## 3 Proposed Methodology
### Sentence Representation
#### 3.1.1 Normalized Representation of Sentence Vector  
In this category, normalization techniques for representing a sentence as a feature vector are discussed. In the following expressions, \(w_{i}\) represents the word vector of _ith_ word and \(n\) is the count of words in the given sentence (s), where \(L1\) and \(L2\) are the mathematical norm of vectors.  
(1) Averaging Difference of Word Vectors.  
\(Avg_{sub}\), is the method used to represent sentence \(S\ =\ (w_{1},\ w_{2},\ldots,w_{n})\), by taking the difference of all the word vectors and dividing by the total number of words count \(n\) in a sentence as shown in Equation (1):  
\[Avg_{Sub}=\frac{\overrightarrow{w_{1}}-\sum_{i=2}^{n}\overrightarrow{w_{i}}}{n} \tag{1}\]  
(2) \(L1\)-Normalized Sum of Average Word Vectors.  
Figure 1: System architecture of aspect category detection (ACD).  
The \(L1\)-norm of average word vector is used to represent a sentence as a feature vector as per Equation (2):  
\[L1-AvgSOW=\frac{1/n\sum_{i=1}^{n}\overrightarrow{w_{i}}}{\left|\left|1/n\sum_{i=1 }^{n}\overrightarrow{w_{i}}\right|\right|} \tag{2}\]  
(3) \(L1\)-Normalized Difference of Average Word Vectors.  
This method is the opposite of the method mentioned as (2). The sum operator is replaced with a difference operator:  
\[L1-AvgDOW=\frac{1/n\Big{(}w_{1}-\sum_{i=2}^{n}\overrightarrow{w_{i}}\Big{)}} {\left|\left|1/n\Big{(}w_{1}-\sum_{i=2}^{n}\overrightarrow{w_{i}}\Big{)} \right|\right|} \tag{3}\]  
(4) \(L2\)-Normalized Average Difference of Word Vectors.  
This method is similar to the L2-NAV (sum) proposed in [30]; here, we have used subtraction of the word vectors instead of addition in a sentence to represent a feature vector of a sentence:  
\[L2-AvgDOW=\frac{1/n\Big{(}w_{1}-\sum_{i=2}^{n}\overrightarrow{w_{i}}\Big{)}} {\left|\left|1/n\Big{(}w_{1}-\sum_{i=2}^{n}\overrightarrow{w_{i}}\Big{)} \right|\right|^{2}} \tag{5}\]  
(5) \(L1\)-Normalized Sum of Word Vectors.  
In this method all the word vectors in a sentence are summed and normalized by \(L1\)-norm. \(L1\)-norm is used to scale vector features in such a way that sum must be equal to 1:  
\[L1-SOW=\frac{\sum_{i=1}^{n}\overrightarrow{w_{i}}}{\left|\left|\sum_{i=1}^{n} \overrightarrow{w_{i}}\right|\right|} \tag{6}\]  
(7) \(L2\)-Normalized Sum of Word Vectors.  
This method all the word vectors in a sentence are summed and normalized by the \(L2\)-norm:  
\[L2-SOW=\frac{\sum_{i=1}^{n}\overrightarrow{w_{i}}}{\left|\left|\sum_{i=1}^{n} \overrightarrow{w_{i}}\right|\right|^{2}} \tag{8}\]  
(7) \(L1\)-Normalized Difference of Word Vectors.  
This method is the opposite to (5). Subtraction of word vectors is used instead of addition:  
\[L1-DOW=\frac{\overrightarrow{w_{1}}-\sum_{i=2}^{n}\overrightarrow{w_{i}}}{ \left|\left|\overrightarrow{w_{1}}-\sum_{i=2}^{n}\overrightarrow{w_{i}} \right|\right|^{2}} \tag{9}\]  
(8) \(L2\)-Normalized Difference of Word Vectors.  
This is the opposite method of (6). Here we use subtraction of word vectors instead of addition:  
\[L2-DOW=\frac{\overrightarrow{w_{1}}-\sum_{i=2}^{n}\overrightarrow{w_{i}}}{ \left|\left|\overrightarrow{w_{1}}-\sum_{i=2}^{n}\overrightarrow{w_{i}} \right|\right|^{2}} \tag{10}\]