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Errata

on

Robust Perron Cluster Analysis

in Conformation Dynamics

by Peter Deuflhard and Marcus Weber

Susanna Kube and Peter Deuflhard

December 5, 2006

Abstract

In the recent paper [2], one of the authors presented an O(?2) pertur-

bation result for the eigenvectors of a generalized symmetric stochastic

matrix, where the parameter ? characterized the departure of the per-

turbed matrix from a completely decomposable Markov chain. Due to

some erroneous interchange of indices, the proof has turned out to be in-

correct. Here, we give the corrected results.

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Introduction

In the recent paper [2], a new cluster algorithm for nearly completely decom-

posable Markov chains had been introduced, the Robust Perron Cluster Cluster

Analysis (named PCCA+). Upon characterizing the departure from completely

decomposable chains by ?, the robustness of the method had been justified by

an O(?2) perturbation result for the eigenvectors of a generalized symmetric

stochastic matrix. Unfortunately, the proof of that result has turned out to be

incorrect, due to some unlucky interchange of indices. However, it can still be

shown that the O(?) bound for metastability, which was originally derived for

a hard {0,1}-clustering (PCCA), carries over to PCCA+. In the following, we

indicate where the error in the previous proof had occurred and how the results

must be corrected. In order to facilitate the orientation, we kept the names of

the sections from [2].

1Perron cluster eigenproblem

Nearly uncoupled Markov chains.

which preceded the proof of the perturbation result.

Denote the number of nearly uncoupled Markov chains by k. In this case the

transition matrix?T will be block diagonally dominant after suitable permuta-

eigenvalues

?λ1= 1,

scale as

? = 1 −?λ2.

?T(?) = T + ?T(1)+ O(?2) ,

and for the Perron cluster eigenvectors? X = [? X1,...,? Xk] ∈ RN×kas

? Xi(?) = Xi+ ?X(1)

In [1], the result

Let us shortly repeat the assumptions

tion. As a perturbation of the k-fold Perron root λ = 1, a Perron cluster of

?λ2= 1 − O(?),...

?λk= 1 − O(?) ,

will arise, where ? > 0 denotes some perturbation parameter, which we here

(1.1)

Let formal ?-expansions be introduced for the stochastic matrix as

(1.2)

i

+ O(?2) .

(1.3)

X(1)

i

=

k

?

?

j=1

bjiχj

???

(I)

+

N

?

j=k+1

?

1

1 − λjΠjT(1)Xi

???

(II)

(1.4)

has been obtained using projections Πj as defined in the book of Kato [4].

Obviously, the term (I) represents just shifts of the locally constant levels to

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be associated with the almost invariant sets. In [2], the term (II) had wrongly

been dropped. Thus, the corresponding Lemma 2.1 therein, which claimed that

X(1)= χB, must be abandoned.

In order to see the mistake, let us shortly revisit the lines at the “proof”. Start-

ing from

?T(?)? Xi(?) =?λi(?)? Xi(?) ,

TXi= Xi

for

i = 1,...,N

and inserting the ?-expansions leads to

i = 1,...,k ,

(1.5)

and

T(1)Xi= (I − T)X(1)

i

− Xiδλi,i = 1,...,k .

(1.6)

Hence, for j = k + 1,...,N and i = 1,...,k, one obtains

?Xj,T(1)Xi?π

= ?Xj,(I − T)X(1)

= ?(I − T)Xj,X(1)

i

− δλiXi?π

?π− δλi?Xj,Xi?π.

i

The last term above vanishes due to the π-orthogonality of the unperturbed

eigenvectors. Using TXj= λjXjresults in

?Xj,T(1)Xi?π= (1 − λj)?Xj,X(1)

i

?π.

The last line is the place where the error occurred in the proof. Due to an

index permutation it was assumed that TXj = Xj which is only satisfied for

j = 1,...,k but not for j = k + 1,...,N.

With the above result, equation (9) from [1] for i = 1,...,k reads correctly

Xi(?) =

k

?

j=1

? αijχj+ ?

N

?

j=k+1

1

1 − λjΠjT(1)Xi+ O(?2)(1.7)

and can thus be rewritten as

Xi(?) =

k

?

j=1

? αijχj+ ?

i

N

?

j=k+1

?Xj,X(1)

i

?πXj+ O(?2).

(1.8)

In general, the terms ?Xj,X(1)

?Xj(?)? = 1. However, if the perturbation matrix T(1)has a special structure,

the result can be improved. This is verified by the following corollary.

?πXj are of order O(1) due to normalization

Corollary 1.1 Under the modeling assumption that T(1)inherits the nearly

completely decomposable structure of T, i.e.

T(1)= κT + O(?)(1.9)

with some constant parameter κ, (1.7) simplifies to

Xi(?) =

k

?

j=1

? αijχj+ O(?2).

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Proof: Replacing T(1)with (1.9) leads to

?ΠjT(1)Xi= ?κΠjTXi+ O(?2) = ?κΠjXi+ O(?2) = O(?2),i = 1,...,k,

due to the orthogonality of the unperturbed eigenvectors. Insertion into the

second summand of (1.7) yields the result.

Thus, the O(?2) perturbation result for the eigenvectors would still be valid.

Equivalently to (1.9), one can write

?

(T(i,j) − T(1)(i,j))/T(i,j) = (1 − κ) + O(?),

Thus, T(1)inherits the structure of T if the element-wise relative error has

nearly the same size for all elements of T.

∀i,j = 1,...,N.

Robustness of the PCCA approach. Although it cannot be shown that

the constant level pattern of the eigenvectors are perturbed in only O(?2), the

newly proposed algorithm PCCA+ is nevertheless more robust than the older

method based on the sign structure. This is due to the fact that it avoids the

generic “dirty zero” problem by allowing the occurrence of transition states.

2Almost characteristic functions

As pointed out in [2], Huisinga and Schmidt [3] had already shown an O(?) lower

bound for the measure of metastability in the case of a strict {0,1}-clustering.

The following theorem shows that the result carries over to the framework of

almost characteristic functions. However, in contrast to the earlier proposition

in [2], this bound cannot be improved. The proof is essentially the same as

before, but only uses

? π = π + O(?).

most characteristic functions ? χ =? X?

is satisfied. Then metastability can be bounded in terms of the perturbation

parameter ? via

k

?

The proof is based on the representation of the coupling matrix

Theorem 2.1 Let?Λ = diag(?λ1,...,?λk). Assume that for a feasible set of al-

Θ = ??

A the inequality

A−1?Λ?

A − Ik?1< 1(2.1)

i=1

?λi− O(?) ≤

k

?

i=1

wii<

k

?

i=1

?λi.

(2.2)

?

W =?D−2?? χ,?T ? χ?π.

AT?? X,?T? X?π?

(2.3)

Proof: Upon reformulating

?? χ,?T ? χ?π=?

A =?

AT?? X,? X?Λ?π?

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A =?

AT?? X,? X?π?Λ?

A

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and using π-orthogonality of the eigenvectors? X we arrive at

?

W =?D−2?

AT?? X,? X?π?

W =?D−2?? χ, ? χ?π

AT?Λ?

A .

(2.4)

Moreover, based on the relation

?? χ, ? χ?π=?

?

A =?

?

AT?

A ,

we may derive the alternative expression

?

???

S

A−1?Λ?

A

? ?? ?

M

.

(2.5)

By a short calculation, the above matrix S = (Sij) can be shown to be stochastic,

which implies that?k

Θ = ?M −Ik?1< 1, where ?·?1denotes the maximum column sum norm; this

implies that Mii> Mjifor j ?= i.

With these properties, the upper bound in (2.2) can be directly verified as

follows:

j=1Sij= 1. The matrix M = (Mij) is obviously spectrally

similar to the eigenvalue matrix?Λ. By assumption, M satisfies the condition

k

?

i=1

wii=

k

?

i=1

k

?

j=1

SijMji<

k

?

i=1

(

k

?

j=1

Sij)Mii=

k

?

i=1

Mii=

k

?

i=1

?λi.

In order to verify the lower bound in (2.2), let D2= diag(π1,...,πk) and observe

that D2= ?χ,χ?π. With this preparation, we derive the perturbation pattern

of the matrix S as:

S =?D−2?? χ, ? χ?π=?D2+ O(?)?−1(?χ,χ?π+ O(?)) = Ik+ O(?) .

tr(?

and therefore confirms (2.2) in particular, which completes the proof.

Note that?Λ(?)|?=0= Ik so that for sufficiently small perturbation parameter

Insertion into the expression (2.5) then immediately yields

W) = tr(SM) = tr(M) + O(?) = tr(?Λ) + O(?) .

This result applies for both the upper and the lower bound of the metastability

?

the above inequality (2.1) will be satisfied.

Conclusion

In contrast to the recent paper [2], the O(?2) perturbation result for eigenvectors

has to be replaced by O(?) bounds. Fortunately, the cluster algorithm PCCA+

maintains its validity because it is not based on this result. However, we still

do not have a satisfactory theoretical explanation for the simplex structure of

the perturbed eigenvectors. This question will be in the focus of future work.

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