Signature Selection for Grouped Features with a Case Study on Exon Microarrays - Feature Selection for Data and Pattern Recognition - page 330

On the other hand, suppose that

ʲ G k =

0 is optimal. From ( 14.10 ), we get

ʻ(

1

− ʱ)
ʲ G k 2 +

1

s k − ʻʱ g G k = ʲ G k .

Here, we note that ( ʲ G k ) j = 0 is optimal whenever | ( s k ) j |≤ ʻʱ since it satisfies an

equality in the multiple equations above. Therefore we have

ʻ(

1

− ʱ)
ʲ G k 2 +

1

soft ( s k ,ʻʱ) = ʲ G k .

This gives that ʲ G k 2 = soft ( s k ,ʻʱ) 2 − ʻ( 1 − ʱ) . Replacing ʲ G k 2 above and

rearranging terms, we get

1 soft

ʻ( 1 − ʱ)

ʲ G k = − ( s k ,ʻʱ).

soft ( s k ,ʻʱ) 2

Combining with the condition in ( 14.11 ), we obtain the claim.

Theorem 2 tells that there are two possibilities for a component ʲ j , j ∈ G k ,to

have the zero value:
• (

s k ,ʻʱ) ≤ ʻ

( · ) + part of the expression of ʲ G k inTheorem 2

soft : in this case, the

ʲ G k ʲ j ) becomes the

becomes zero, and therefore the entire subvector (including

zero vector. If this is the case, the group G k is not selected,
•

soft ( s k ,ʻʱ) >ʻ and | ( s k ) j |≤ ʻ : in this case the group G k is selected, but

the component j ∈ G k is not selected because ( soft ( s k ,ʻʱ)) j = 0, and therefore

0.
Comparing to Theorem 1 for group lasso in Sect. 14.2.1 , group lasso has a similar
property to the first one above for groupwise selection, but lacks the second property
for within group feature selection.

ʲ j =

14.3 A Case Study on Exon Microarray Data

As a case study of selecting grouped features, we consider identifying alternative
splicing of genes from high-throughput genomic data. We use a specific type of data
acquired by using the GeneChip Human Exon 1.0 ST Arrays from Affymetrix, 2 a
popular platform for profiling gene expression of entire human genome in exon level.
For background information, a gene is a sequence inDNA (deoxyribonucleic acid)
composed of four letters A, T, G, and C, and it has coding regions (called exons) and
non-coding regions (called introns). After its sequence is copied into a messenger
RNA (ribonucleic acid), only coding regions are combined together and later used to

2 http://www.affymetrix.com