Question: How is the fluctuation in the sense of the word and its neighbors measured?
Evidence: Distributed representation of word sense provides us with the ability to perform several operations on the word. One of the most important operations on a word is to obtain the set of words whose meaning is similar to the word, or whose usage in text is similar to the word. We call this set the neighbor of the word. When a word has several senses, it is called a polysemic word. When a word has only one sense, it is called a monosemic word. We have observed that the neighbor of a polysemic word consists of words that resemble the primary sense of the polysemic word. We can explain this fact as follows. Even though a word may be a polysemic, it usually corresponds to a single vector in distributed representation. This vector is primarily determined by the major sense, which is most frequently used. The information about a word's minor sense is subtle, and the effect of a minor sense is difficult to distinguish from statistical fluctuation.
To measure the effect of a minor sense, this paper proposes to use the concept of surrounding uniformity. The surrounding uniformity roughly corresponds to statistical fluctuation in the vectors that correspond to the words in the neighbor. We have found that there is a difference in the surrounding uniformity between a monosemic word and a polysemic word. This paper describes how to compute surrounding uniformity for a given word, and discuss the relationship between surrounding uniformity and polysemy.
We choose the uniformity of vectors, which can be regarded as general case of triangle inequality. The uniformity of a set of vectors is a ratio, i.e., the size of the vector of the vector addition of the vectors divided by the scalar sum of the sizes of the vectors. If and only if all directions of the vectors are the same, the uniformity becomes 1.0. We compute this uniformity for the neighbors, including the word itself. Surrounding Uniformity (SU) can be expressed as follows: $SU(\vec{w}) = \frac{|\vec{s}(\vec{w})|}{|\vec{w}| + \sum _{i}^{N}|\vec{a_i}(\vec{w})|}$
where $\vec{s}(\vec{w}) = \vec{w} + \sum _{i}^{N} \vec{a_i}(\vec{w}).$
One of the most important operations on a word is to obtain the set of words whose meaning is similar to the word, or whose usage in text is similar to the word. We call this set the neighbor of the word.
We have observed that the neighbor of a polysemic word consists of words that resemble the primary sense of the polysemic word.
 Even though a word may be a polysemic, it usually corresponds to a single vector in distributed representation. This vector is primarily determined by the major sense, which is most frequently used. The information about a word's minor sense is subtle, and the effect of a minor sense is difficult to distinguish from statistical fluctuation.
To measure the effect of a minor sense, this paper proposes to use the concept of surrounding uniformity.
The surrounding uniformity roughly corresponds to statistical fluctuation in the vectors that correspond to the words in the neighbor
Surrounding Uniformity (SU) can be expressed as follows: $SU(\vec{w}) = \frac{|\vec{s}(\vec{w})|}{|\vec{w}| + \sum _{i}^{N}|\vec{a_i}(\vec{w})|}$

where $\vec{s}(\vec{w}) = \vec{w} + \sum _{i}^{N} \vec{a_i}(\vec{w}).$
Answer: Our method performs a statistical test to determine whether a given word is used polysemously in the text, according to the following steps:
1) Setting N, the size of the neighbor.
2) Choosing N neighboring words ai in the order whose angle with the vector of the given word w is the smallest.
3) Computing the surrounding uniformity for ai(0 < i ≤ N) and w.
4) Computing the mean m and the sample variance σ for the uniformities of ai .
5) Checking whether the uniformity of w is less than m − 3σ. If the value is less than m − 3σ, we may regard w as a polysemic word.