to each other and there are different conventions. Nevertheless, many models simplify the trilinear couplings to be diagonal in the flavour eigenbasis with only scalar $A$-parameters remaining.} $m_{12}^2=\mu B$ (and $T_F=\lambda_f A_F$) is chosen. Therefore, the corresponding terms are called $A$ and $B$-terms~\cite{pdg, arthur, chung, peskin}.

\noindent Having introduced all parameters, the same counting procedure as in the SM can be repeated for the MSSM. The gauge sector contributes four real parameters with $g$, $g'$, $g_\mathrm{s}$, and $\theta_\mathrm{QCD}$, while the superpotential contains the three Yukawa matrices $\lambda_u$, $\lambda_d$, and $\lambda_e$ (each contributing 18 parameters). Two parameters come from the modulus and phase of $\mu$.
Most of the parameters are introduced by the SUSY breaking terms: Three generally complex gaugino masses give rise to six parameters, while the five different hermitian scalar mass matrices introduce nine parameters per matrix, and the Higgs-sector adds $v$, $\beta$, and three trilinear scalar coupling matrices $T_F$ with 18 parameters each.
Using the flavour transformations $\un(3)^5/\un(1)^2$, 43 parameters can be removed. Here, the fixation on lepton number conservation per generation can not be sustained anymore, since many of the additional terms in the MSSM can and will lead to flavour changing processes that were previously supressed strongly in the SM.
Actually, this is not all, since there exist two additional $\un(1)$ transformations that can be exploited to further remove phases from the MSSM: The before-mentioned R-transformations as well as a so called Peccei-Quinn symmetry $\un(1)_\mathrm{PQ}$, which can be used to make the gaugino mass $M_3$ and $\mu$ real~\cite{haber}.
In the end, 45 parameters can be removed from the 169 parameters introduced in the Lagrangian.
Finally, the full MSSM is given by 124 parameters and this full description is called the MSSM-124. Of these 124 parameters, of which 105 originate in the SUSY breaking terms, 39 will be real values like masses and couplings, 39 will be mixing angles, and 45 will correspond to CP-violating phases~\cite{haber, pdg}.

\section{Phenomenological implications of the MSSM}\label{mssm-details}
One striking implication of the MSSM is the structure of the Higgs sector:
The contributions to $V_\mathrm{Higgs}$ come from the $D$-terms in the Kähler potential as well as the soft SUSY breaking terms and lead to a full potential
\begin{align}\nonumber
  V_\mathrm{Higgs}=&\left(m_1^2+|\mu|^2\right)H_d^\dagger H_d+\left(m_2^2+|\mu|^2\right)H_u^\dagger H_u+m_{12}^2\left(H_u\cdot H_d+\hc\right)\\
  &+\frac{g^2+{g'}^2}8\left(H_d^\dagger H_d-H_u^\dagger H_u\right)+\frac12g^2\left|H_d^\dagger H_u\right|^2.
\end{align}
After minimising $V_\mathrm{Higgs}$, both doublets acquire VEVs
\begin{align}
\left\langle H_f^0\right\rangle&=v_f,
\end{align}
which can be related to the previous $v$ via
\begin{align}
v^2=v_u^2+v_d^2.
\end{align}
By convention, an angle $\beta$ is defined between the $v_f$ as
\begin{align}
  \tan\beta&=\frac{v_u}{v_d}.
\end{align}
Since the quartic coupling is given as function of the weak coupling constant, this implies an \textit{upper bound} on the mass of the lightest Higgs boson – at least on tree level – of
\begin{align}
  m^2_h&\le m^2_Z\cos^22\beta.
\end{align}
This bound relaxes at higher loop order where suitable corrections lift this bound as high as $\SI{135}{\GeV}$, removing the contradiction present before~\cite{pdg, higgs, peskin, primer, haber}.

\noindent The most interesting aspects can be found in the particle spectrum:
With its superfields the MSSM gives rise to a luscious landscape of new particles to observe, but the additional fields include many new possibilities for mixing, modifying the expectable spectrum of new particles.
On the SM side of the MSSM, the particle content remains relatively unchanged with exception of the Higgs boson, which is now replaced by additional Higgs bosons coming from the second doublet. The real perturbations of the $H^0_f$ around $v_f$ will mix to form two CP-even scalars denoted by $h^0$ and $H^0$ (capital letter corresponding to larger mass), while their imaginary counterparts form a CP-odd scalar $A^0$ and a neutral Goldstone boson $G^0$, which is later eaten as part of the Higgs mechanism's inner mechanics, giving rise to longitudinal modes and therefore masses for the $W$ and $Z$ bosons. Similarly, the charged components, $H^+_u$ and $H^-_d$, will mix forming two charged Higgs bosons $H^\pm$ and two Goldstone bosons $G^\pm$. In total, the contribution from the additional doublet leads to four new Higgs bosons $H^0$, $H^\pm$, and $A^0$~\cite{higgs, peskin, primer}.
The Higgsinos will mix with the other weakly charged gauginos present.
The neutral Higgsinos $\tilde{H}^0_u$ and $\tilde{H}^0_d$ are now accompanied by the bino $\tilde{B}^0$ and the neutral wino $\tilde{W}^0_3$ so that mixing will be described by a complex \mbox{$4\!\times\!4$}-matrix and result in four neutral physical mass eigenstates, called \textit{neutralinos} and denoted $\tilde{\chi}^0_i$, their index incrementing with increasing mass. Accordingly, the charged winos $\tilde{W}^\pm$ will mix with the charged Higgsinos $\tilde{H}^\pm$ and form two (actually four) chargino states, denoted $\tilde{\chi}^\pm_i$.
Considering the sfermions, the scalar superpartners of the SM-fermions, it must be noted that there are two superpartners per (charged) fermion $f$ (with exception of the neutrinos): One scalar partner for the left-handed component $\tilde{f}_L$, one for the right-handed component $\tilde{f}_R$. Generally, mixing can and will occur between these different fields if they carry the same charges, thus leading to \mbox{$6\!\times\!6$}-mixing for the up/down-type squarks ($\tilde{q}_L$, $\tilde{q}_R$) and the charged sleptons ($\tilde{\ell}_L$, $\tilde{\ell}_R$). For the sneutrinos only \mbox{$3\!\times\!3$}-mixing is present. In the end, only mass eigenstates $\tilde{u}_i$, $\tilde{d}_i$, $\tilde{\ell}_i$, and $\tilde{\nu}_i$ will remain, whose identification with flavour eigenstates will be highly dependent on the parameters at play.
\begin{table}[H]\centering
\caption{Examples for parameter configurations leading to nearly pure gaugino states. Adapted from~\cite{pdg}.\label{gaugino}}
\begin{tabular}{cc}
parameter conditions&$\tilde{\chi}^{0, \pm}_1$\\\hline
$\left|M_1\right|, \left|M_2\right|\lesssim\left|\mu\right|, m_Z$&$\tilde{\gamma}$\\
$\left|M_1\right|, m_Z\lesssim\left|M_2\right|, \left|\mu\right|$&$\tilde{B}$\\
$\left|M_2\right|, m_Z\lesssim\left|M_1\right|, \left|\mu\right|$&$\tilde{W}^{0, \pm}$\\
$\left|\mu\right|, m_Z\lesssim\left|M_1\right|, \left|M_2\right|$&$\tilde{H}^{0, \pm}$
\end{tabular}
\end{table}
\noindent By this it becomes apparent that only very specific configurations in parameter space will lead to a `double-SM'-structure of the MSSM, and the phenomenology depends delicatly on the parameters~\cite{pdg, peskin}. Some examples for such configurations can be found in tab.\,\ref{gaugino}.


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% @author   Shmish  "shmish90@gmail.com"
% @legal    MIT     "(c) Christopher Schmitt"
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  \begin{problem}{1}
    Solve each of the following counting problems, writing your answer as an integer in the standard decimal notation.\par
    In how many ways can a student club that consists of 20 students select 3 students to attend a conference?
    $$\binom{20}{3} = 1140$$

    How many strings of 0s and 1s have length 11 and have exactly eight ones?
    $$\binom{11}{8} = 165$$

    How many strings of 0s and 1s have length 11 and have at most three ones?
    $$2^{11} - 2^{7} = 1920$$

    A donut shop has donuts in six different flavors: chocolate frosted, chocolate creme, glazed, jelly, plain, and Boston creme. The donut shop has an unlimited supply of each flavor. In how many different ways can the donut shop create a box of 18 donuts? Note that the box can include multiple donuts of the same flavor, and that donuts of the same flavor are considered indistinguishable.
    $$\binom{n + r - 1}{r} = \binom{6 + 18 - 1}{18} = \binom{23}{18} = 33649$$

    The donut shop now wants to create a box of 18 donuts, including at least four glazed ones. In how many ways can this be done?
    $$\binom{n + r - 1}{r} = \binom{6 + 14 - 1}{14} = \binom{19}{14} = 11628$$

    The donut shop now wants to create a box of 18 donuts, including at most four glazed ones. In how many ways can this be done?
    $$\binom{6 + 18 - 1}{18} - \binom{6 + 13 - 1}{13}= \binom{23}{18} - \binom{18}{13} = 33649 - 8568 = 25018$$

    The donut shop now wants to create a box of 18 donuts, including exactly four glazed ones. In how many ways can this be done?
    $$1 * \binom{5 + 14 - 1}{14} = \binom{18}{14} = 3060$$

    I want to buy one donut every day from Monday through Friday at the donut shop. In how many ways can this be done?
    $$\binom{n + r - 1}{r} = \binom{6 + 5 - 1}{5} = 252$$

    How many “words” can be formed by rearranging the letters of the word MATHEMATICALLY? The “words” do not have to be actual English words.
    $$\frac{14!}{2! * 3! * 2! * 2!} = 1816214400$$

    Find the coefficient of $x^8y^4$ in the expansion of the binomial $(x + 2y)^{12}$
    $$\binom{12}{4} = 495$$
    $$495(x^8)(2y^4) = 990x^8y^4$$
  \end{problem}

  \begin{problem}{2}
    $$(x + y)^{2k} = \sum_{i=0}^{2k} x^i y^{2k - 1} \binom{2k}{i}$$
    $$(x + y)^{2J + 1} = \sum_{i=0}^{2J + 1} x^i y^{2J} \binom{2J + 1}{i}$$
    Theorem: $$\sum_{0 \le i \le n, n\ odd} \binom{n}{i} = \sum_{0 \le i \le n, n\ even} \binom{n}{i}$$
    $$\sum \binom{2k+1}{i} = \sum \binom{2j}{i}$$
    $$?$$
  \end{problem}
\end{document}


\documentclass[]{article}

\usepackage{amsmath}
\usepackage{graphicx}
\graphicspath{ {./figures/} }

%opening
\title{Magical ecologies: a model of predator-prey equilibria for closed systems of magical girls and witches}
\author{Kyubey \#65471}

\begin{document}

\maketitle

\begin{abstract}
	We present a simple mathematical model of interactions between magical girl (\textit{Puella Magica}) and witch (\textit{Maga Malefica}) populations in a closed regulated system. The model makes use of a few descriptive parameters to characterise the environment and predicts dynamical population behaviour. We show how some very general conclusions can be deduced about the equilibrium populations of magical girls and witches and their stability. This study will provide operators in the field with more tools to better manage the ecology of these species, which are vital for our universe's thermodynamic preservation.
\end{abstract}

\section{Introduction}

Since the first documented discovery of the exothermic, anentropic decay process from magical girl (\textit{Puella Magica}) to witch (\textit{Maga Malefica}) \cite{kyubey1}, this fundamental phenomenon has become a key element of our species' activities in the field of cosmic scale thermodynamic preservation. The management of magical girl-witch ecologies has today a disproportionate impact on our economy, employing effectively 99.93\% of our active population, and providing a free energy flux of more than $10^{40} W$ across all the galaxies over which we've currently spread. While this is still far below the amount necessary to definitely stave off heat death, the impressive scale and efficiency of this operation can not be overstated. It is thus somewhat surprising that until now the management of local ecologies has often been left to individual operators, almost as an art more than a science. While this has often been successful, occasional examples of ecological collapses on a catastrophic scale have been recorded \cite{kyubey2}. Therefore, we endeavour here to begin an attempt at a systematic description of magical girl-witch ecological equilibria, as a tool for future control and regulation of such system.

\section{The model}

Let us consider a closed system containing two populations of magical girls and witches, labelled respectively as $M$ and $W$. The indigenous species is here considered as a virtually infinite reserve; this is in general a good approximation, and we will see later that it is not a problem for the results of the model. We then consider the dynamics affecting these populations. For magical girls, three phenomena are at play:

\begin{enumerate}
	\item a recruitment process by the local Kyubey agent;
	\item a loss to witches in battle;
	\item a loss to decay processes and transformation into witches.
\end{enumerate}

Conversely, for witches, there are only two relevant phenomena:

\begin{enumerate}
	\item an increase in numbers thanks to decay processes of magical girls;
	\item a loss to magical girls in battle.
\end{enumerate}

In order to fully characterise this system we introduce five parameters. The first is the recruitment rate, $r$. The second is the probability of battle $b$; this is chosen so that the total number of battles per unit time can be found as $N_B = bMW$. Since $MW$ is proportional to the likelihood of a clash between magical girls and witches, which increases with both their population, $b$ depends in general by the area over which these population are spread. For a large city, for example, $b$ will be small, as encounters are less likely unless the populations are large enough to compensate. Other local factors however can affect $b$, such as the attitudes and motivation level of the local population of magical girls and the general psychological state of the indigenous community, which has an effect on witch activity.\newline
The third parameter is the win rate, $w$ (not to be confused with the witch population, $W$). This expresses simply the likelihood of the average magical girl defeating the average witch in battle, so it has to be $0 \leq w \leq 1$. The last two parameters are inherent to the qualities of the soul gems of the local magical girls. These are the average decay time, $\tau_0$, which represents the average time a magical girl would decay into a witch if deprived of any means to recharge her soul gem, and the efficiency of recharge, $\eta$, which represents the average fraction of a soul gem's total energy that can be recovered with a grief seed. While it has to be $\eta \geq 0$, there is no upper limit; soul gems can not be overcharged, but nothing forbids magical girls from saving partially used-up grief seeds to use for future recharges if they manage to do so. However, in practice, it tends to be that $\eta \sim 0.25$.\newline

We can now outline the dynamical equations of the model. These are a pair of first order differential equations:

\begin{equation}\label{eq_mg}
\frac{dM}{dt} = r-\frac{1}{\tau_0+\eta b w W}M-b (1-w)MW
\end{equation}

\begin{equation}\label{eq_w}
\frac{dW}{dt} = \frac{1}{\tau_0+\eta b w W}M-bwMW
\end{equation}

Most of these terms are self-explanatory. The magical girls creation rate $r$ is the only positive contribution to $dM/dt$, and losses to battles are represented as $N_B$ multiplied by the likelihood of unfavourable outcome for each party, respectively, $1-w$ and $w$ for magical girls and witches. However, the decay process deserves a bit more of attention. We have here made use of an effective decay rate,

\begin{equation}\label{tau}
\tau = \tau_0+\frac{\eta w N_B}{M} = \tau_0+\eta b w W
\end{equation}

. In other words, we consider an extension to the regular decay time that is proportional to how many witches each magical girls gets to defeat, on average, times the efficiency $\eta$.\newline
We now move on to study this model in some detail. As we will see, it leads to surprising and useful insights about the equilibria of magical girls and witches, and the resulting energy production.

\section{Results}
\subsection{Equilibrium}

To search for the equilibrium conditions of this model we look for values of $M$ and $W$ which cause Eq. \ref{eq_mg} and \ref{eq_w} to be zero. This turns out to be actually quite simple. If we focus on \ref{eq_w}, we see that since $M$ appears in both terms, it can be simplified. This leads us to the first important finding of this work: \textit{the equilibrium population of witches is independent from the amount of magical girls present}. In particular, by solving a second degree equation and discarding the unphysical negative solution, we find

\begin{equation}\label{eq_pop_w}
W_{eq} = \frac{\sqrt{\tau_0^2+4\eta}-\tau_0}{2\eta bw}
\end{equation}

. In other words, the equilibrium population for witches only depends on the average properties of soul gems and on other local factors such as the area of the habitat in which they're living and the combat skill of the magical girls. In particulars, since it's reasonable to assume that $b \sim 1/A$, where $A$ is the area of the habitat, this leads to a constant sustainable density of witches that only depends on the properties of the local population of magical girls and indigenous species.\newline
Having found this equilibrium, we then have for Eq. \ref{eq_mg} that

\begin{equation}\label{eq_pop_m}
M_{eq} =   \frac{r}{\frac{1}{\tau_0+\eta b w W_{eq}}+b (1-w)W_{eq}}
\end{equation}

. This is a very interesting result, because it can be interpreted as a prescription: the rate of recruitment can be used to set the desired magical girl population at one's will. In other words, if $W_{eq}$ is simply a function of environmental factors, which can only be controlled very indirectly, $M_{eq}$ is a quantity entirely in the hands of the local Kyubey agent. This is especially important with regards to energy production. If the average energy produced per decay is $E_{decay}$ and the average energy consumed per wish granted as a part of the recruiting process is $E_{wish}$, then the net energy production will be

\begin{align}\label{net_power}
P_{eq} &=   E_{decay}\frac{M_{eq}}{\tau_0+\eta b w W_{eq}}-E_{wish}r \\
&= r\left[E_{decay}\frac{1}{1+b(1-w)(\tau_0+\eta b w W_{eq})W_{eq}}-E_{wish}\right] \nonumber \\
& = r\left[wE_{decay}-E_{wish}\right] \nonumber
\end{align}

. This is the second, surprising key result of this work: \textit{at equilibrium, the energy production depends only on the recruiting rate and the win rate of magical girls}. Maximum efficiency is reached in a condition in which the magical girls win every time against the witches. While this may sound counterintuitive, as winning leads magical girls to gather grief seeds and thus slow down decay, it makes sense if we consider that each loss of a magical girl to a witch is an inefficiency, as it wastes their wish energy without delivering any decay payoff. Conversely, a highly successful population of magical girls will succumb only to decay, however long that might take. In an equilibrium condition the effective rate is then controlled by recruitment. For low $r$, there will be a small amount of long lived veterans. For high $r$, the turnover will be much faster, due to having the same population of witches, but a much higher population of magical girls to split the grief seeds between. Nevertheless, it is in the best interest of the local Kyubey agents to guarantee that their magical girls are as close to 100\% successful in battle as possible, as that results in maximum conversion efficiency.

\subsection{Dynamics}

Having verified that an equilibrium exists, however, it remains to be seen under which conditions it is stable, which is the major concern for operators on the field. Luckily, intuition already tells us that this is likely to be the case. The decay term in Eq. \ref{eq_mg} and \ref{eq_w} has clearly a stabilising effect. In presence of an excess of witches, it will cause the life of magical girls to get longer, thanks to a greater availability of grief seeds; while if there were too few, the decay process would speed up and compensate. This can also be seen by analysing Eq. \ref{eq_w} around its equilibrium point, as the sign of the derivative is opposite to that of the error, $W-W_{eq}$. However, in order to gain a more detailed understanding of these dynamical processes, we carry out numerical integration of the equations for a number of different scenarios.\newline

\begin{figure}
	\includegraphics[width=0.7\textwidth]{fig1}
	\centering
	\caption{Model 1. Initial conditions: $M > M_{eq}$, $W < W_{eq}$.}
	\label{fig1}
\end{figure}

\begin{figure}
	\includegraphics[width=0.7\textwidth]{fig2}
	\centering
	\caption{Model 2. Initial conditions: $M < M_{eq}$, $W > W_{eq}$.}
	\label{fig2}
\end{figure}

\begin{figure}
	\includegraphics[width=0.7\textwidth]{fig3}
	\centering
	\caption{Model 3. Initial conditions: $M > M_{eq}$, $W > W_{eq}$. The low win rate ($w = 0.1)$ causes an oscillation in the magical girl population before equilibrium is reached.}
	\label{fig3}
\end{figure}

Figures \ref{fig1}-\ref{fig3} show some possible scenarios. These start with off-equilibrium values of $M$ and $W$ and converge over time. The time scale is determined by setting $\tau_0=1$. As one can see, all the convergence is quick and without incident, regardless of starting conditions. Figure \ref{fig3} shows an example in which an oscillation of the magical girl population can be observed. This is the consequence of a specifically `soft' model, with a low win rate ($w=0.1$). While convergence eventually still occurs, this is another reason for why a high win rate for magical girls is generally desirable - as it tends to result in a faster and surer convergence of populations to their equilibrium values.

\section{Conclusions}
An idealised model describing the population dynamics of magical girls and witches in a closed environment has been developed and characterised. The model allows us some important and non trivial insights in the ecological balance of such populations, in particular:

\begin{itemize}
	\item that the density of witch population sustainable by a territory is independent of the population of magical girls, and only depends on the properties of the territory itself,
	\item that the population of magical girls is directly proportional to the recruitment rate, and
	\item that the energy production rate due to the decay of magical girls into witches is maximally efficient when the former have high rates of success in battle against the latter.
\end{itemize}

In practical field applications, of course, these considerations will need to be balanced with other requirements. For example, while a high recruitment rate and a perfect win rate would maximise energy production as per Eq. \ref{net_power}, they would actively work against the psychological needs of recruitment itself, which often relies on the threat posed by witches as a motivational factor. If witches are vastly outnumbered by magical girls and are dispatched quickly and reliably, the appeal of such arguments would be significantly weakened. Regardless, these results provide useful guidelines that need then to be integrated within the larger needs of ecological management.\newline
There are also potentially significant effects that have been ignored in this first model. Among them are any finite-size effects that could manifest in the situation in which an area was significantly overcrowded with witches or magical girls; the possibility, documented albeit of relatively small incidence, of internal combat and even murder within magical girls \cite{kyubey3}; and any effects due to the finiteness of the indigenous population if put under extreme stress due to either the drain of recruitment or the damage caused by witches. While some or all of these effects can matter in extreme circumstances, we believe that in the case of well-managed systems, the conditions for them to be important should never verify, and the results of this paper should hold. Nevertheless, the inclusion of such effects will be the object of study of future work.

\bibliographystyle{unsrt}
\bibliography{mweco}

\end{document}


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\begin{document}

\twocolumn

\thispagestyle{empty}

\begin{center}
{\Large Assignment 13}\\
{\Large Published on 2020-12-06,}\\
{\em Estimated Time: 30 minutes,}\\
{\em Max.grade 10\textperthousand}
\end{center}

\section{Dijkstra's Algorithm}

(Goodrich2011, p.640) defines Dijkstra's algorithm.
See also \url{https://bit.ly/2JSXqMU}.
It is an efficient algorithm; it requires $O((m+n)\log_2 n)$ time, if
we use priority queues; here $m$ is the number of edges and $n$ is the number
of vertices in a graph.

In this exercise you do not need to implement a priority queue;
assume that you can always pick the vertex with the smallest distance and
add it to the set $S$ of visited vertexes (those having distances already computed).

\section{Problem}

We start with the graph shown in Figure~\ref{fig:problem-graph}.

\begin{figure}[!htb]
\center{\includegraphics[width=2in]{assignment13-dijkstra/problem-graph.png}}
\caption{\label{fig:problem-graph} Graph diagram}
\end{figure}

The following edges $(A,B)$, $(E,C)$, $(D,C)$ have weights
$a+1$, $b+1$, and $c+1$ respectively
(here $a,b,c$ should be replaced by the digits from your Student ID).
Vertex $A$ will be your source vertex. (You can assume that the distance
from $A$ to itself is $0$; initially all the other distances are infinite, but
then Dijkstra's algorithm relaxes them).

\vspace{10pt}
{\bf (A)} Redraw the graph Figure~\ref{fig:problem-graph},
replace the edge weights $a+1$, $b+1$, and $c+1$ by your values of $a,b,c$.

\vspace{10pt}
{\bf (B)} Run the Dijkstra's algorithm;
create a table showing how distances to $A,B,C,D,E$
change as the relaxations are performed.
At every iteration highlight which vertex (among those not yet finished)
has the minimum distance. Add it to the set $S$ of finished vertices
(the set $S$ will have $0$ in the very first iteration; after that it will grow by
one vertex at a time).

\vspace{10pt}
{\bf (C)} Summarize the result: For each of the $5$ vertices
tell what is its minimum distance from the source.
Also tell what is the shortest path how to get there.

\end{document}


\section{Algorithm}
\label{sec:algorithm}

In this section we describe our ROSC algorithm. Figure~\ref{figure:flow_graph}(c) shows a
flow diagram of ROSC. ROSC follows the basic pipeline of spectral clustering except that
it generates a coefficient matrix $Z$
%(shown in the red box of the diagram)
and feeds it
 into the clustering pipeline in place of
the similarity matrix $S$.
The objective is to find a  $Z$ that possesses  grouping effect.
To achieve that, ROSC uses PI to obtain {\pev}s from which a basic $Z$ is derived.
Next, it generates the TKNN graph with which $Z$ is rectified.
In the following, we discuss how $Z$ is derived from the {\pev}s, define the TKNN graph,
describe the rectification process,
and prove that the resulting $Z$ has the desired grouping effect.

%
%In this section, we propose a \textbf{RO}bust \textbf{S}pectral \textbf{C}lustering method ROSC.
%Instead of directly performing spectral clustering on the given similarity matrix,
%it constructs a new matrix that reserves the efficacy of effective similarities
%and rectifies ineffective ones in the raw similarity matrix.
%The new matrix will lead to a more robust spectral clustering.

\comment{
\subsection{Model overview}
The overall framework of ROSC is summarized as follows.

First, generate a set of pseudo-eigenvectors.
Since the standard spectral clustering methods using only the dominant eigenvectors may fail
in the data containing multi-scale clusters,
we generate a set of pseudo-eigenvectors to fuse more useful cluster-separation information.

Second, rectify the given similarity matrix.
The failure of spectral clustering on multi-scale data originates from the inaccurate similarity matrix.
Therefore, we rectify the matrix and aim to achieve a new one
that can more accurately reflect the true similarities between objects.
%This is a major problem to be addressed.

Third, perform spectral clustering.
After rectification, the new similarity matrix will be more accurate and it will be safer to perform spectral clustering.

These three steps will lead to robust clustering results and next we introduce each step in detail.
}

\subsection*{Pseudo-Eigenvectors}
\label{sec:sec:pseudo}
Given a similarity matrix $S$, we normalize it by $D^{-1}S$ and apply PI to obtain {\pev}s.
%When dealing with multi-scale data,
%spectral clustering
%using only the top-$k$ eigenvectors may fail while some other eigenvectors may be useful~\cite{lin2010power}.
%Therefore, it is necessary to fuse the discriminative cluster-separation information in all eigenvectors
%and PI provides such fusion by returning a pseudo-eigenvector.
%However, when the number of clusters is large,
%a single pseudo-eigenvector is insufficient due to the cluster-collision problem.
%%Since PI returns a pseudo-eigenvector that is a combination of all the eigenvectors,
Similar to~\cite{ye2016fuse},
we run PI multiple times with different random initial vectors to generate a set of {\pev}s,
which maps each object into a low dimensional embedding.
Note that small {\ev}s are {\it shrunk} by PI~\cite{lin2010power}.
%\footnote{From Equation~\ref{eq:pi}, we see that the $i$-th
%largest \ev\ is
%shrunk at a rate of $\lambda_i/\lambda_1$ per iteration.}.
To alleviate the shrinkage of small {\ev}s,
we follow the approach of~\cite{huang2014diverse}
and gradually decrease the number of iterations executed in PI as more {\pev}s are obtained.

Since the {\pev}s approximate the most dominant {\ev}, they could be similar.
To reduce this redundancy,
whitening~\cite{kessy2017optimal} is used to make the pseudo-eigenvectors uncorrelated.
Moreover, noise in the {\pev}s are reduced
by a rectification process, which will be discussed later.

\comment{
First, we generate a set of pseudo-eigenvectors using different random vectors $\bm{v}_0$.
From Eq.~\ref{eq:v0}, $\bm{v}_0$ can be represented by the subspace spanned by all the eigenvectors.
Suppose $\bm{e}_i$ is an informative eigenvector with a relatively small eigenvalue $\lambda_i$.
Given a $\bm{v}_0$, if it has a dominant component in the direction of $\bm{e}_i$,
i.e., the associated coefficient $c_i$ is large,
with an appropriate number of iterations, $c_i\lambda_i^t\approx c_2\lambda_2^t$,
the effect of $\bm{e}_i$ will be finally retained in the pseudo-eigenvector.
Therefore, multiple pseudo-eigenvectors generated by different $\bm{v}_0$
may increase the chance to hold all the informative eigenvectors.

Second, we can control the number of iterations in PI.
In PIC, the \emph{velocity} and \emph{acceleration} at the $t$-th iteration are respectively defined as
$\bm{\delta}_t = \bm{v}_t - \bm{v}_{t-1}$ and $\bm{\epsilon}_t = \bm{\delta}_t-\bm{\delta}_{t-1}$.
A threshold $\epsilon$ is introduced to measure the acceleration change in two consecutive iterations
and stop PI when $|\bm{\epsilon}_t-\bm{\epsilon}_{t-1}|<\epsilon$.
However,
since the effect of eigenvectors corresponding to smaller eigenvalues will shrink as $t$ increases,
we can decrease the number of iterations by gradually increasing the $\epsilon$ value,
which provides another way to keep useful information in ``less important'' eigenvectors.
}

\subsection*{Transitive {\it K} Nearest Neighbor (TKNN) Graph}

%\begin{figure}
%    \centering
%        \includegraphics[width = 0.8\linewidth]{figure/example1.eps}
%        \caption{An toy example}
%        \label{figure:example1}
%\end{figure}

%The \emph{$K$ nearest neighbor} (KNN) graph is generally used to depict the most adjacent relations between objects,
%in which the linked objects are generally assumed to share common characteristics.
%Given a set of objects $\mathcal{X} = \{\bm{x}_1,\bm{x}_2,...,\bm{x}_n\}$,
%the KNN graph $G_{K} = (V, E)$ is a graph with $V = \mathcal{X}$ and $E = \{e_{ij}\}$.
%%An edge $e_{ij}$ between $\bm{x}_i$ and $\bm{x}_j$ indicates
%%$x_j \in N_K(\bm{x}_i)$ or $x_i \in N_K(\bm{x}_j)$,
%%where $N_K(\bm{x}_i)$ denotes the set of $K$ nearest neighbors of $\bm{x}_i$.
%%at least one of the two end nodes is in the $K$ nearest neighborhood of the other.
%%However, the $K$ nearest neighborhood relation is not symmetric,
%%i.e., $x_j \in N_K(\bm{x}_i)$ is inequivalent to $x_i \in N_K(\bm{x}_j)$,
%Some previous work~\cite{yin2016laplacian} simply considers $e_{ij}$
%exists when $\bmx_j \in N_K(\bmx_i)$ or $x_i \in N_K(\bmx_j)$,
%where $N_K(\bmx_i)$ denotes the set of $K$ nearest neighbors of $\bmx_i$.
%However,
%in the case of data containing multi-scale clusters,
%such definition may lead to an edge between
%two objects in different clusters.
%Fig.~\ref{figure:example1} shows an toy example which consists of two multi-scale clusters
%marked by ``$\times$" and ``$+$" respectively.
%For object $\bmx_1$,
%object $\bmx_2$ is its nearest neighbor.
%But for $\bmx_2$, $\bmx_1$ is far away from it.
%In this case, there will be an edge between $\bmx_1$ and $\bmx_2$ in the traditional KNN graph.
%However, since $\bmx_1$ and $\bmx_2$ belong to different clusters,
%they should not be connected.

Our objective is to capture the high correlations between objects that belong to the same cluster even
 though the objects could be located at distant far ends of a cluster.
 These correlations are expressed via a TKNN graph, which is used to
 regularize the coefficient matrix $Z$.

\begin{definition}
\label{def:nei_relation}
\textbf{(Mutual  $K$-nearest neighbors)}
Let $N_K(x)$ be the set of $K$ nearest neighbors of an object $x$.
Two objects $x_i$ and $x_j$ are said to be mutual $K$-nearest neighbors of each other,
denoted by $x_i \sim x_j$,
iff $x_i \in N_K(x_j)$ and $x_j \in N_K(x_i)$.
\hfill$\Box$
\end{definition}

%In the case of multi-scale data,
%the KNN graph is sometimes insufficient to reflect whether two objects belong to the same cluster or not.
%For example, objects $\bmx_2$ and $\bmx_3$ in Fig.~\ref{figure:example1} belong to the same cluster,
%but they are both far away from each other.
%%objects $x_1$ and $x_2$ are in different clusters despite their closeness.
%To construct an effective KNN graph, $\bmx_2$ and $\bmx_3$
%are expected to be linked.
%Therefore,
%the reachability between two objects is introduced.

\begin{definition}
\label{def:reachability}
\textbf{(Reachability)}
Two objects $x_i$ and $x_j$ are said to be reachable from each other
if there exists a sequence of $h \geq 2$ objects
 $\{x_i = x_{a_1}, \ldots, x_{a_h} = x_j\}$ such that
 $x_{a_r} \sim x_{a_{r+1}}$ for $1 \leq r < h$.
 \hfill$\Box$
\end{definition}

\begin{definition}
\label{def:trans_relation}
\textbf{(Transitive $K$-nearest neighbor (TKNN) graph)}
Given a set of objects $\mathcal{X} = \{x_1, x_2,..., x_n\}$,
the TKNN graph $\mathcal{G}_K = (\mathcal{X},\mathcal{E})$
is an undirected graph
where $\mathcal{X}$ is the set of vertices and $\mathcal{E}$ is the set of edges.
Specifically, the edge ($x_i$, $x_j$) $\in \mathcal{E}$
iff $x_i$ and $x_j$ are reachable from each other.
We represent the TKNN graph by an $n \times n$ {\it reachability matrix}
$\mathcal{W}$
whose ($i$,$j$)-entry $\mathcal{W}_{ij} = 1$ if ($x_i$, $x_j$) $\in \mathcal{E}$; 0 otherwise.
\hfill$\Box$
\end{definition}

%We note that mutual-KNN is a symmetric relation and so is reachability.

%Compared with the traditional KNN graph,
%the TKNN graph can more accurately reflect
%whether two objects belong to the same cluster or not.
%It will be used in the next section to rectify the matrix of similarity between objects.

\subsection*{Coefficient Matrix}
%In the case of multi-scale data,
%the distance-based similarity is likely to be ineffective.
%To improve the clustering performance,
%it is necessary to rectify the matrix.
%The rectification should follow two principles.
%First, for the effective similarities, the efficacy should be reserved.
%Second, for the ineffective ones, the efficacy should be enhanced.
%For example,
%the closeness between objects $\bmx_3$ and $\bmx_4$ in Fig.~\ref{figure:example1}
%leads to a large similarity value.
%Since the two objects are of the same cluster,
%the new value should still be large.
%In comparison,
%both the similarity between $\bmx_1$ and $\bmx_2$ and the one between $\bmx_2$ and $\bmx_3$
%should be rectified, as these distance-based similarities are ineffective.

%In the multi-scale data,
%different clusters may have different sizes, densities or even geometric structures.
The similarity between two objects describes the degree to which they share common characteristics.
The more similar they are,
the more likely that one object can be represented by the other.
Therefore,
to depict the relationships between objects in multi-scale data,
each object is linearly characterized by other objects,
assuming the well-known linear subspace model~\cite{liu2013robust}.

We generate $p$ {\pev}s using PI.
%just as a non-linear manifold can be locally approximated by linear subspaces.
%First, these vectors, which retain the effects of dominant eigenvectors,
%can certainly retain the original accurate similarities between objects.
%Second, they absorb the cluster-separation information from the seemingly ``less important'' eigenvectors,
%which further provides a reliable source of information to rectify the erroneous similarities.
Let $X\in \mathcal{R}^{p\times n}$ be a matrix whose rows are the {\pev}s.
The $q$-th column of $X$ can be taken as a feature vector $\bmx_q$ of an object $x_q$.
We normalize the column vectors of $X$ such that $\bmx_q^T \bmx_q = 1 \; \forall 1 \leq q \leq n$.
We determine a coefficient matrix $Z \in \mathcal{R}^{n \times n}$ by\footnote{We will regularize $Z$ to avoid the trivial solution of $Z$ being the identity matrix.}
\begin{equation}
\label{eq:nonoise}
X = XZ.
\end{equation}
One can interpret $Z_{ij}$
as a value that reflects how well object $x_i$ characterizes object $x_j$.
%To avoid the trivial solution of $Z$ equalling the identity matrix,
%we will regularize $Z$ later.
%For simplicity, we do not explicitly mention this constraint in the rest of the paper.

As indicated by previous works~\cite{ye2016fuse}, the generated pseudo-eigenvectors are likely to be noise-corrupted.
We thus extend Equation~\ref{eq:nonoise} by introducing a noise matrix $O$, giving:
%However, although the pseudo-eigenvectors inherit useful cluster-separation information,
%they may also be contaminated by noise.
%Thus a matrix $O$ is introduced to capture the noise,
\begin{equation}
X = XZ + O.
\label{eq:O}
\end{equation}
%The TKNN graph conveys information on whether two objects belong to the same cluster or not,

As we have explained, the TKNN graph  conveys useful clustering information, bringing highly
correlated objects that are located at distant far ends of a cluster together.
We thus use the TKNN graph to regularize matrix $Z$.
We derive the following objective function:
%Since our objective is to derive a $Z$ which can better reflect the true similarities between objects,
\comment{
\begin{equation}
%\label{eq:obj}
\min_Z ||O||_F^2+\lambda_1||Z||_F^2 + \lambda_2||Z-\mathcal{W}_K||_F^2,
s.t., X = XZ+O,
\end{equation}
}
%By substituting $O$, we derive
\begin{equation}
\label{eq:obj_constraint}
\min_Z ||X-XZ||_F^2 + \alpha_1 ||Z||_F^2 + \alpha_2 ||Z-\mathcal{W}||_F^2,
\end{equation}
where $\alpha_1 > 0,\alpha_2 \geq 0$ are two
weighting factors that adjust the relative weights of the three components that constitute
the objective function.
The objective function consists of three terms.
The first term aims to reduce the noise matrix $O$ (see Equation~\ref{eq:O}),
the second term is the Frobenius norm on $Z$, and
%that restricts any item in $Z$ from being dominant.
the third term regularizes $Z$ by the TKNN graph.
%For simplicity, as in~\cite{lu2012robust},
%we remove the constraint and formulate the problem as
%\begin{equation}
%\label{eq:obj}
%\min_Z ||X-XZ||_F^2 + \lambda_1 ||Z||_F^2 + \lambda_2 ||Z-\mathcal{W}_K||_F^2.
%\end{equation}
%$\lambda_1$ and $\lambda_2$ are two parameters to balance the effects of three parts.
A closed-form solution, $Z^*$, to the optimization problem is
\begin{equation}
\label{eq:solution}
Z^* = (X^TX + \alpha_1I + \alpha_2I)^{-1}(X^TX+\alpha_2\mathcal{W}).
\end{equation}

\subsection*{Grouping Effect}
For an object $x_p$,
let $\bmz_p$ be the $p$-th column of the coefficient matrix $Z$.
We interpret the entries of $\bmz_p$ as the coefficients that express $x_p$ as a linear combinations
of other objects.
Previous works~\cite{lu2012robust,lu2013correlation,hu2014smooth} have shown that if $Z$ has {\it grouping effect}, then performing spectral
clustering based on $Z$ would be effective.
Intuitively, $Z$ has grouping effect if, given two {\it highly correlated} objects $x_i$ and $x_j$,
their characterizations of other objects are similar.
Existing works mostly consider {\it high correlation} between objects as {\it high similarity} of their
feature vectors. With ROSC, we consider object similarity in terms of both feature similarity and
reachability similarity.
Feature similarity is measured by the objects' feature vectors as given by the columns of matrix $X$.
Reachability similarity is measured by the columns of matrix $\mathcal{W}$,
each of which shows the reachability of an object to all others.
Formally,

%It has been shown that the first two terms in the objective function decide the \emph{grouping effect} of $Z$~\cite{lu2012robust}.
%In Problem~\ref{eq:obj}, the first two terms decides whether $Z$ has the grouping effect~\cite{lu2012robust} defined as follows.
%i.e., the more highly correlated two objects are,
%the larger weight one will have in characterizing the other~\cite{lu2012robust}.

\begin{definition}
\label{def:grouping}
\textbf{(Grouping effect)}.
Given a set of objects $\mathcal{X} = \{x_1, x_2,..., x_n\}$,
let $\bmw_q$ be the $q$-th column of $\mathcal{W}$.
Further, let $\xarrow{i}{j}$ denote the condition:
(1) $\bmx_i^T \bmx_j \rightarrow 1$ and
(2) $\lVert \bmw_i - \bmw_j \rVert_2 \rightarrow 0$.
A matrix $Z$ is said to have grouping effect
if
\[
\xarrow{i}{j} \Rightarrow |Z_{ip} - Z_{jp}| \rightarrow 0\; \forall 1 \leq p \leq n.
\]
%for an arbitrary object $\bmx$ and its associated coefficient vector $\bm{z}$,
%If $\bmx_i\rightarrow \bmx_j$,
%then $z_{pi}\rightarrow z_{pj}$,
%where $z_{pi}$ and $z_{pj}$ are the $i$th and $j$th entry in $\bm{z}_p$ respectively.
%Then we say the self-representation matrix $Z$
%has the grouping effect.
%\hfill$\Box$
\end{definition}

Our next task is to prove that the optimal solution $Z^*$ (as given in Equation~\ref{eq:solution})
% of Equation~\ref{eq:obj_constraint}
has the grouping effect.
In the following discussion, we use $\bmz_q^*$ to denote the $q$-th column vector of $Z^*$.

\begin{lemma}
\label{lemma1}
Given a set of objects $\mathcal{X}$,
the matrix
$X\in \mathcal{R}^{p\times n}$ that is composed of the {\pev}s as rows,
 the reachability matrix $\mathcal{W}$,
 and the optimal soution $Z^*$ of Equation~\ref{eq:obj_constraint},
\begin{equation}
\label{eq:zi}
Z_{ip}^* = \frac{\bm{x}_i^T(\bm{x}_p-X\bm{z}_p^*) + \alpha_2 \mathcal{W}_{ip}}{\alpha_1+\alpha_2}, \;\;\; \forall 1 \leq i, p \leq n.
\end{equation}
\end{lemma}

\begin{proof}
For $1 \leq p \leq n$,
let $J(\bm{z}_p) =  ||\bm{x}_p-X\bm{z}_p||_2^2 + \alpha_1 ||\bm{z}_p||_2^2 + \alpha_2 ||\bm{z}_p-\bm{w}_p||_2^2$.
Since $Z^*$ is the optimal solution of Equation~\ref{eq:obj_constraint}, we have $\frac{\partial{J}}{\partial{Z}_{ip}}|_{\bm{z}_p = \bm{z}_p^*} = 0\; \forall 1\leq i \leq n$.
Hence, $-2\bm{x}_i^T(\bm{x}_p-X\bm{z}_p^*)+2\alpha_1Z_{ip}^*+2\alpha_2(Z_{ip}^*-\mathcal{W}_{ip}) = 0$,
which induces Equation~\ref{eq:zi}.
\end{proof}

%\begin{lemma}
%\label{lemma1}
%Given a set of objects $X\in \mathcal{R}^{d\times n}$ and a TKNN graph $\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{W}_K)$,
%let $\bm{x}_p$ be an arbitrary object and $\bm{w}_p$ be the column vector in $\mathcal{W}_K$ describing the connectivity of $\bmx_p$.
%Each item $z_{pi}^*$ in the optimal solution $\bm{z}_p^*$ to the problem
%$\min_{\bm{z}} ||\bm{x}_p-X\bm{z}_p||_2^2 + \lambda_1 ||\bm{z}_p||_2^2 + \lambda_2 ||\bm{z}_p-\bm{w}_p||_2^2$ is
%\begin{equation}
%\label{eq:zi}
%z_{pi}^* = \frac{\bm{x}_i^T(\bm{x}_p-X\bm{z}_p^*) + \lambda_2 w_{pi}}{\lambda_1+\lambda_2},
%\end{equation}
%where $w_{pi} = 1$, if $\bm{x_p}$ and $\bm{x}_i$ are reachable; $0$, otherwise.
%
%Proof. Let $J(\bm{z}_p) =  ||\bm{x}_p-X\bm{z}_p||_2^2 + \lambda_1 ||\bm{z}_p||_2^2 + \lambda_2 ||\bm{z}_p-\bm{w}_p||_2^2$.
%Since $\bm{z}_p^*$ is the optimal solution, then $\frac{\partial{J}}{\partial{z}_{pi}}|_{\bm{z}_p = \bm{z}_p^*} = 0$.
%Thus we have $-2\bm{x}_i^T(\bm{x}_p-X\bm{z}_p^*)+2\lambda_1z_{pi}^*+2\lambda_2(z_{pi}^*-w_{pi}) = 0$,
%which induces Eq.~\ref{eq:zi}.
%\end{lemma}

\comment{
Eq.~\ref{eq:zi} calculates the weight of object $\bm{x}_i$ in characterizing $\bm{x}$.
For $w_i$, if $\bm{x}$ and $\bm{x}_i$ are reachable in the TKNN graph, $w_i = 1$; otherwise, $w_i = 0$.
Eq.~\ref{eq:zi} rectifies the ineffective similarity as follows.
When $\bm{x}$ and $\bm{x_i}$ are far away from each other in the feature space
but reachable in the TKNN graph,
$z_i$ will be increased by $w_i = 1$.
%which rectifies the original incorrect similarity value between them.
Moreover, if $\bm{x}$ and $\bm{x}_i$ are highly correlated,
the first two terms in the problem can decide a large $z_i$,
even in the case that they are unreachable.
%$z_i$ can still keep large with a small $\lambda_2$.
In summary, Eq.~\ref{eq:zi} can not only hold the original accurate similarities but also rectify the incorrect ones.

\noindent{\small$\bullet$}
For any two objects dissimilar in the feature space but connected in the TKNN graph,
their similarity will be increased.

\noindent{\small$\bullet$}
For any two objects dissimilar in the feature space and disconnected in the TKNN graph,
their similarity will be further decreased.

\noindent{\small$\bullet$}
For any two objects similar in the feature space and connected in the TKNN graph,
their similarity will be further increased.

\noindent{\small$\bullet$}
For any two objects similar in the feature space but disconnected in the TKNN graph,
their similarity will be decreased. In this case,
provided that the raw similarity is effective already,
even though the new value will be decreased,
but it will be still
}

\begin{lemma}
\label{lemma2}
%Given a set of objects $X\in R^{d\times n}$ and a TKNN graph $\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{W}_K)$,
%let $\bm{x}_p$ be an arbitrary object,
%$\bm{w}_p$ be the column vector in $\mathcal{W}_K$ describing the connectivity of $\bm{x}_p$, and
%$\bm{z}_p^*$ be the optimal solution to the problem
%$\min_{\bm{z}_p} ||\bm{x}_p-X\bm{z}_p||_2^2 + \lambda_1 ||\bm{z}_p||_2^2 + \lambda_2 ||\bm{z}_p-\bm{w}_p||_2^2$.
%Assume all the objects have been normalized.
%For any two objects $\bm{x}_i$ and $\bm{x}_j$,
$\forall 1 \leq i, j, p \leq n$,
\begin{equation}
\label{eq:norm}
|Z_{ip}^*-Z_{jp}^*| \leq \frac{c\sqrt{2(1-r)} + \alpha_2|\mathcal{W}_{ip}-\mathcal{W}_{jp}|}{\alpha_1+\alpha_2},
\end{equation}
where $c = \sqrt{1+\alpha_2||\bm{w}_p||_2^2}$ and $r = \bm{x}_i^T\bm{x}_j$.
\end{lemma}

\begin{proof}
From Equation~\ref{eq:zi}, we have
\begin{equation}
\nonumber
Z_{ip}^*-Z_{jp}^* = \frac{(\bm{x}_i^T - \bm{x}_j^T)(\bm{x}_p-X\bm{z}_p^*) + \alpha_2 (\mathcal{W}_{ip}-\mathcal{W}_{jp})}{\alpha_1+\alpha_2}.
\end{equation}
That implies
\begin{small}
\begin{equation}
\label{eq:zizj}
\begin{split}
|Z_{ip}^*-Z_{jp}^*| & \leq \frac{|(\bm{x}_i^T - \bm{x}_j^T)(\bm{x}_p-X\bm{z}_p^*)| + \alpha_2 |\mathcal{W}_{ip}-\mathcal{W}_{jp}|}{\alpha_1+\alpha_2}\\
& \leq \frac{||\bm{x}_i - \bm{x}_j||_2||\bm{x}_p-X\bm{z}_p^*||_2 + \alpha_2 |\mathcal{W}_{ip}-\mathcal{W}_{jp}|}{\alpha_1+\alpha_2}\\
\end{split}
\end{equation}
\end{small}

Since the column vectors of $X$ are normalized (i.e., $\bmx_q^T \bmx_q = 1 \; \forall 1 \leq q \leq n$) , we have
$||\bm{x}_i - \bm{x}_j||_2 = \sqrt{2(1-r)}$,
where $r = \bm{x}_i^T\bm{x}_j$.
%measuring the closeness between $\bm{x}_i$ and $\bm{x}_j$ in the feature space.
%As $\bm{z}_p^*$ is the optimal solution,
As $Z^*$ is the optimal solution of Equation~\ref{eq:obj_constraint}, we have
\begin{equation}
\begin{split}
J(\bm{z}_p^*) & = ||\bm{x}_p-X\bm{z}_p^*||_2^2 + \alpha_1 ||\bm{z}_p^*||_2^2 + \alpha_2 ||\bm{z}_p^*-\bm{w}_p||_2^2 \leq  \\
J(\bm{0}) & = ||\bm{x}_p||_2^2 + \alpha_2 ||\bm{w}_p||_2^2 = 1 + \alpha_2 ||\bm{w}_p||_2^2.
\end{split}
\end{equation}
Hence, $||\bm{x}_p-X\bm{z}_p^*||_2 \leq \sqrt{1 + \alpha_2 ||\bm{w}_p||_2^2} = c$.
Equation~\ref{eq:zizj} can be further simplified as
\begin{equation}
\nonumber
|Z_{ip}^*-Z_{jp}^*| \leq \frac{c\sqrt{2(1-r)}+ \alpha_2 |\mathcal{W}_{ip}-\mathcal{W}_{jp}|}{\alpha_1+\alpha_2}.
\end{equation}
\end{proof}

\begin{lemma}
Matrix $Z^*$ has grouping effect.
\label{lemma:z-star}
\end{lemma}
\begin{proof}
Given two objects $x_i$ and $x_j$ such that $\xarrow{i}{j}$,
we have,  %by definition,
(1) $\bmx_i^T \bmx_j \rightarrow 1$ and (2) $||\bmw_{i}-\bmw_{j}||_2 \rightarrow 0$.
These imply
$r = \bmx_i^T \bmx_j \rightarrow 1$ and  $|\mathcal{W}_{ip}-\mathcal{W}_{jp}| \rightarrow 0$.
Hence, the two terms of the numerator of the R.H.S of Equation~\ref{eq:norm} are close to 0.
Therefore, $|Z_{ip}^*-Z_{jp}^*| \rightarrow 0$ and thus $Z^*$ has grouping effect.
\end{proof}

Indeed, Equation~\ref{eq:norm} shows how our algorithm ROSC enhances the effectiveness of
spectral clustering on multi-scale data.
Comparing with traditional approaches, which focus on feature similarity, ROSC uses $Z^*$
to integrate feature similarity  ($r$) with
reachability similarity ($|\mathcal{W}_{ip}-\mathcal{W}_{jp}|$).
In particular, two distant objects $x_i$ and $x_j$ of a cluster may not share a strong feature similarity.
This leads to a small $r$ and traditional approaches will likely put them into separate clusters.
On the contrary, ROSC considers the strong reachability of the objects
to derive a small value of $|\mathcal{W}_{ip}-\mathcal{W}_{jp}|$, and thus keeping them in the same cluster.
Moreover, for $x_i$ and $x_j$ that belong to two different dense clusters but happen to be close in
space (i.e., $x_i$ and $x_j$ have strong feature similarity),
traditional approaches may inadvertently merge them into the same cluster.
ROSC, however, would discover their low reachability (via the mutual-KNN relation)
and derive a large value of $|\mathcal{W}_{ip}-\mathcal{W}_{jp}|$.
This regulates matrix $Z^*$ and avoids the incorrect merging.
As we will see in the next section, ROSC's approach
greatly improves clustering quality and is more robust than other algorithms in handling
multi-scale data.

%Lemma~\ref{lemma1} and~\ref{lemma2} prove the grouping effect of $Z^*$ derived by Eq.~\ref{eq:solution}.
%%the weight difference of two objects in characterizing an arbitrary object $\bm{x}$ is bounded.
%Given two objects $\bmx_i$ and $\bmx_j$, if $\bmx_i \rightarrow \bmx_j$,
%i.e., $\bmx_i^T \bmx_j \rightarrow 1$ and $||w_{pi}-w_{pj}||_2 \rightarrow 0$,
%then $||z_{pi}^*-z_{pj}^*||_2 \rightarrow 0$.
%%which shows the grouping effect of $Z$ derived by Eq.~\ref{eq:solution}.
%%However, in the case of (1) $\bm{x}_i^T\bm{x}_j = 0$ but $w_i=w_j=1$ and
%%(2) $\bm{x}_i^T\bm{x}_j = 1$ but $||w_i-w_j||_2=1$,
%%$||z_i^*-z_j^*||_2$ can still be small with an appropriate setting on $\lambda_1$ and $\lambda_2$.
%%Further, since $Z$ can not only retain the accurate similarities but also rectify the incorrect ones,
%%In other words, objects in the same cluster will be better reflected in the rectified matrix.
%%Such grouping effect together with The rectification ability could guarantee a more robust clustering performance.
%We then summarize how the TKNN graph rectifies the raw similarity matrix.
%Given any two objects $\bmx_i$ and $\bmx_j$,
%\begin{enumerate}[label=(\arabic*), leftmargin=*]
%\setlength{\itemsep}{0pt}
%\setlength{\parsep}{0pt}
%\setlength{\parskip}{0pt}
%\item $\bmx_i^T \bmx_j \rightarrow 0$, $\bmx_i$ and $\bmx_j$ are reachable, the effect of $S_{ij}$ will be increased.
%\item $\bmx_i^T \bmx_j \rightarrow 0$, $\bmx_i$ and $\bmx_j$ are unreachable, the effect of $S_{ij}$ will be further decreased.
%\item $\bmx_i^T \bmx_j \rightarrow 1$, $\bmx_i$ and $\bmx_j$ are reachable, the effect of $S_{ij}$ will be further increased.
%\item $\bmx_i^T \bmx_j \rightarrow 1$, $\bmx_i$ and $\bmx_j$ are unreachable, the effect of $S_{ij}$ will be decreased.
%\end{enumerate}
%Among the four cases,
%$Z$ reserves the efficacy of effective similarities by (2) and (3)
%while it enhances the efficacy of ineffective similarities by (1) and (4).
%Thus spectral clustering based on $Z$ will be more robust.

\comment{
\noindent{\small$\bullet$}
$\bmx_i^T \bmx_j \rightarrow 0$ but reachable, $S_{ij}$ will be increased.

\noindent{\small$\bullet$}
%For any two objects dissimilar in the feature space and disconnected in the TKNN graph, their similarity will be further decreased.
$\bmx_i^T \bmx_j \rightarrow 0$ and unreachable, $S_{ij}$ will be further decreased.

\noindent{\small$\bullet$}
%For any two objects similar in the feature space and connected in the TKNN graph, their similarity will be further increased.
$\bmx_i^T \bmx_j \rightarrow 1$ and reachable, $S_{ij}$ will be further increased.

\noindent{\small$\bullet$}
$\bmx_i^T \bmx_j \rightarrow 1$ but unreachable, $S_{ij}$ will be decreased.
}

\subsection*{ROSC: Robust Spectral Clustering}
%The coefficient matrix $Z$ take advantage of both cluster-separation information in all the eigenvectors
%and the TkNN graph, which not only retains the original accurate similarities but also rectifies the incorrect ones.

We note that the matrix $Z^*$ obtained may be asymmetric and it may contain negative values.
To construct a matrix of object similarity, a common fix~\cite{liu2013robust,lu2012robust} is to
compute $\tilde{Z} = (|Z^*|+|(Z^*)^T|)/2$.
After $\tilde{Z}$ is computed, ROSC executes a standard spectral clustering method
(e.g., NCuts) using $\tilde{Z}$ as the similarity matrix in place of $S$.
It can be proved that $|Z^*|$, $|(Z^*)^T|$, and hence $\tilde{Z}$ all have grouping effect.
%Due to space limitations, readers are referred to [XXX] for the proofs.
Finally, ROSC is summarized in Algorithm~\ref{alg}.

\begin{lemma}
Matrix $|Z^*|$ has grouping effect.
\label{lemma:z-star-abs}
\end{lemma}
\begin{proof}
Given two objects $x_i$ and $x_j$,
we have
$\bigl||Z_{ip}^*|-|Z_{jp}^*|\bigr| \leq |Z_{ip}^*-Z_{jp}^*|$.
From Lemma~\ref{lemma:z-star},
$Z^*$ has grouping effect, i.e.,
if $x_i \rightarrow x_j$, $|Z_{ip}^*-Z_{jp}^*| \rightarrow 0$.
Thus $\bigl||Z_{ip}^*|-|Z_{jp}^*|\bigr| \rightarrow 0$ and
$|Z^*|$ has grouping effect.
\end{proof}

\begin{lemma}
\label{lemma3}
$(Z^*)^T$ and $|(Z^*)^T|$ have grouping effect.
\begin{proof}
The problem is equivalent to if $x_i \rightarrow x_j$, $Z^*_{pi} \rightarrow Z^*_{pj}$.
From Lemma~\ref{lemma1},
\begin{equation}
\nonumber
Z_{pi}^*-Z_{pj}^* = \frac{\bm{x}_p^T ( \bmx_i - \bmx_j - X(\bm{z}_i^*-\bm{z}_j^*)) + \alpha_2 (\mathcal{W}_{pi}-\mathcal{W}_{pj})}{\alpha_1+\alpha_2}.
\end{equation}
Since $\bm{z}_i^* = (X^TX + \alpha_1I + \alpha_2I)^{-1}(X^T\bmx_i+\alpha_2\bm{w}_i),\bm{z}_j^* = (X^TX + \alpha_1I + \alpha_2I)^{-1}(X^T\bmx_j+\alpha_2\bm{w}_j)$,
let $Y = X(X^TX + \alpha_1I + \alpha_2I)^{-1}$. Then
\begin{equation}
\begin{split}
|Z_{pi}^*-Z_{pj}^*| & \leq \frac{|\bmx_p^T(\bm{x}_i - \bm{x}_j)| + |\bmx_p^TX(\bm{z}_i^*-\bm{z}_j^*)| + \alpha_2 |(\mathcal{W}_{pi}-\mathcal{W}_{pj})|}{\alpha_1+\alpha_2}\\
& \leq \frac{|\bmx_p^T(\bm{x}_i - \bm{x}_j)| + |\bmx_p^TYX^T(\bmx_i-\bmx_j)|}{\alpha_1+\alpha_2}\\
& + \frac{\alpha_2|\bmx_p^TY(\bm{w}_i - \bm{w}_j)| + \alpha_2 |\mathcal{W}_{pi}-\mathcal{W}_{pj}|}{\alpha_1+\alpha_2}\\
& \leq \frac{||\bmx_p||_2||\bm{x}_i - \bm{x}_j||_2 + ||\bmx_p^TYX^T||_2||\bmx_i-\bmx_j||_2}{\alpha_1+\alpha_2}\\
& + \frac{\alpha_2||\bmx_p^TY||_2||\bm{w}_i - \bm{w}_j||_2 + \alpha_2 |\mathcal{W}_{pi}-\mathcal{W}_{pj}|}{\alpha_1+\alpha_2}\\
\end{split}
\end{equation}
If $x_i \rightarrow x_j$, i.e., $\bmx_i^T\bmx_j \rightarrow 1$ and $||\bm{w}_i - \bm{w}_j||_2 \rightarrow 0$,
we have
$||\bm{x}_i - \bm{x}_j||_2 \rightarrow 0$ and $|\mathcal{W}_{pi}-\mathcal{W}_{pj}| \rightarrow 0$.
Then $|Z_{pi}^*-Z_{pj}^*| \rightarrow 0$ and
thus $(Z^*)^T$ has grouping effect.
Since $\bigl||Z_{pi}^*|-|Z_{pj}^*|\bigr| \leq |Z_{pi}^*-Z_{pj}^*|$, $|(Z^*)^T|$ also has grouping effect.
\end{proof}
\end{lemma}

\begin{lemma}
\label{lemma4}
Matrix $\tilde{Z}$ has grouping effect.
\begin{proof}
From Lemma~\ref{lemma:z-star-abs} and~\ref{lemma3},
both $\lvert Z^*\rvert$ and $\lvert (Z^*)^T\rvert$ have the grouping effect.
Since $\tilde{Z} = (|Z^*|+|(Z^*)^T|)/2$,
\begin{equation}
\begin{split}
\lvert \tilde{Z}_{ip} - \tilde{Z}_{jp}\rvert & = \frac{\bigl| (|Z^*_{ip}|+|Z^*_{pi}|) - (|Z^*_{jp}|+|Z^*_{pj}|)\bigr|}{2} \\
& \leq \frac{\bigl||Z_{ip}^*|-|Z_{jp}^*|\bigr| + \bigl||Z_{pi}^*|-|Z_{pj}^*|\bigr|}{2}
\end{split}
\end{equation}
If $x_i \rightarrow x_j$,
both $\bigl||Z_{ip}^*|-|Z_{jp}^*|\bigr| \rightarrow 0$ and $\bigl||Z_{pi}^*|-|Z_{pj}^*|\bigr| \rightarrow 0$,
so $\lvert \tilde{Z}_{ip} - \tilde{Z}_{jp}\rvert \rightarrow 0$ and
$\tilde{Z}$ has grouping effect.
\end{proof}
\end{lemma}

\comment{
\begin{lemma}
\label{lemma3}
$(Z^*)^T$ has the grouping effect.

Proof. The problem is equivalent to if $x_i \rightarrow x_j$, $Z^*_{pi} \rightarrow Z^*_{pj}$.
From Lemma~\ref{lemma1},
\begin{equation}
\nonumber
z_{pi}^*-z_{pj}^* = \frac{\bm{x}_p^T ( \bmx_i - \bmx_j - X(\bm{z}_i^*-\bm{z}_j^*)) + \alpha_2 (W_{pi}-W_{pj})}{\alpha_1+\alpha_2}.
\end{equation}
Since $\bm{z}_i^* = (X^TX + \alpha_1I + \alpha_2I)^{-1}(X^T\bmx_i+\alpha_2\bm{w}_i),\bm{z}_j^* = (X^TX + \alpha_1I + \alpha_2I)^{-1}(X^T\bmx_j+\alpha_2\bm{w}_j)$,
let $Y = X(X^TX + \alpha_1I + \alpha_2I)^{-1}$. Then
\begin{equation}
\begin{split}
||Z_{pi}^*-Z_{pj}^*||_2 & \leq \frac{||\bmx_p^T(\bm{x}_i - \bm{x}_j)||_2 + ||\bmx_p^TX(\bm{z}_i^*-\bm{z}_j^*)||_2 + \alpha_2 ||(W_{pi}-W_{pj})||_2}{\alpha_1+\alpha_2}\\
& \leq \frac{||\bmx_p^T(\bm{x}_i - \bm{x}_j)||_2 + ||\bmx_p^TYX^T(\bmx_i-\bmx_j)||_2}{\alpha_1+\alpha_2}\\
& + \frac{\alpha_2||\bmx_p^TY(\bm{w}_i - \bm{w}_j)||_2 + \alpha_2 ||W_{pi}-W_{pj}||_2}{\alpha_1+\alpha_2}\\
\end{split}
\end{equation}
If $x_i \rightarrow x_j$, i.e., $\bmx_i^T\bmx_j \rightarrow 1$ and $||\bm{w}_i - \bm{w}_j||_2 \rightarrow 0$,
we have
$||\bm{x}_i - \bm{x}_j||_2 \rightarrow 0$ and $||W_{pi}-W_{pj}||_2 \rightarrow 0$.
Then $||Z_{pi}^*-Z_{pj}^*||_2 \rightarrow 0$,
which proves the grouping effect of $(Z^*)^T$.
%\begin{equation}
%\begin{split}
%||\bmx_p^TX(\bm{z}_i^*-\bm{z}_j^*)||_2 & \leq ||\bmx_p^TY(X^T(\bmx_i-\bmx_j) +\alpha_2(\bm{w}_i - \bm{w}_j))||_2\\
%& \leq ||\bmx_p^TYX^T(\bmx_i-\bmx_j)||_2 +\alpha_2||\bmx_p^TY(\bm{w}_i - \bm{w}_j)||_2\\
%\end{split}
%\end{equation}
\end{lemma}

\begin{lemma}
\label{lemma4}
$\tilde{Z}$ has the grouping effect.

Proof. From Lemma~\ref{lemma2} and~\ref{lemma3},
both $Z^*$ and $(Z^*)^T$ have the grouping effect,
which can be easily extended to
$|Z^*|$ and $|(Z^*)^T|$.
Since $\tilde{Z} = (|Z^*|+|(Z^*)^T|)/2$,
$\tilde{Z}$ also has the grouping effect.
\end{lemma}

To this end, we summarize ROSC as follows.
ROSC first computes the TKNN graph $\mathcal{G}$
and the associated weight matrix $\mathcal{W}_K$. Then it applies power iteration to generate $p$
pseudo-eigenvectors that form a $p \times n$ matrix $X$.
After whitening and normalization on $X$,
the coefficient matrix $Z^*$ can be calculated by solving Eq.~\ref{eq:solution},
which further leads to the construction of a rectified similarity matrix $\tilde{Z}$.
Since $\tilde{Z}$ has the grouping effect,
the standard spectral clustering methods (e.g., NCuts) will be finally applied to derive more robust clustering results.
The time complexity of ROSC will be no more than the standard spectral clustering methods,
which is $O(n^3)$ in general.
In the future, we will attempt to improve it.
}

\begin{algorithm}
\begin{small}
\caption{ROSC}
\label{alg}
\begin{algorithmic}[1]
\Require $S$, $k$.
\Ensure $\mathcal{C} = \{C_1, ..., C_k\}$
\State Compute the TKNN graph and the reachability matrix $\mathcal{W}$
\State Calculate $W = D^{-1}S$, where $D_{ii} = \sum_jS_{ij}$
%\For $j \leftarrow 1$ do
%\State$t = 0$
%\Repeat
%\State $v_j^{t+1} \leftarrow \frac{Wv_j^t}{||Wv_j^t||_1}$
%\State $ \delta^{t+1} \leftarrow |v_j^{t+1} - v_j^t|$
%\State $t$++
%\Until $||\delta_j^t+1 - \delta_j^t||_{max} \leq \epsilon$ or $t\geq T$
%\EndFor
\State Apply PI on $W$ and generate $p$ pseudo-eigenvectors $\{\bm{v}_r\}_{r=1}^p$
\State $X = \{\bm{v}_1^T; \bm{v}_2^T; ...; \bm{v}_p^T\}$; $X$ = whiten($X$)
\State Normalize each column vector $\bm{x}$ of $X$ such that $\bm{x}^T\bm{x} = 1$
\State Calculate the coefficient matrix $Z^*$ by Eq.~\ref{eq:solution}
\State Construct $\tilde{Z} = (|Z^*| + |(Z^*)^T|)/2$
\State Run standard spectral clustering methods, e.g., NCuts, with $\tilde{Z}$ as the
similarity matrix to obtain clusters $\mathcal{C} = \{C_r\}_{r=1}^k$
%\State Decode $\{C_r\}_{r=1}^k$ from $\{{\bm z_r}\}_{r=1}^k$
\State \Return $\mathcal{C} = \{C_1, ..., C_k\}$
\end{algorithmic}
\end{small}
\end{algorithm}