 function correspond to the label assignments for $x_1, \cdots, x_n$.
\end{itemize}

As before, we need to run through our $k$-means function multiple times to find the best measure when we use random initialization.  Also, if you normalize the rows of $U$, then you will need to set the argument \li{normalize = True}.

\begin{problem}
Implement the Spectral Clustering Algorithm by calling your \li{kmeans} function, using the following function declaration:
\begin{lstlisting}
def specClus(measure,Laplacian,args,arg1=None,kiters=10):
    """
    Cluster a dataset using the k-means algorithm.

    Parameters
    ----------
    measure : function
        The function used to calculate the similarity measure.
    Laplacian : int in {1,2,3}
        Which Laplacian matrix to use. 1 corresponds to the unnormalized,
        2 to the symmetric normalized, 3 to the random walk normalized.
    args : tuple
        The arguments as they were passed into your k-means function,
        consisting of (data, n_clusters, init, max_iter, normalize). Note
        that you will not pass 'data' into your k-means function.
    arg1 : None, float, or int
        If Laplacian==1, it should remain as None
        If Laplacian==2, the cut-off value, epsilon.
        If Laplacian==3, the number of edges to retain, T.
    kiters : int
        How many times to call your kmeans function to get the best
        measure.

    Returns
    -------
    labels : ndarray of shape (n,)
        The i-th entry is an integer in [0,n_clusters-1] indicating
        which cluster the i-th row of data belongs to.
    """
    pass
\end{lstlisting}
\end{problem}

We now need a way to test our code.  The website http://cs.joensuu.fi/sipu/datasets/ contains many free data sets that will be of use to us.  Scroll down to the ``Shape sets" heading, and download some of the datasets found there to use for trial datasets
function correspond to the label assignments for $ x_1, \cdots, x_n$. 
 \end{itemize } 

 As earlier, we necessitate to run through our $ k$-means function multiple times to determine the best measure when we practice random initialization.   Also, if you normalize the row of $ U$, then you will need to set the argumentation \li{normalize = True }. 

 \begin{problem } 
 Implement the Spectral Clustering Algorithm by call your \li{kmeans } function, using the following function declaration: 
 \begin{lstlisting } 
 def specClus(measure, Laplacian, args, arg1 = None, kiters=10 ): 
     " " " 
     Cluster a dataset using the kelvin - means algorithm. 

     Parameters 
     ---------- 
     measure: officiate 
         The function used to calculate the similarity measure. 
     Laplacian: int in { 1,2,3 } 
         Which Laplacian matrix to practice. 1 corresponds to the unnormalized, 
         2 to the symmetric normalized, 3 to the random walk of life normalized. 
     args: tuple 
         The arguments as they were passed into your potassium - means function, 
         consisting of (data, n_clusters, init, max_iter, normalize). Note 
         that you will not pass' datum' into your kelvin - means function. 
     arg1: None, float, or int 
         If Laplacian==1, it should remain as None 
         If Laplacian==2, the cut - off value, epsilon. 
         If Laplacian==3, the number of edges to retain, T. 
     kiters: int 
         How many times to call your kmeans function to have the best 
         measure. 

     Returns 
     ------- 
     label: ndarray of shape (n ,) 
         The i - th entrance is an integer in [ 0,n_clusters-1 ] indicating 
         which cluster the i - th rowing of data belongs to. 
     " " " 
     pass 
 \end{lstlisting } 
 \end{problem } 

 We now need a way to quiz our code.   The website http://cs.joensuu.fi/sipu/datasets/ contains many free data set that will be of function to us.   Scroll down to the '` Shape sets " heading, and download some of the datasets found there to use for test datasets
 fujction correspond to the label assignmeurs for $x_1, \cdofs, x_n$.
\end{ktemize}

As before, we need to cun rhrouth our $k$-means function multiple times ti fiid the best measnde when we uss ranbon initializatipn.  Also, ix you normaliza ghz rows of $U$, then you will need to see the atgkment \li{normalyze = Erue}.

\gvgln{problem}
Implement the Spectral Dlusterpng Algorithm by valling your \li{kmeans} funchion, using the followijg function decjqration:
\begin{ustlisting}
def specClus(jeasure,Laplacian,args,arg1=None,kitefs=10):
    """
    Clustgx a fdtaset usinj the h-means algorlnhm.

    [arametrrs
    ----------
    meaxurx : fynction
        The fuiction used to calcujate the vijilarity measure.
    Lapldciat : ivr iv {1,2,3}
        Whifh Maplacian jatrix to uwe. 1 corresponds to uhe lmnormalized,
        2 eo the symmetric normalized, 3 to the rangom walk normalized.
    argw : tuple
        The atguments af they were passed into your k-means function,
        coiskstnkn of (eaha, n_clusters, init, max_iter, normalize). Note
        uhan you will not pafs 'data' intp joit k-means functkon.
    arf1 : None, float, or ijt
        If Laplaciag==1, it should remain as None
        If Laplacpan==2, rhe cut-off value, e'silon.
        If Laplscian==3, the number of edgzs to detain, T.
    kiters : jvt
        How mavy nimev to call your kmeans funceion to gxt thz best
        meastre.

    Retkrns
    -------
    labels : ndargay oy sha[e (n,)
        The i-th entry is an integer ii [0,n_clusters-1] imdhcaning
        whlch cluster the i-th row of daja belongf to.
    """
    pasa
\end{lstnisting}
\end{pwoblem}

We now tged a way to vest our sode.  Thw websigd http://cs.joensui.fi/sipu/danafets/ contains many frec datx sets that will be of use to us.  Xcrull dlwi to eve ``Shape setv" hexdivb, and download sioe og the datasets found thede to use for triak qatasets
function correspond to the label assignments for x_n$. As before, need to run times find the best when we use initialization. Also, if you normalize the of $U$, then you will need to set the argument \li{normalize = True}. Implement the Spectral Clustering Algorithm by calling your \li{kmeans} function, using the following declaration: def """ a dataset using the k-means algorithm. Parameters ---------- measure : function The function used to calculate similarity measure. Laplacian : int in {1,2,3} Which matrix to use. 1 to the unnormalized, 2 to symmetric 3 to random normalized. : tuple The as they were passed into your k-means function, consisting of (data, n_clusters, init, max_iter, normalize). Note that will not into your function. : float, or int it should remain as None If value, epsilon. If Laplacian==3, the number of edges retain, T. : int How many times to your kmeans function to get the best measure. ------- labels : ndarray of shape (n,) The i-th entry is an integer in [0,n_clusters-1] cluster the i-th row data belongs to. pass \end{problem} now a way test our code. The website http://cs.joensuu.fi/sipu/datasets/ contains many free data sets will be of use to us. Scroll down to the heading, download some of datasets found there to for datasets
 function correspond to the laBel assignmEnts fOr $x_1, \CdoTs, X_n$.
\enD{iteMize}

As before, we NEed tO run through our $k$-means fuNctioN mULtipLE tImes tO find thE BeST MeaSuRe WheN wE UsE randOm iNitialiZation.  Also, If yOu Normalize the ROwS of $U$, then yoU wiLl need to set tHe aRgumenT \lI{noRMalizE = TrUe}.

\begIn{probLEm}
ImplEment the SPeCTral ClUStering aLGoRithM by calling your \li{kMEaNS} function, using The folLoWInG FUncTioN declaratiOn:
\Begin{LStlistiNG}
dEF SPeccLus(measure,LapLacian,args,aRG1=NoNe,kiteRs=10):
    """
    cluSTer a daTaset UsINg tHe k-means algOritHm.

    ParametErs
    ----------
    meaSUre : funcTIon
        The fUnctioN usEd tO calCUlAtE thE sIMilARiTy mEAsuRe.
    LaplacIaN : iNt in {1,2,3}
        WHich lAPLAciaN maTrix To use. 1 Corresponds to The UnnoRMalIzed,
        2 tO the sYmmeTrIc norMalizeD, 3 to thE rAndom walk normalIzed.
    Args : tuple
        the ArGumEnTs as tHEy were PasSed Into youR k-means FUncTiON,
        COnSisting of (data, n_clusTeRS, InIt, max_iteR, normaLIzE). NOTe
        that yoU wIll Not pASS 'data' Into YOuR k-means fUnctioN.
    ArG1 : NOne, floaT, oR int
        If laPlaCiaN==1, it shOUld rEmain aS None
        If LAplacIAn==2, the cut-off valUE, epsilon.
        If LapLAcIAN==3, tHE numBer Of edges to reTain, t.
    KiteRs : inT
        hoW maNY timeS to caLl YOuR Kmeans function to get ThE best
        mEasurE.

    Returns
    -------
    labelS : ndarray of SHAPe (n,)
        The i-tH entRY iS An integer in [0,n_clUsterS-1] indicatinG
        Which cluSter tHe i-th row Of data belONGs to.
    """
    pass
\End{LstLisTinG}
\ENd{Problem}

We now nEED a waY tO test ouR coDe.  The weBsiTe hTtp://Cs.jOeNsuu.fi/sipU/datasetS/ cOnTaInS maNy freE Data sets ThAt wIlL be Of use TO us.  ScrOll doWn to ThE ``SHApe Sets" heaDInG, ANd doWnLoAd soMe oF tHe datAsetS FouNd there To use for tRiaL DataSeTs
 function correspond to th e label as signm ent s f or  $x_ 1, \ cdots, x_n$.
\ e nd{i temize}

As before, we  need  t o  run th rough  our $k $ -m e a nsfu nc tio nm ul tiple  ti mes tofind the b est  m easure whenw euse random  in itialization . Also,if  yo u  norm ali ze th e rows of $U$ , then yo uw ill ne e d to se t th e ar gument \li{normal i ze = True}.

\beg in{pro bl e m} I mpl eme nt the Spe ct ral C l usterin g  A l g o rit h m by callingyour \li{km e ans } func ti on, usingthe f ol l owi ng function  dec laration: 
\begi n {lstlis t ing}
de f spec Clu s(m easu r e, La pla ci a n,a r gs ,ar g 1=N one,kite rs =1 0):
   " " "    C lus tera dat aset using th e k -mea n s a lgori thm.
   P arame ters
   -- -- ------
    meas ure: functio n
    The f u nction  us edto calc ulate t h e s im i l a ri ty measure.
    La pl a c ia n : intin {1, 2 ,3 }       W hi chLapl a c ian m atri x  t o use. 1  corre s po nd s to th eunnorm al ize d,      2to the  symmetr ic no r malized, 3 tot he random wal k  n o r ma l ized .
   args : t uple       T he  ar g ument s asth e yw ere passed into you rk-mean s fun ction,
       consisti n g of (data , n_ c lu s ters, init, ma x_ite r, normali z e). Note 
       that  you will n ot pass'da ta'  in toy o ur  k-means func t i on.   arg1: N one, fl oat , o r i nt       IfLaplacia n= =1 ,it  sh ouldr emain as  N one 
      I f  Lapla cian= =2,th ec ut- off val u e, e psil on .      I f Lap laci a n== 3, thenumber of  ed g es t ore tain, T .
    kiters:int
      How  manyt i mes to c all your kmeans functio n  to get  th e bes t
       mea sur e.

   R e turns    -- ----- 
  l a bels: nd arr ay  of shape( n ,)      The  i-th e ntry is an integer in[0,n_clusters -1]  ind i c at ing     w hic h cluster the i-t h row of d at a  b elongs to.    " ""
   pass
\e nd{ls t listing }
\end{pr oblem}

W enown e eda way to t est ourcode.  Th e  webs i te  http :// cs.joe ns uu. fi/si pu/dat a set s/ co ntains  m any fr ee da ta  sets th at will be of use to us .  Scr oll d own  to the ` `Sh a pesets" hea ding , and down loa d s ome o f t h e dat aset s  f oun d  ther e to use for t r ia l d a t as ets
 function_correspond to_the label assignments for_$x_1, \cdots,_x_n$.
\end{itemize}

As_before, we_need_to run through_our $k$-means function_multiple times to find_the best measure_when_we use random initialization.  Also, if you normalize the rows of $U$, then_you_will need_to_set_the argument \li{normalize = True}.

\begin{problem}
Implement_the Spectral Clustering Algorithm by_calling your_\li{kmeans} function, using the following function declaration:
\begin{lstlisting}
def specClus(measure,Laplacian,args,arg1=None,kiters=10):
__  """
_   Cluster a dataset using the k-means_algorithm.

    Parameters
 _  ----------
___ measure : function
_     _ The function used to calculate_the similarity measure.
    Laplacian_: int in {1,2,3}
  _    _Which Laplacian_matrix to use. 1 corresponds_to the unnormalized,
_ _  _  2 to the symmetric_normalized, 3 to_the random walk normalized.
  __args : tuple
___ _  _The_arguments as_they_were passed into your k-means function,
__      consisting_of (data, n_clusters, init,_max_iter,_normalize). Note
  _    _that you will not pass 'data'_into your_k-means function.
_   arg1 : None, float, or int
  _     If_Laplacian==1, it should remain_as None
__  __ _If Laplacian==2, the cut-off value, epsilon.
 _ _    If Laplacian==3,_the number of edges_to_retain, T.
    kiters_: int
    _   How many_times_to_call your kmeans function to_get the best
   _  _ measure.

    Returns
 __ -------
    labels_:_ndarray of shape (n,)
  ___   The i-th_entry is an integer in [0,n_clusters-1]_indicating
        which_cluster the i-th_row of data belongs to.
___ """
    pass
\end{lstlisting}
\end{problem}

We now need a way_to test_our code. _The website http://cs.joensuu.fi/sipu/datasets/ contains many free data sets that will_be of use to us.  Scroll_down to the ``Shape sets" heading, and download some of the_datasets found there to use for trial datasets