Stoltz, Dustin S., and Marshall A. Taylor. 2019. “Concept Mover’s Distance.” Journal of Computational Social Science 2(2):293-313.

  title={Concept Mover's Distance},
  author={Stoltz, Dustin S and Taylor, Marshall A},
  journal={Journal of Computational Social Science},

Load text2map

Install and load the text2map package:

Document-Term Matrix

To use the Concept Mover’s Distance function (CMDist) , you will need to transform your corpus into a document-term matrix (DTM). The preferred DTM is a sparse matrix in Matrix format (class “dgCMatrix”) as output by text2vec, Quanteda, udpipe, and tidytext’s cast_sparse function, but we have tried to make the package accommodate DTMs made with the tm package (technically a “simple_triplet_matrix”) or just a regular old base R dense matrix (see this for a comparison of DTM creation methods). You can also use text2map’s super fast unigram DTM builder dtm_builder().

Word Embedding Matrix

You will also need a matrix of word embedding vectors (with the “words” as rownames), and ultimately, CMDist is only as good as the word embeddings used. Word embedding vectors can be from a pre-trained source, for example:

It will take a little data wrangling to get these loaded as a matrix in R with rownames as the words (feel free to contact us if you run into any issues loading embeddings into R), or you can just download these R-ready fastText English Word Vectors trained on the Common Crawl (crawl-300d-2M.vec) hosted on Google Drive:

You can also download the file directly into your R session using the following:

library(googledrive) # (see
temp <- tempfile()
drive_download(as_id("17H4GOGedeGo0urQdDC-4e5qWQMeWLpGG"), path = temp, overwrite = TRUE)
my.wv <- readRDS(temp)
# save them to your project file so you don't have to re-download
saveRDS(my.wv, "data/")

Instead of pretrained embeddings, you can create your own embeddings trained locally on the corpus on which you are using CMDist. For example, the text2vec R package provides a nice workflow for using the GloVe method to train embeddings. As we discuss in our original paper, the decision to use pre-trained vs local corpus-trained is understudied as applied in the social-scientific context. Pretrained embeddings are a simple way to get an analysis started, but researchers should always be critical about the corpus used to train these models.

One important caveat: the terms used to denote a concept or build a semantic direction need not be in the corpus, but it must be in the word embeddings matrix. If it is not, the function will stop and let you know. This means, obviously, that corpus-trained embeddings cannot be used with words not in the corpus (pre-trained must be used). As the fastText method can be trained on “subword” character strings, it is possible to average the character strings that make up an out-of-vocabulary term.

Measuring Conceptual Engagement

Terms Denoting Focal Concepts

The most difficult and important part of using Concept Mover’s Distance is selecting anchor terms. This should be driven by (a) theory, (b) prior literature, (c) domain knowledge, and (d) the word embedding space. One way of double-checking that selected terms are approriate is to look at the term’s nearest neighbors. Here we use the sim2 function from text2vec to get the cosine distance between “thinking” and its top 10 nearest neighbors.


cos_sim <- sim2(x = my.wv, 
                y = my.wv["thinking", , drop = FALSE], 
                method = "cosine")

head(sort(cos_sim[,1], decreasing = TRUE), 10)

Single Word

Once you have a DTM and word embedding matrix, the simplest use of CMDist involves finding the closeness to a focal concept denoted by a single word. Here, we use the word “thinking” as our concept word (“cw”):

doc_closeness <- CMDist(dtm = my.dtm, cw = "thinking", wv = my.wv)

Compound Concepts

However, we may be interested in specifying the concept with additional words: for example, we might want to capture “critical thinking.” To handle this, it is as simple as including both words separated by a space in the quotes. This creates a pseudo-document that contains only “critical” and “thinking.”

doc_closeness <- CMDist(dtm = my.dtm, cw = "critical thinking", wv = my.wv)


What if instead of a compound concept we are interested in a common concept represented with two words (i.e., a bigram)? First, just like with any other word, the ngram must be in the embedding matrix (the fastText pre-trained embeddings have a lot of n≥1 grams). As long as the ngram is in the embeddings, the only difference for CMDist is that an underscore rather than a space needs to be placed between words:

doc_closeness <- CMDist(dtm = my.dtm, cw = "critical_thinking", wv = my.wv)

Semantic Directions and Centroids

In the original JCSS paper, we discussed the “binary concept problem,” where documents that are close to a concept with a binary opposite will likely also be close to both opposing poles. For example if a document is close to “love” it will also be close to “hate.” But, very often an analyst will want to know whether a document is close to one pole or the other of this binary concept. To deal with this we incorporate insights from Kozlowski et al’s (2019) paper “Geometry of Culture” to define “semantic directions” within word embeddings – i.e. pointing toward one pole and away from the other. We outline this procedure in more detail in a subsequent JCSS paper “Integrating Semantic Directions with Concept Mover’s Distance to Measure Binary Concept Engagement”.

The procedure involves generating a list of antonym pairs for a given binary concept – that is, terms which may occur in similar contexts but are typically juxtaposed. Then we get the differences between these antonyms’ respective vectors, and averaging the result (with get_direction()). The resulting vector will be the location of one “pole” of this binary concept, and CMDist calculates the distance each document is from this concept vector (“cv”). As this creates a one-dimensional continuum, documents that are far from this “pole” will be close to the opposing “pole.”

# first build the semantic direction:
additions  <- c("death", "casualty", "demise", "dying", "fatality")
substracts <- c("life", "survivor", "birth", "living", "endure")

pairs <- cbind(additions, substracts) <- get_direction(pairs, my.wv)

# input it into the function just like a concept word:
doc_closeness <- CMDist(dtm = my.dtm, cv =, wv = my.wv)

Instead of building a pseudo-document with several terms, as in the compound concept section above, one could also average several terms together to arrive at a “centroid” in the word embedding space – this can be understood as referring to the center of a “semantic region.” Again, CMDist will calculate the distance each document is from this concept vector (“cv”).

# first build the semantic centroid:
terms  <- c("death", "casualty", "demise", "dying", "fatality") <- get_centroid(terms, my.wv)

# input it into the function just like a concept word:
doc_closeness <- CMDist(dtm = my.dtm, cv =, wv = my.wv)

Other Options and Considerations

Multiple Distances at Once

An analysis might suggest multiple concepts are of interest, and so it is useful to get multiple distances at the same time. This, in effect, is adding more rows to our pseudo-document-term matrix. For example, in our JCSS paper, we compare how Shakespeare’s plays engage with “death” against 200 other concepts. CMDist also incorporate both concept words and vectors into the same run.

# example 1
concept.words <- c("thinking", "critical", "thought")
doc_closeness < CMDist(dtm = my.dtm, cw = concept.words, wv = my.wv)

# example 2
concept.words <- c("critical thought", "critical_thinking")
doc_closeness < CMDist(dtm = my.dtm, cw = concept.words, wv = my.wv)

# example 3
concept.words <- c("critical thought", "critical_thinking")
doc_closeness < CMDist(dtm = my.dtm, cw = concept.words, cv = concept.vectors, wv = my.wv)

Performance and Parallel CMDist

Calculating CMD (as of version 0.5.0) relies on Linear Complexity RWMD or LC-RWMD. The performance gain over the previous RWMD implementation is considerable. Nevertheless, it may be desirable to boost performance further on especially large projects.

One way to reduce complexity and thus time (without a noticeable drop in accuracy) is by removing very sparse terms in your DTM. Parallelizing is another option, so we built it in. To use parallel calculations just set parallel = TRUE. The default number of threads is 2, but you can set it as high as you have threads/cores.

doc_closeness <- CMDist(dtm = my.dtm, 
                      cw = "critical_thinking", 
                      wv = my.wv, 
                      parallel = TRUE, 
                      threads = 2)

Scaling Output and Vector Comparison Metric

By default, the outputs are normalized using the scale() function in R. If this is not desired, set scale = FALSE.

 doc_closeness <- CMDist(dtm = my.dtm, cw = "thinking", wv = my.wv, scale = FALSE)

CMDist relies text2vec which currently uses cosine as the distance metric to solve the underlying transportation problem. Kusner et al. (2015) and Atasu et al. (2017) papers used Euclidean distance to compare word embeddings vectors. Cosine and Euclidean are mathematically similar (and in many cases identical). In all of our comparisons of Concept Mover’s Distance using Euclidean and cosine the results were nearly identical.

Note About Linear Complexity Relaxed Word Mover’s Distance

Rather than the Relaxed Word Mover’s Distance (RWMD) as discussed in Kusner et al’s (2015) “From Word Embeddings To Document Distances”, text2vec uses the Linear-Complexity Relaxed Word Mover’s Distance (LC-RWMD) as described by Atasu et al. (2017). LC-RWMD not only reduces the computational demands considerably, if we take the maximum of each document pair across the upper and lower triangles (i.e., the maximum of each [i,j]-[j,i] pair), the resulting symmetric distance matrix is precisely the same as the previous algorithm using Kusner et al’s approach. As far as our tests have shown, the triangle corresponding to treating the pseudo-document as a “query” and the documents as a “collection” (see the text2vec documentation, p. 28) will always return this set of maximums, and therefore there is no reason for the additional comparison step. If you have reason to believe this is not the case, please contact us and we will share the code necessary to reproduce results with the original Kusner et al. approach to check.

For more discussion of the math behind Concept Mover’s Distance see Stoltz and Taylor (2019) “Concept Mover’s Distance” in the Journal of Computational Social Science.