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Word embedding is one of the most important components in natural language processing, but interpreting high-dimensional embeddings remains a challenging problem. ICA-transformed word embeddings reveal interpretable semantic axes; however, the order of these axes are arbitrary. In this study, we focus on this property and propose a novel method, Axis Tour, which optimizes the order of the axes. Inspired by Word Tour, a one-dimensional word embedding method, we aim to improve the clarity of the word embedding space by maximizing the semantic continuity of the axes. Furthermore, we show through experiments on downstream tasks that Axis Tour yields better or comparable low-dimensional embeddings compared to both PCA and ICA. Embedding is an important tool in natural language processing, but interpreting high-dimensional embeddings is challenging. In this study, inspired by a one-dimensional word embeddings, Word Tour Sato , which leverages the Traveling Salesman Problem TSP , we aim to improve the clarity of the word embedding space by maximizing the semantic continuity of the axes. In Axis Tour, the top words of the axes are far from the center, with the meanings of the axes changing continuously. Conversely, in Skewness Sort, the top words are nearer to the center, and the axes with different meanings are placed adjacently. In fact, the average distance from the origin to the top words in Fig. We also assume that the consecutive axes in the Axis Tour embeddings can be considered as a subspace whose axes have similar meanings. Based on this, we project each subspace onto a single dimension for dimensionality reduction. Some studies transform embeddings by rotation Park et al. TICA relaxes the assumption of statistical independence and assumes higher-order correlations between adjacent axes. TICA then estimates the order of the axes. The resulting one-dimensional embeddings have similar meanings if they are close in order. This section explains Axis Tour and the dimensionality reduction method using Axis Tour. As mentioned in Section 3. We define axis embedding for use in Word Tour. As we saw in Fig. This approach then maximizes the sum of cosine similarities between adjacent axis embeddings. The sum of cosine similarities between adjacent axis embeddings can be considered as a metric of the semantic continuity of the axes. Thus, Axis Tour determines the order of the axes by maximizing this metric. For more details, refer to Appendix B. Helsgaun as the TSP solver. We computed the axis embeddings 6 6 6 See Appendix E. For baselines, we used whitened PCA -transformed embeddings 7 7 7 Whitened ICA-transformed embeddings are obtained by applying an orthogonal matrix to these embeddings. See Appendix E for additional experiments, including those of other embeddings. Table 1 shows the illustrative examples of the meaning of countries , law , and art by the consecutive axes of the Axis Tour embeddings. These examples illustrate how the meanings of the axes change continuously. For example, in the top row, the axis meaning changes from Eastern Europe to Germany and France , followed by Canada which shares a connection with France , then to Australia English-speaking regions , the regions in England , and finally to soccer a popular sport in England , demonstrating geographic and cultural continuity. This result is consistent with the formulation in 4. In Fig. However, no such trend is observed in Fig. This indicates that in Axis Tour, the axes with higher skewness are more likely to be ordered adjacent to the axes with higher similarity. Using Word Embedding Benchmark Jastrzebski et al. PCA selects the axes in descending order of eigenvalue. Random Order and Skewness Sort select the axes sequentially from the first to last. Axis Tour adopts the dimensionality reduction 11 11 11 Fig. This result suggests that the dimensionality reduction for Axis Tour efficiently merges the axes with similar meanings. For more details, refer to Appendix D. In this study, we proposed a novel method, Axis Tour, which optimizes the order of axes in ICA-transformed word embeddings. We focus on the fact that the word embeddings reveal interpretable semantic axes while the order of these axes are arbitrary. Axis Tour aims to improve the clarity of the word embedding space by maximizing the semantic continuity of the axes. We also showed through experiments on downstream tasks that Axis Tour yields better or comparable low-dimensional embeddings compared to both PCA and ICA. While the dimension reduction experiments showed the improvement of the downstream task performance for the Axis Tour embeddings, there are three aspects that could be further improved:. In addition, nonlinear transformations beyond linear ones could be considered for dimension reduction. The method in Section 4. However, adaptively determining the division points could allow selecting more semantically coherent groups of axes. In this case, the overall optimized axis order may not be determined as in Axis Tour, but performing Axis Tour within each cluster and then concatenating these could determine an axis order depending on the number of clusters. However, this study focuses on a method for maximizing the semantic continuity of axes in ICA-transformed embeddings, leaving detailed investigation of the effective low-dimensional vector as future work. In Axis Tour, while adjacent axes may have similar meanings, axes with similar meanings may not be in close order. This is due to the fact that in Word Tour, high-dimensional embeddings result in one-dimensional embeddings, and the meanings of words are similar when the word order is close, but semantically similar words are not always embedded close to each other. As seen in Fig. However, as the number of axes increases, the angles between the axes become small, resulting in crowded axes. This can cause problems such as the top words of the axes being closer to the origin, which can be difficult to interpret. Therefore, as the dimension of the embeddings increases, the computation time for Axis Tour becomes longer. Note that for the dimensional GloVe used in this study, the computation time for Axis Tour is about one second. A potential risk of this method is that we interpret the meanings of the axes of the ICA-transformed embeddings by the top words of each axis. If the embeddings contain personal information, such as email addresses or phone numbers, and these are contained in the top words, this can be problematic. Therefore, in this study, URLs, email addresses, and phone numbers were anonymized to avoid revealing such information. We would like to thank Momose Oyama for the discussion. This section explains the two-dimensional projection method used for the scatterplots in Fig. We then present similar scatterplots for the three examples in Table 1. Finally, we define the metrics used in Section 1 to evaluate the quality of the scatterplots. We will explain the scatterplot drawing method using the Axis Tour embeddings. First, we define a set of axis indices for projection. For example, in Fig. Similarly, we can apply the same procedure to the Skewness Sort embeddings and obtain the two-dimensional scatterplot. Similar to Fig. In Section 1 , we compared the quality of the scatterplots for Axis Tour and Skewness Sort by calculating the average distance of the top words from the origin. In this section, we first explain this metric and then, based on 4 , define a new metric derived from the average of the cosine similarities between adjacent axis embeddings. We then compare these metrics for the scatterplots in Figs. Similar to Appendix A. In addition to the results for several subintervals in Table 2 , we also see the semantic continuity of all axes. This section details and supplements Section 4. This is based on the assumption that the axis becomes more meaningful as the skewness increases 14 14 14 While not specific to ICA-transformed embeddings, it is known from a study of sparse coding for language models that skewness correlates with interpretability Cunningham et al. This section explains the projection from a subspace to a one-dimensional space using a specific example. Consider the subspace spanned by three consecutive axes 89, 90, 91 from Fig. The projection direction is in the direction representing the subspace, and the top words of each axis are projected close together. This is easily shown as follows. PCA Rand. All Analogy capital-common-countries 0. We used MEN Bruni et al. In the word similarity tasks, the quality of the embeddings is evaluated by measuring the cosine similarity of the word embeddings and comparing it to the human-rated similarity scores. In the categorization tasks, the quality of the embeddings is evaluated by clustering them in the setting where each word is assigned a class label. As the evaluation metric, we used Purity, which shows the proportion of the most frequent class in the clusters. As already seen, Fig. The Axis Tour embeddings showed superior performance in the word similarity tasks and the categorization tasks for almost all dimensions compared to other methods. First, as we saw in Appendix B. Then, by definition, Axis Tour, Random Order, and Skewness Sort are the embeddings obtained by reordering the axes of the ICA-transformed embeddings and flipping their signs as needed. Thus, these three can be seen as the embeddings obtained by applying an orthogonal matrix to the ICA-transformed embeddings. In Table 1 , we used dimensional GloVe and showed the semantic continuity of the axes of the Axis Tour embeddings, with illustrative examples of the meaning of countries the 23 23 23 23 rd axis to the 31 31 31 31 st axis , law the st axis to the th axis , and art the th axis to the th axis. This section presents the top five words of the normalized embeddings across all axes in Tables 4 , 5 ,and 6. For example, in Table 4 , the 45 45 45 45 th and 46 46 46 46 th axes are related to soccer , the 47 47 47 47 th axis to golf , the 48 48 48 48 th axis to tennis , the 49 49 49 49 th and 50 50 50 50 th axes to scores , the 51 51 51 51 st axis to American football , the 52 52 52 52 nd axis to basketball , and the 53 53 53 53 rd axis to baseball. These axes illustrate the semantic continuity across sports. In Table 5 , the th to the th axes are related to numbers. It is also interesting to note that the top words of each axis have similar numerical scales, highlighting how well the axis of the ICA-transformed embeddings captures meaning. In addition, the meaning of each axis changes continuously, much like a game of word association: the th axis relates to colors , the th axis to light , the st axis to space , the nd axis to airplanes , the rd axis to ships , the th axis to storms , the th axis to weather , the th axis to biomes , the th axis to plants. In Table 6 , the nd to the th axes are related to personal names from different linguistic regions as if this were a cluster of the meaning. Note that due to space limitations, the top 1 and top 3 words on the th axis are truncated because they are repetitive symbols, and that URLs, email addresses, and phone numbers are anonymized. From Fig. In this case, it impossible to find the order that maximizes the semantic continuity of the axes. Abstract Word embedding is one of the most important components in natural language processing, but interpreting high-dimensional embeddings remains a challenging problem.

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Ongoing investigations The heads of several cartels have been arrested in the United States and Mexico in recent months, and they are due to stand trial soon. Aliaume Leroy. Likes Subscribers Followers.

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