cos(a, b) = at b a b. To get a distance measure, subtract the cosine similarity from one. dist(a, b) =1 cos(a, b)

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1 8 Clusterng 8.1 Some Clusterng Examples Clusterng comes up n many contexts. For example, one mght want to cluster journal artcles nto clusters of artcles on related topcs. In dong ths, one frst represents adocumentbyavector. Thscanbedoneusngthevectorspacemodelntroducedn Chapter 2. Each document s represented as a vector wth one component for each term gvng the frequency of the term n the document. Alternatvely, a document may be represented by a vector whose components correspond to documents n the collecton and the th vector s j th component s a 0 or 1 dependng on whether the th document referenced the j th document. Once one has represented the documents as vectors, the problem becomes one of clusterng vectors. Another context where clusterng s mportant s the study of the evoluton and growth of communtes n socal networks. Here one constructs a graph where nodes represent ndvduals and there s an edge from one node to another f the person correspondng to the frst node sent an emal or nstant message to the person correspondng to the second node. A communty s defned as a set of nodes where the frequency of messages wthn the set s hgher than what one would expect f the set of nodes n the communty were a random set. Clusterng parttons the set of nodes of the graph nto sets of nodes where the sets consst of nodes that send more messages to one another than one would expect by chance. Note that clusterng generally asks for a strct partton nto subsets although n realty n ths case for nstance, a node may well belong to several communtes. In these clusterng problems, one defnes ether a smlarty measure between pars of objects or a dstance measure (a noton of dssmlarty). One measure of smlarty between two vectors a and b s the cosne of the angle between them: cos(a, b) = at b a b. To get a dstance measure, subtract the cosne smlarty from one. dst(a, b) =1 cos(a, b) Another dstance measure s the Eucldean dstance. There s an obvous relatonshp between cosne smlarty and Eucldean dstance. If a and b are unt vectors, then a b 2 =(a b) T (a b) = a 2 + b 2 2a T b =2(1 cos (a, b)). In determnng the dstance functon to use, t s useful to know somethng about the orgn of the data. In problems where we have to cluster nodes of a graph, we may represent each node as a a vector, namely, as the row of the adjacency matrx correspondng to the node. One noton of dssmlarty here s the square of the Eucldean dstance. For

2 vectors, ths measure s just the number of uncommon 1 s, whereas, the dot product s the number of common 1 s. In many stuatons one has a stochastc model of how the data was generated. An example s customer behavor. Suppose there are d products and n customers. A reasonable assumpton s that each customer generates from a probablty dstrbuton, the basket of goods he or she buys. A basket specfes the amount of each good bought. One hypothess s that there are only k types of customers, k << n. Each customer type s characterzed by a probablty densty used by all customers of that type to generate ther baskets of goods. The denstes may all be Gaussans wth dfferent centers and covarance matrces. We are not gven the probablty denstes, only the basket bought by each customer, whch s observable. Our task s to cluster the customers nto the k types. We may dentfy the customer wth hs or her basket whch s avector. Onewaytoformulatetheproblemmathematcallysvaaclusterngcrteron whch we then optmze. Some potental crtera are to partton the customers nto k clusters so as to mnmze 1. the sum of dstances between all pars of customers n the same cluster, 2. the sum of dstances of all customers to ther cluster center (any pont n space may be desgnated as the cluster center), or 3. mnmze the sum of squared dstances to the cluster center. The last crteron s called the k-means crteron and s wdely used. A varant called the k-medan crteron mnmzes the sum of dstances (not squared) to the cluster center. Another possblty, called the k-center crteron, s to mnmze the maxmum dstance of any pont to ts cluster center. The chosen crteron can affect the results. To llustrate, suppose we have data generated accordng to a equal weght mxture of k sphercal Gaussan denstes centered at µ 1,µ 2,...,µ k,eachwthvarance1neverydrecton.thenthedenstyofthemxture s F (x) =Prob[x] = 1 1 k e x µ t 2. k (2π) d/2 Denote by µ(x) thecenternearesttox. Sncetheexponentalfunctonfallsofffast,we have the approxmaton F (x) 1 1 e x µ(x) 2. k (2π) d/2 So, gven a sample of ponts x (1), x (2),...x (n) drawn accordng to the mxture, the lkelyhood of a partcular µ 1,µ 2,...,µ k,namely,the(posteror)probabltyofgeneratng the sample f µ 1,µ 2,...,µ k were n fact the centers, s approxmately 1 1 k n (2π) nd/2 n =1 t=1 e x() µ(x () ) 2 = ce n =1 x() µ(x () )

3 So, mnmzng the sum of squared dstances to cluster centers fnds the maxmum lkelyhood µ 1,µ 2,...,µ k and ths suggests the crteron : sum of dstance squared to the cluster centers. On the other hand, f the generatng process had an exponental probablty dstrbuton, wth the probablty law Prob[(x 1,x 2,...,x d )] = 1 2 d e x µ = 1 2 e =1 d =1 x µ = 1 2 e x µ 1, one would use the L 1 norm (not the L 2 or the square of the L 1 )sncetheprobabltydensty decreases as the L 1 dstance from the center. The ntuton here s that the dstance used to cluster data should be related to the actual dstrbuton of the data. The choce of whether to use a dstance measure and cluster together ponts that are close or use a smlarty measure and cluster together ponts wth hgh smlarty and what partcular dstance or smlarty measure to use can be crucal to the applcaton. However, there s not much theory on these choces; they are determned by emprcal doman-specfc knowledge. One general observaton s worth makng. Usng dstance squared nstead of dstance, favors outlers more snce the square functon magnfes large values, whch means a small number of outlers may make a clusterng look bad. On the other hand, dstance squared has some mathematcal advantages; see for example Corollary 8.3 whch asserts that wth the dstance squared crteron, the centrod s the correct cluster center. The wdely used k-means crteron s based on sum of squared dstances. In the formulatons we have dscussed so far, we have one number (eg. sum of dstances squared to the cluster center) as the measure of goodness of a clusterng and we try to optmze that number (to fnd the best clusterng accordng to the measure). Ths approach does not always yeld desred results, snce often, t s hard to optmze exactly (most clusterng problems are NP-hard). Often, there are polynomal tme algorthms to fnd an approxmately optmal soluton. But such a soluton may be far from the optmal (desred) clusterng. We wll n secton (8.4) how to formalze some realstc condtons under whch an approxmate optmal soluton ndeed gves us a desred clusterng as well. But frst we see some smple algorthms for gettng a good clusterng accordng to some natural measures. 8.2 A Smple Greedy Algorthm for k-clusterng There are many algorthms for clusterng hgh-dmensonal data. We start wth a smple one. Suppose we use the k-center crteron. The k-center crteron parttons the ponts nto k clusters so as to mnmze the maxmum dstance of any pont to ts cluster center. Call the maxmum dstance of any pont to ts cluster center the radus of the clusterng. There s a k-clusterng of radus r f and only f there are k spheres, each of radus r whch together cover all the ponts. Below, we gve a smple algorthm to fnd k spheres coverng a set of ponts. The lemma followng shows that ths algorthm needs to 242

4 use a radus that s off by a factor of at most two from the optmal k-center soluton. The Greedy k-clusterng Algorthm Pck any data pont to be the frst cluster center. At tme t, fort =2, 3,...,k,pck any data pont that s not wthn dstance r of an exstng cluster center; make t the t th cluster center. Lemma 8.1 If there s a k-clusterng of radus r, then the above algorthm fnds a k- 2 clusterng wth radus at most r. Proof: Suppose for contradcton that the algorthm usng radus r fals to fnd a k- clusterng. Ths means that after the algorthm chooses k centers, there s stll at least one data pont that s not n any sphere of radus r around a pcked center. Ths s the only possble mode of falure. But then there are k +1dataponts,wtheachparmore than dstance r apart. Clearly, no two such ponts can belong to the same cluster n any k-clusterng of radus r contradctng the hypothess. 2 There are n general two varatons of the clusterng problem for each of the crtera. We could requre that each cluster center be a data pont or allow a cluster center to be any pont n space. If we requre each cluster center to be a data pont, the problem can be solved n tme n k tmes a polynomal n the length of the data. Frst, exhaustvely enumerate all sets of k data ponts as the possble sets of k cluster centers, then assocate each pont to ts nearest center and select the best clusterng. No such nave enumeraton procedure s avalable when cluster centers can be any pont n space. But, for the k-means problem, Corollary 8.3 shows that once we have dentfed the data ponts that belong to a cluster, the best choce of cluster center s the centrod. Note that the centrod mght not be a data pont. 8.3 Lloyd s Algorthm for k-means Clusterng In k-means crteron, a set A = {a 1, a 2,...,a n } of n ponts n d-dmensons s parttoned nto k-clusters, S 1,S 2,...,S k,soastomnmzethesumofsquareddstancesof each pont to ts cluster center. That s, A s parttoned nto clusters, S 1,S 2,...,S k,and a center s assgned to each cluster so as to mnmze d (S 1,S 2,...,S k )= k j=1 a S j (c j a ) 2 where c j s the center of cluster j. Suppose we have already determned the clusterng or the parttonng nto S 1,S 2,...,S k. What are the best centers for the clusters? The followng lemma shows that the answer s the centrods, the coordnatewse means, of the clusters. 243

5 Lemma 8.2 Let {a 1, a 2,...,a n } be a set of ponts. The sum of the squared dstances of the a to any pont x equals the sum of the squared dstances to the centrod plus the number of ponts tmes the squared dstance from the pont x to the centrod. That s, a x 2 = a c 2 + n c x 2 where c s the centrod of the set of ponts. Proof: a x 2 = = a c + c x 2 a c 2 +2(c x) (a c)+n c x 2 Snce c s the centrod, (a c) =0. Thus, a x 2 = a c 2 + n c x 2 AcorollaryofLemma8.2sthatthecentrodmnmzesthesumofsquareddstances snce the second term, n c x 2,salwayspostve. Corollary 8.3 Let {a 1, a 2,...,a n } be a set of ponts. The sum of squared dstances of the a to a pont x s mnmzed when x s the centrod, namely x = 1 a n. Another expresson for the sum of squared dstances of a set of n ponts to ther centrod s the sum of all parwse dstances squared dvded by n. Frst, a smple notatonal ssue. n n For a set of ponts {a 1, a 2,...,a n }, a a j 2 counts the quantty a a j 2 once =1 j=+1 for each ordered par (, j), j>. However, a a j 2 counts each a a j 2 twce, so,j the later sum s twce the frst sum. Lemma 8.4 Let {a 1, a 2,...,a n } be a set of ponts. The sum of the squared dstances between all pars of ponts equals the number of ponts tmes the sum of the squared dstances of the ponts to the centrod of the ponts. That s, a a j 2 = n a c 2 j> where c s the centrod of the set of ponts. Proof: Lemma 8.2 states that for every x, a x 2 = a c 2 + n c x

6 Lettng x range over all a j and summng the n equatons yelds a a j 2 = n a c 2 + n c a j 2,j j =2n a c 2. Observng that yelds the result that a a j 2 =2,j a a j 2 j> a a j 2 = n a c 2. j> The k-means clusterng algorthm Anaturalalgorthmfork-means clusterng s gven below. There are two unspecfed aspects of the algorthm. One s the set of startng centers and the other s the stoppng condton. k-means algorthm Start wth k centers. Cluster each pont wth the center nearest to t. Fnd the centrod of each cluster and replace the set of old centers wth the centrods. Repeat the above two steps untl the centers converge (accordng to some crteron). The k-means algorthm always converges but possbly to a local mnmum. To show convergence, we argue that the cost of the clusterng, the sum of the squares of the dstances of each pont to ts cluster center, always mproves. Each teraton conssts of two steps. Frst, consder the step whch fnds the centrod of each cluster and replaces the old centers wth the new centers. By Corollary 8.3, ths step mproves the sum of nternal cluster dstances squared. The other step reclusters by assgnng each pont to ts nearest cluster center, whch also mproves the nternal cluster dstances. 8.4 Meanngful Clusterng va Sngular Value Decomposton Optmzng a crteron such as k means s often not an end n tself. It s a means to fndng a good (meanngful) clusterng. How do we defne a meanngful clusterng? Here s a possble answer: an optmal clusterng s meanngful f t s unque, n the sense that any other nearly optmal clusterng agrees wth t on most data ponts. We wll formalze ths below. But the bad news s that we wll soon see that ths s too much to ask for and many common data sets do not admt such a clusterng. Luckly though, the dscusson 245

7 wll lead us to a weaker requrement whch has the twn propertes of beng met by many data sets as well as admttng an effcent (SVD-based) algorthm to fnd the clusterng. We start wth some notaton. We denote by n the number of (data) ponts to be clustered; they are lsted as the rows A of a n d matrx A. A clusterng (partton) of the data ponts s represented by a matrx of cluster centers C whch s also a n d matrx; the th row of C s the center of the cluster that A belongs to. So C has only k dstnct rows. We refer to A as the data and to C as the clusterng. Defnton: The cost of the clusterng C s the sum of dstances squared to the cluster centers; so we have cost(a, C) = A C 2 F. The mean squared dstance (MSD) of a clusterng C s just cost(a, C)/n. We wll say two clusterngs of A dffer n s ponts f s s the mnmum number of data ponts to be reassgned to get from one clusterng to the other. [Note: a clusterng s specfed by just a partton; the cluster centers are just the centrods of data ponts n a cluster.] Here s the frst attempt at defnng when a clusterng s meanngful: A k clusterng C s meanngful f every k clusterng C whch dffers from C n at least n ponts has cost at least (1 + Ω()) cost(c). [ s a small constant.] Clam 8.1 If C s meanngful under ths defnton, and each of ts clusters has Ω(n) data ponts n t, then for any two cluster centers µ, µ of C, µ µ 2 MSD(C) : Proof: We wll prove the clam by showng that f two cluster centers n C are too close, then we may move n ponts from cluster to the other wthout ncreasng the cost by more than a factor of (1 + O()), thus contradctng the assumpton that C s meanngful. Let T be the cluster wth cluster center µ. Project each data pont A T,onto the lne through µ, µ and let d be the dstance of the projected pont to µ. Let T 1 be the subset of T whose projectons land on the µ sde of µ and let T 2 = T \ T 1 be the ponts whose projectons land on the other sde. Snce µ s the centrod of T,wehave T 1 d = T 2 d. Snce each A T s closer to µ than to µ,for T 1,wehave d µ µ /2 andso T 1 d T µ µ /2; hence also, T 2 d T µ µ /2. So, d T µ µ. T Now from the assumpton that T Ω(n), we have T 2n. So,then th smallest d s T at most T n µ µ 2 µ µ.wecannowgetanewclusterngc as follows: move the n A n T wth the smallest d to the cluster wth µ as center. Recompute centers. The recomputataton of the centers can only reduce the cost (as we saw n Corollary (8.3)). What about the move? The move can only add cost (dstance squared) n the drecton of the lne jonng µ, µ and ths extra cost s at most 4n µ µ 2.Bytheassumptonthat C s meanngful (under the proposed defnton above), we must thus have µ µ 2 MSD(C) asclamed. 246

8 But we wll now see that the condton that µ µ 2 MSD(C) stoostrongforsome common data sets. Consder two sphercal Gaussans n d space, each wth varance 1 n each drecton. Clearly f we have data generated from a (equal weght) mxture of the two, the correct 2-clusterng one would seek s to splt them nto the Gaussans they were generated from wth the actual centers of the Gaussans (or a very close pont) as the cluster centers. But the MSD of ths clusterng s approxmately d. So,bytheClam,for C to be meanngful, the centers must be Ω(d) apart.itseasytoseehoweverthatfthe separaton s Ω( ln n), the clusters are dstnct : the projecton of each Gaussan to the lne jonng ther centers s a 1 dmensonal Gaussan of varance 1, so the probablty that any data pont les more than a tenth of the way (n the projecton) to the wrong center s at most O(1/n 2 ), so by unon bound, all data ponts have ther dstance to the wrong center at least 10 tmes the dstance to the correct center, provded, we measure dstances only n the projecton. Snce the varance s 1 n each drecton, themeansquared Dstance to the cluster center n each drecton s only O(1). So, n ths example, t would make more sense to requre a nter-center separaton of the maxmum mean squared dstance n any one drecton to the cluster center. We wll be able to acheve ths n general. Defnton: Let A, C be the data and cluster centers matrces respectvely. The mean squared dstance n a drecton (denoted MSDD(A, C)) s the maxmum over all unt length vectors v of the mean squared dstance of data ponts to ther cluster centers n the drecton v, namely, MSDD(A, C) = 1 n Max v: v =1 (A C)v 2 = 1 n A C 2 2, where, we have used the basc defnton of the largest sngular value to get the last expresson. Theorem 8.5 Suppose there exsts a k clusterng C of data ponts A wth () Ω(n) data ponts per cluster and () dstance at least Ω( MSDD(A, C)) between any two cluster centers. Then any clusterng returned by the followng smple algorthm (whch we henceforth call the SVD clusterng of A) dffers from C s at most n ponts (here, the hdden constants n Ω depend on.) Fnd the SVD of A. Project data ponts to the space of the top k (rght) sngular vectors of A. Return a 2 approxmate 8 k means clusterng n the projecton. Remark: Note that f we dd the approxmate optmzaton n the whole space (wthout projectng), we wll not succeed n general. In the example of the two sphercal Gaussans above, for the correct clusterng C, MSDD(A, C) =O(1). But f the nter-center separaton s just O(1), then n ponts of the frst Gaussan may be put nto the second at an added cost (where cost s dstance squared n the whole space) of only O(n) asweargued, whereas, the cost of clusterng C s O(nd). 8 A 2-approxmate clusterng s one whch has cost at most twce the optmal cost. 247

9 Remark: The Theorem also mples a sort of unqueness of the clusterng C, namely, that any other clusterng satsfyng both () and () dffers from C n at most 2n ponts, as seen from the followng: The Theorem apples to the other clusterng as well, snce t satsfes the hypothess. So, t also dffers from the SVD clusterng n at most n ponts. Thus, C and the other clusterng cannot dffer n more than 2n ponts. The proof of the Theorem wll use the followng two Lemmas, whch llustrate the power of SVD. The frst Lemma says that for any clusterng C wth Ω(n) pontspercluster,the SVD clusterng descrbed n the Theorem fnds cluster centers farly close to the cluster centers n C, where,closesmeasuredntermsofmsdd(a, C). The argument wll be that one canddate clusterng n the SVD projecton s to just use the same centers as C (projected to the space) and f the SVD clusterng does not have a cluster center close to a partcular cluster of C, t ends up payng too hgh a penalty compared to ths canddate clusterng to be 2-optmal. Lemma 8.6 Suppose A s the data matrx and C a clusterng wth Ω(n) ponts n each cluster. 9 Suppose C s the SVD clusterng (descrbed n the Theorem) of A. Then for every cluster center µ of C, there s a cluster center of C wthn dstance O( MSDD(A, C)). Proof: Let α =MSDD(A, C). Let T be the set of data ponts n a cluster of C and suppose for contradcton that centrod µ of T has no cluster center of C at dstance O( kα). Let Ā denote projecton of A onto the SVD subspace; so the SVD clusterng actually clusters ponts Ā. RecallthenotatonthatC s the cluster center of C closest to Ā. The cost of a data pont A T n the SVD soluton s Ā C 2 = (µ C ) (µ Ā) µ C 2 µ Ā 2 Ω(α) µ Ā 2, where, we have used a b a 2 b 2 for any two vectors a and b. Addng over all ponts n T,thecostofC s at least Ω(nα) Ā C 2 F. Now, one way to cluster the ponts Ā s to just use the same cluster centers as C; thecostofthsclusterng s Ā C 2 F. So the optmal clusterng of ponts Ā costs at most Ā C 2 F and snce the algorthm fnds a 2-approxmate clusterng, the cost of the SVD clusterng s at most 2 Ā C 2 F.Soweget 2 Ā C 2 F Ω(nα) Ā C 2 F = Ā C 2 F Ω(nα). We wll prove that Ā C 2 5kαn n Lemma 8.7. By (), we have k O(1), so F Ā C 2 F = O(α). So, we get a contradcton (wth sutable choce of constants under the Ω.) Note that n the Lemma below, the man nequalty has Frobenus norm on the left hand sde, but only operator norm on the rght hand sde. Ths makes t stronger than havng ether Frobenus norm on both sdes or operator norm on both sdes. 9 Note that C s not assumed to satsfy condton () of the Theorem. 248

10 Lemma 8.7 Suppose A s an n d matrx and suppose C s an n d rank k matrx. Let Ā be the best rank k approxmaton to A found by SVD. Then, Ā C 2 F 5k A C 2 2. Proof: Let u 1, u 2,...,u k be the top k sngular vectors of A. Extend the set of the top k sngular vectors to an orthonormal bass u 1, u 2,...,u p of the vector space spanned by the rows of Ā and C. Note that p 2k snce Ā s spanned by u 1, u 2,...,u k and C s of rank at most k. Then, Ā C 2 F = k p (Ā C)u 2 + (Ā C)u 2. =1 =k+1 Snce {u 1 k} are the top k sngular vectors of A and snce Ā s the rank k approxmaton to A, for1 k, Au = Āu and thus Ā C u 2 = (A C) u 2.For >k, Āu =0,thus Ā C u 2 = Cu 2. From ths t follows that Ā C 2 F = k (A C)u 2 + =1 k A C p =k+1 p =k+1 Cu 2 Au +(C A)u 2 Usng a + b 2 2 a 2 +2 b 2 Ā C 2 F k A C p =k+1 Au 2 +2 p =k+1 (C A)u 2 k A C (p k 1) σ 2 k+1(a)+2(p k 1) A C 2 2 Usng p 2k mples k>p k 1 Ā C 2 F k A C kσ 2 k+1(a)+2k A C 2 2. (8.1) As we saw n Chapter 4, for any rank k matrx B, A B 2 σ k+1 (A) andsoσ k+1 (A) A C 2 and pluggng ths n, we get the Lemma. Proof: (of Theorem (8.5)) Let β = MSDD(A, C). We wll use Lemma (8.6). To every cluster center µ of C, theresaclustercenterν of the SVD clusterng wthn dstance O(β). Also, snce, C satsfes () of the Theorem, the mappng from µ to ν s 1-1. C and the SVD clusterng dffer on a data pont A f the followng happens: A belongs to cluster center µ of C whose closest cluster center n C s ν, butā (the projecton of A 249

Subspace clustering. Clustering. Fundamental to all clustering techniques is the choice of distance measure between data points;

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