CUERPOS INTRUSIVOS

Let t= ({1, . . . , t}, At) denote a cycle network with t nodes such that the domain of the

dissimilarity function dom(At) ={(i, i+ 1)}it−=11∪(t,1) contains all pairs of subsequent nodes

plus the pair (t,1). Further require all pairs (i, j)∈dom(At) to have unit dissimilarities, At(1,2) =At(2,3) =. . .=At(t−1, t) =At(t,1) = 1. (5.39) The first three elements of the class of cycle networks,2,3, and4, are illustrated in Fig. 5.6. In this section we study representable methods where the collections of representers contain cycle networks.

We first note that the method defined by a representer collection that contains a finite number of cycle networks is equivalent to the method defined by the singleton collection that contains as representer the longest of the cycles. Indeed, consider a finite collection

Ωt1,...,tn of cyclic representers Ωt1,...,tn = {t1,t2, . . . ,tn} and assume, without loss of

generality, that t1 > t2 > . . . > tn. We can always find a dissimilarity reducing map from t1 toti for all i= 2, . . . , n. For example, define the mapφt1→ti from t1 toti as

φt1→ti(j) = min(j, ti), (5.40)

forj ∈ {1, . . . , t1}. The mapφt1→tiis dissimilarity reducing sincetiof the unit dissimilarities

t1

ti

N

φt1→ti

φ

φ◦φt1→ti

Figure 5.7: Given a dissimilarity reducing mapφfromti toN, one can always find a dissimilarity reducing map fromt1 toN.

in t1 are mapped to null dissimilarities in ti. Thus, given a dissimilarity reducing map φ between ti and an arbitrary network N, we may construct the dissimilarity reducing

map φ◦φt1→ti from t1 to N; see Fig. 5.7. Moreover, φ◦φt1→ti has the same image as φsinceφt1→ti is surjective by construction [cf. (5.40)]. The map φt1→ti being dissimilarity

reducing ensures that L(φ◦φt1→ti;t1, N) ≤ L(φ;ti, N), which, by (5.20) implies that λt1

X (x, x0)≤λ

_ti

X (x, x0) and from (5.23) we conclude thatλ

Ωt₁,...tn

X (x, x0) =λ

t₁

X (x, x0). This

means that the method represented by Ωt1,...,tn is equivalent to the method represented by

the longest cyclet1.

Therefore, any method defined by a finite collection of cycle representers is equivalent to a method that is defined by a single cycle representer. Consider then the singleton collections{t} and denote the corresponding method as Ht :=H{t}. The methodHt

is referred to as thetth cyclic method. Cyclic methodsHt _{for allt≥2 are admissible and}

scale preserving as shown in the following corollary of Theorem 11.

Corollary 3 Cyclic methods, defined as representable methods Ht _{associated with the cy-}

cle networkst= ({1, . . . , t}, At)having dissimilarities as in (5.39)satisfy axioms (A1) and

(A2) and the Scale Preservation Property (P2).

Proof: Since networkstare strongly connected and structure representers, the hypotheses

of Theorem 11 are satisfied.

The first cyclic methodH2_{was used to introduce the concept of representable clustering}

in (5.15)-(5.17) and shown to coincide with the reciprocal clustering method HR _{in (5.18).}

Interpreting2 as a basic cluster unit we can then think of reciprocal clusteringHR_{≡ H}2

as a method that allows propagation of influence through cycles that contain at most two nodes. Likewise, the methodH3 _{represented by the cycle network 3 can be interpreted}

as a method that allows propagation of influence through cycles that contain at most three nodes. To see the consistency of this interpretation consider the application ofH3 _{to deter-}

mine the ultrametricu3

X (x, x

0_{) between pointsxandx}0 _{of the networkN = (X, AX) shown}

in Fig. 5.8. In the figure, undrawn edges have dissimilarities greater than 5. Due to the scarcity of bidirectional paths linkingxandx0, a quick inspection reveals thatu2

X (x, x

since the minimizing bidirectional path is given by [x, x2, x3, x4, x0]. In order to compute

u3

X (x, x

0_{) as defined by (5.24) focus on the minimizing pathP}

xx0 = [x, x₂, x₃, x0]. Consider

the map φx,x2 from 3 to N such that φx,x2(1) = x, φx,x2(2) = x2 and φx,x2(3) = x1.

By computing (5.19), we have that L(φx,x2;3, N) = 2. Moreover, 2 is the minimum

multiple that allows the construction of a dissimilarity reducing map that contains in its image the nodes x and x2. From (5.20), we then have that λ_X3(x, x2) = 2. Similarly, we

can construct maps φx2,x3 and φx3,x0 containing in their images the second and the third

pair of consecutive nodes in Pxx0 respectively. The map φx

2,x3 goes from 3 to N with φx2,x3(1) = φx2,x3(2) = x2 and φx2,x3(3) = x3. The map φx3,x0 goes from 3 to N with φx3,x0(1) =x3,φx3,x0(2) =x4 andφx3,x0(3) =x

0_{. By computing the corresponding Lipschitz}

constants as done for φx,x2, we have that λ

3

X (x2, x3) = 3 and λ

3

X (x3, x

0_{) = 1. From (5.24)}

the ultrametric value is the maximum of these three multiples, i.e.,

u3 X (x, x 0 ) = max[λ3 X(x, x2), λ 3 X(x3, x 0 ), λ3 X(x2, x3)] = 3. (5.41)

Moving on to the fourth cyclic method consider the ultrametric u4

X (x, x

0_{) generated by}

the method H4 _{where influence cycles of up to 4 nodes are allowed. Focus on the same}

minimizing path as in the previous case Pxx0 = [x, x₂, x₃, x0] and join the pairs x, x₂ and

x3, x0 using the same maps as in the case for H3, i.e. λ_X4(x, x2) = 2 and λ_X4(x3, x0) = 1.

These maps exist because for every dissimilarity reducing map from 3 to N we may construct a dissimilarity reducing map from 4 to N using (5.40). However, in order to join x2 and x3 inPxx0 we replace the mapping to the subnetwork formed byx₂ and x₃ by

a map from 4 to the four node subnetwork formed by x2, x1, x4, and x3. In this case,

we obtain λ4

X (x2, x3) = 2 and the ultrametric value becomes the maximum of the three

multiples, u4 X (x, x 0_{) = max[λ}4 X(x, x2), λ 4 X(x3, x 0_{), λ}4 X(x2, x3)] = 2. (5.42)

Observe that the third and fourth cyclic methods yield ultrametric distances smaller than the reciprocal ultrametric distance. This is not only consistent with Theorem 4 but also indicative of the status of these methods as relaxations of the condition of direct mutual influence. As we keep allowing for increasingly long cycles of influence the question arises of whether we end up recovering nonreciprocal clustering. This is not true in general for any t where t is finite. However, if we define C∞ as the following infinite collection of

representers

C∞:={t}∞t=1, (5.43)

we can show that the method HC∞ _{is equivalent to the nonreciprocal clustering method}

x x2 x1 x3 x4 x0 1 2 3 2 2 2 4 5 1 3 2 1 2 4 1 1 2 2 1 1 1 λ3 X (x, x2) = 2 λ 3 X (x3, x 0 ) = 1 λ3_X(x2, x3) = 3 N Nx,x₂ Nx2,x3 Nx3,x0 3 Figure 5.8: Computation ofu3

X . The minimizing path corresponds toPxx0 = [x, x₂, x₃, x0]. Three

dissimilarity reducing maps are constructed from multiples of 3 to N. The images of the maps,

marked by dashed ellipses, contain pairs of consecutive nodes inPxx0. The ultrametric value corre-

sponds to the maximum multiple of3, i.e. uX3(x, x

0_{) = max(2,3,1) = 3.}

Proposition 18 The clustering methodHC∞ represented by the family of all cycle networks

C∞ defined in (5.43) is equivalent to the nonreciprocal clustering method HNR with output

ultrametrics as defined in (3.8).

Proof: In order to show the equivalence HC∞ ≡ HNR_{, we have to show that the outputs}

(X, uC∞_X ) = HC∞(N) and (X, uNR_X ) = HNR_{(N) coincide for every network N = (X, AX).}

From Theorem 11 we know that HC∞ _{is an admissible method since it is represented by a}

collection of strongly connected structure representers, thus by Theorem 4, we have that

uNR_X (x, x0)≤uC∞_X (x, x0), (5.44)

for arbitrary nodes x, x0 in any network N = (X, AX).

Given a networkN = (X, AX), pick any pair of nodesxandx0and defineδ:=uNR_X (x, x0) as the nonreciprocal ultrametric between these nodes. From definition (3.8), this implies that we can find a pathPxx0 = [x=x₀, x₁, . . . , x_l=x0] fromxtox0 and a pathP_x0_x= [x0=

x0, x1, ..., xl0 =x] in the opposite direction both paths of cost not greater thanδ. Thus, the

loop Pxx = Pxx0 ]P_x0_x generated by the concatenation of the aforementioned paths has a

cost not exceeding δ and it contains l+l0 nodes. Consequently, we may construct a map

φ from l+l0∈ C_∞ toN mapping the nodes in the cycle of δ∗_l+l0 to the loop Pxx. From

(5.20) and (5.24), this implies that

Combining (5.44) and (5.45) we obtain that

uC∞_X (x, x0) =uNR_X (x, x0), (5.46)

for every pair of nodes x, x0 in any network N = (X, AX), as wanted.

Proposition 18 provides a generative reformulation of nonreciprocal clustering. More- over, it can be shown that any method represented by a collection of countably infinitely many distinct cycle representers is equivalent toHC∞ _{as we show next.}

Corollary 4 Given any collection Ω of countably infinitely many distinct cycle represen- ters, the represented methodHΩ _{≡ H}C∞ _{where C}

∞ is defined in (5.43).

Proof: An analogous proof to the one of Proposition 18 can be done to show HΩ _{≡ H}NR_.

Combining this with the result in Proposition 18, it follows thatHΩ _{≡ H}C∞_.

Combining the result in Corollary 4 with the fact that a method represented by a finite collection of cycles is equivalent to the method represented by the longest cycle in the collection, it follows that by considering methods Ht _{for finite t and method H}C∞ _we

are considering every method that can be represented by a countable collection of cyclic representers.

As is intended of representable clustering methods, the reformulation in Proposition 18 expresses nonreciprocal clustering through the consideration of particular cases, namely cycles of arbitrary length. This reformulation uncovers the drawback of nonreciprocal clus- tering – propagating influence through cycles of arbitrary length is perhaps unrealistic – but also offers alternative formulations that mitigate this limitation – restrict the propagation of influence to cycles of certain length. In that sense, cyclic methods of length t can be interpreted as a tightening of nonreciprocal clustering. This interpretation is complemen- tary of their interpretation as relaxations of reciprocal clustering that we discussed above. Given this dual interpretation, cyclic clustering methods are of practical importance.

Algorithms for the computation of the output ultrametrics associated with cyclic meth- ods follow from matrix operations in the dioid algebra ( ¯R+,min,max) defined in Section

3.5. Explicit expressions are given in the following proposition.

Proposition 19 Consider a given network N = (X, AX) with n nodes. Denote by u_Xt

the t-th cyclic ultrametric generated by the method Ht _{represented by the cycle network}

t= ({1, . . . , t}, At) with dissimilaritiesAt as in (5.39). Then, we can compute u_Xt as ut X = "_t−1 M k=1 maxAk_X,(AT_X)t−k #n−1 , (5.47)

where the matrix powers are computed in the dioid algebra ( ¯R+,min,max) as defined in

(3.71). Equivalently, the expression in (5.47) can be simplified to

ut X = max AX,(A T X)t−1 n−1 . (5.48)

Proof: We begin by showing validity of (5.47). Notice that if we show that "_t−1 M k=1 maxAk_X,(AT_X)t−k # ij =λt X(xi, xj), (5.49)

then we are done since the outmost (n−1) dioid power in (5.47) corresponds to computing single linkage [cf. (3.91)] and, thus, (5.35) completes the proof.

Recall thatAk_X contains the minimum path cost of paths of length at mostknodes, i.e. h Ak_Xi ij = minPk xixj max m|xm∈Pk xixj AX(xm, xm+1), (5.50) and similarly, h (AT_X)t−ki ij=min_Pt−k xj xi max m|xm∈P_{xj xi}t−k AX(xm, xm+1). (5.51)

Hence, the maximum of (5.50) and (5.51), i.e. max([Ak_X]ij,[(ATX)t−k]ij), gives us the min-

imum cost of a loop containing xi and xj in which the path from xi to xj has at most k

nodes and the path in the opposite direction has at mostt−knodes. Further, notice that

max h Ak_X i ij, h (AT_X)t−k i ij = h max Ak_X,(AT_X)t−k i ij. (5.52)

Finally, by minimizing over all possiblek, we find the minimum cost of every loop with at mosttnodes that contains xi and xj. Moreover,

min k h max Ak_X,(AT_X)t−k i ij = t−1 M k=1 h max Ak_X,(AT_X)t−k i ij = "_t−1 M k=1 maxAk_X,(AT_X)t−k # ij . (5.53)

Hence, we know that the left hand side of (5.49) contains the minimum cost of a loop of at mosttnodes containing nodesxi andxj and this is exactly the minimum multiple for which

we should multiply the cycle with unit dissimilarities t such that a dissimilarity reducing

map containingxi and xj in its image can be formed, which is the definition ofλt

X(xi, xj),

In order to show (5.48), first observe that the difference with (5.47) is that instead of minimizing for every k – recall that ⊕ represents minimization – we only consider the case k = 1. Thus, the right hand side of (5.48) must be greater than or equal to (5.47). Consequently, if we show that the every element of the right hand side matrix in (5.48) is not greater than its corresponding optimal multiple λt

X(xi, xj), we are done. By the

argument preceding (5.52) we know that [max AX,(ATX)t−1

]ij contains the minimum cost

of a loop containing xi and xj where the forward path from xi to xj consists of only one

link and the path in the opposite direction contains at mostt−1 nodes. Suppose that the minimum cost loop of at mosttnodes containingxi and xj is formed by the concatenation

of Pxixj = [xi =xi0, xi1, . . . , xil =xj] and Pxjxi = [xj =xi00, xi01, . . . , xi0l0 =xi] and its cost

is, by definition,λt

X (xi, xj). Focus on consecutive pairs of nodes in pathPxixj. It must be

that max AX,(AT_X)t−1_imim +1 ≤λ t X(xi, xj), (5.54)

form= 0, . . . , l−1. To see why (5.54) holds, note that the same concatenated loopPxixj] Pxjxi contains nodes xim and xim+1. Moreover, for these two nodes, the aforementioned

loop is formed by a forward path from xim toxim+1 of just one link and thus its cost must

be stored in max AX,(ATX)t−1

. Since (5.54) is true for everym, once we apply the (n−1) dioid power in (5.48), we are assured that the strong triangle inequality is satisfied. Hence,

h max AX,(ATX)t−1 n−1i ij ≤max m max AX,(ATX)t−1 imim+1≤λ t X(xi, xj), (5.55)

completing the proof.

Observe that when t= 2 (5.47) and (5.48) reduce to (3.76). This is as it ought to be given the equivalence HR _{≡ H}2 _{between the reciprocal and the first cyclic method. The}

reduction of (5.48) to (3.77) which corresponds to the equivalence HNR _{≡ H}C∞ _{is not as}

immediate but follows from recalling that in the dioid algebra limt→∞(AT_X)t−1 = (AT_X)n−1

followed by simple algebraic manipulations. Further observe that the expression in (5.48) is more efficient than (5.47) in terms of number of operations and memory requirements. Implementation of (5.47) requires computing and storing the dioid matrix powers Ak

X and

(AT_X)k for allk≤t−1 to compute the minimum indicated by the ⊕operation. Implemen- tation of (5.48) requires the computation of (AT_X)t−1 only. The latter expression is therefore preferable for implementation.

Remark 14 Its algorithmic handicap notwithstanding, we present (5.47) as an illustra- tion of Proposition 17 that constructs representable methods as the composition of a symmetrizing operation and single linkage clustering. Indeed, observe that the operation

Lt−1

k=1max AkX,(ATX)t−k

in (5.47) outputs a symmetric matrix because for anylthe terms max Al_X,(AT_X)t−land its transpose max At_X−l,(AT_X)lare both part of the dioid sum as it follows from substitutingk=landk=t−l in (5.47). We can then define the symmetriza- tion operation Λt _{: ˜N → N as the one that transforms the possibly asymmetric network} N = (X, AX) into the symmetric network Λt_{(X, AX) defined as}

Λt_{(X, A} X) = X, t−1 M k=1 maxAk_X,(AT_X)t−k ! . (5.56)

Further recalling that by (3.91) the (n−1) dioid power of a symmetric network computes the corresponding single linkage ultrametric it follows that (5.47) is equivalent to

(X, ut

X) =H

SL_◦Λt_{(X, AX), (5.57)}

which has the form in (5.35) of Proposition 17. The operator Λt _{symmetrizes possibly}

asymmetric network (X, AX), which is rendered into the ultrametric (X, ut

X) by the single

linkage operator HSL_{, [cf. Fig. 5.5]. Single linkage is the only admissible operator for this}

latter mapping as shown in Corollary 2.

The intuition behind (5.56) is the following: the matrix Ak_X stores the cost of the optimal forward paths of length at mostknodes and (AT

X)t−kstores the cost of the optimal

backward paths of length at most t−k nodes. Therefore, the componentwise maximum between these two matrices, max(Ak_X,(AT_X)t−k), corresponds to the optimal cost of cycles of length at most t nodes where the forward paths contain no more than k nodes. We then calculate the componentwise minimum over all k through the ⊕ operation in (5.56) to obtain the optimal cost of cycles of length at most tnodes for every length kof forward paths. Consequently, Λt_{(X, A}

X) is a symmetric network where the dissimilarity between xand x0 coincides with the minimum cost of a loop of length at most tnodes that contains nodesxandx0. The single linkage operation in (5.57) transforms this network of minimum cost loops with at mostt nodes into the ultrametric ut

X.

Remark 15 The family of cyclic methods includes reciprocal clustering HR _{≡ H}2 _and

nonreciprocal clustering HNR _{≡ H}C∞ _{as its extreme methods. The semi-reciprocal family}

of clustering methods introduced in Section 3.3 also harbored these methods as extreme. Despite this similarity, semi-reciprocal and cyclic methods are not equivalent in general. Semi-reciprocal methods are a family of intermediate methods from an algorithmic point of view whereas cyclic methods are an intermediate family from a generative perspective. In terms of influence propagation, the semi-reciprocal method HSR(t) _{allows influence to}

that each of the two paths contains no more than t nodes. In contrast, the cyclic method

Ht _{allows influence to propagate from node x to x}0 _{by concatenating intermediate loops}

of at most tnodes containingx and x0. I.e., for each of the intermediate pathsPxixi+1 and Pxi+1xi there is an upper bound on the total number of nodes in both paths but no bound

on the number of nodes in each of them.

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