TESI DOCTORAL - La Salle

λ 1 λ 11 λ 12 λ 13 … λ 1m
λ 2 λ 21 λ 22 λ 23 … λ 2m
λ 3 λ 31 λ 32 λ 33 … λ 3m
λ 4 λ 41 λ 42 λ 43 … λ 4m
λ 5 λ 51 λ 52 λ 53 … λ 5m
…
λ l λ l1 λ l2 λ l3 … λ lm
Chapter 3. Hierarchical consensus architectures
Consensus
function
(a) Flat construction of a consensus clustering solution
on a hard cluster ensemble
1 11 12 13 … 1m Consensus 1
function 1
2 21 22 23 … 2m c 1
3 31 32 33 … 3m
4 41 42 43 … 4m
5 51 52 53 … 5m
l l1 l2 l3 … lm
Consensus
function
Consensus
function
1
c 2
1
c K K1
Consensus
ffunction i
Consensus
function
λ c
1
s
c1 1
1
s
cKs Consensus
function
(b) Hierarchical construction of a consensus clustering solution on a hard cluster ensemble
Figure 3.1: Flat vs hierarchical construction of a consensus clustering solution on a hard
cluster ensemble.
– recursively solves these subproblems
– appropriately combines their outcomes
Transferring this strategy to the consensus clustering problem is equivalent to segmenting
the cluster ensemble into subsets (referred to as mini-ensembles hereafter), building
an intermediate consensus solution upon each mini-ensemble, and subsequently combining
these halfway consensus clusterings into the final consensus clustering solution λc —see
figure 3.1(b). Due to the fact that successive layers (or stages) of consensus solutions are
created, we give this approach the name of hierarchical consensus architecture (HCA), as
opposed to the traditional flat topology of consensus clustering processes.
The rationale of hierarchical consensus architectures is pretty simple. By reducing the
time and space complexities of each intermediate consensus clustering –which is achieved by
creating it upon a smaller ensemble–, we aim to reduce the overall execution requirements
(i.e. memory and specially, CPU time), although a larger number of low cost consensus
clustering processes must be run. However, this strategy is capable of yielding computational
gains, as for large enough values of l, the execution of the original problem becomes
slower than the recursive execution of the subproblems into which it is divided (Dasgupta,
Papadimitriou, and Vazirani, 2006).
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c