Is Minimax Linkage a Lance-Williams hierarchical clustering?
$begingroup$
I found the following article on "Hierarchical Clustering With Prototypes
via Minimax Linkage".
It is stated in Property 6 that
Minimax linkage cannot be written using Lance–Williams updates.
A succinct proof using a counter-example is given:
Proof.
Figure 9 shows a simple one-dimensional example that could not
arise if minimax linkage followed Lance– Williams updates. The upper
and lower panels show two configurations of points for which the
right side of (4) is identical but the left side differs; in
particular, $d(G_1 cup G_2,H) = 9$ for the upper panel, whereas
$d(G_1 cup G_2,H) = 8$ for the lower panel.
But I do not understand their proof. For both cases (upper and lower panels), $d(G_1,H) = 16$, $d(G_2,H) = 7$, $d(G_1,G_2) = 5$.
I cannot see any reason that $alpha(G_2)$ in the first case must equal $alpha(G_2)$ in the second case. For instance, $G_2$ has not the same cardinal.
machine-learning clustering algorithms
edited 13 mins ago
Chaitanya Bapat
1318
asked Sep 2 '15 at 13:34
micmic
298213
$endgroup$
add a comment |
$begingroup$
I found the following article on "Hierarchical Clustering With Prototypes
via Minimax Linkage".
It is stated in Property 6 that
Minimax linkage cannot be written using Lance–Williams updates.
A succinct proof using a counter-example is given:
Proof.
Figure 9 shows a simple one-dimensional example that could not
arise if minimax linkage followed Lance– Williams updates. The upper
and lower panels show two configurations of points for which the
right side of (4) is identical but the left side differs; in
particular, $d(G_1 cup G_2,H) = 9$ for the upper panel, whereas
$d(G_1 cup G_2,H) = 8$ for the lower panel.
But I do not understand their proof. For both cases (upper and lower panels), $d(G_1,H) = 16$, $d(G_2,H) = 7$, $d(G_1,G_2) = 5$.
I cannot see any reason that $alpha(G_2)$ in the first case must equal $alpha(G_2)$ in the second case. For instance, $G_2$ has not the same cardinal.
machine-learning clustering algorithms
edited 13 mins ago
Chaitanya Bapat
1318
asked Sep 2 '15 at 13:34
micmic
298213
$endgroup$
add a comment |
$begingroup$
I found the following article on "Hierarchical Clustering With Prototypes
via Minimax Linkage".
It is stated in Property 6 that
Minimax linkage cannot be written using Lance–Williams updates.
A succinct proof using a counter-example is given:
Proof.
Figure 9 shows a simple one-dimensional example that could not
arise if minimax linkage followed Lance– Williams updates. The upper
and lower panels show two configurations of points for which the
right side of (4) is identical but the left side differs; in
particular, $d(G_1 cup G_2,H) = 9$ for the upper panel, whereas
$d(G_1 cup G_2,H) = 8$ for the lower panel.
But I do not understand their proof. For both cases (upper and lower panels), $d(G_1,H) = 16$, $d(G_2,H) = 7$, $d(G_1,G_2) = 5$.
I cannot see any reason that $alpha(G_2)$ in the first case must equal $alpha(G_2)$ in the second case. For instance, $G_2$ has not the same cardinal.
machine-learning clustering algorithms
edited 13 mins ago
Chaitanya Bapat
1318
asked Sep 2 '15 at 13:34
micmic
298213
$endgroup$
I found the following article on "Hierarchical Clustering With Prototypes
via Minimax Linkage".
It is stated in Property 6 that
Minimax linkage cannot be written using Lance–Williams updates.
A succinct proof using a counter-example is given:
Proof.
Figure 9 shows a simple one-dimensional example that could not
arise if minimax linkage followed Lance– Williams updates. The upper
and lower panels show two configurations of points for which the
right side of (4) is identical but the left side differs; in
particular, $d(G_1 cup G_2,H) = 9$ for the upper panel, whereas
$d(G_1 cup G_2,H) = 8$ for the lower panel.
But I do not understand their proof. For both cases (upper and lower panels), $d(G_1,H) = 16$, $d(G_2,H) = 7$, $d(G_1,G_2) = 5$.
I cannot see any reason that $alpha(G_2)$ in the first case must equal $alpha(G_2)$ in the second case. For instance, $G_2$ has not the same cardinal.
machine-learning clustering algorithms
machine-learning clustering algorithms
edited 13 mins ago
Chaitanya Bapat
1318
asked Sep 2 '15 at 13:34
micmic
298213
edited 13 mins ago
Chaitanya Bapat
1318
asked Sep 2 '15 at 13:34
micmic
298213
edited 13 mins ago
Chaitanya Bapat
1318
edited 13 mins ago
Chaitanya Bapat
1318
edited 13 mins ago
Chaitanya Bapat
1318
1318
asked Sep 2 '15 at 13:34
micmic
298213
asked Sep 2 '15 at 13:34
micmic
298213
asked Sep 2 '15 at 13:34
micmic
298213
298213
add a comment |
add a comment |
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