Finding similar images via tags - image

Question

tl;dr
Fundamentally I'm looking for reasonable ways to implement a similarity rank among tag groups, where a tag group is 2 to 9 tags. Similar to ranking the similarity of 2 to 9 word sentences where the vocabulary is 200,000 words, except word order doesn't matter.
I have a collection of tagged images and I want to implement a couple of search functions:
Similar images
Similar but different images
where the similarity is based solely on the tags.
Finding identically tagged images isn't that hard, but after that I'm at a bit of a loss as to the best way to proceed. We have a hundreds of thousands of tags and no metadata on them, so we don't know that "Outlook" is related to "Microsoft" or "Windows" or "Email" and therefore cannot appreciate the difference in relatedness of an image tagged "Microsoft,Excel,Bar Graph" to an image tagged "Excel,Spreadsheet" versus one tagged "Visio,Bar Graph".
For "Similar images" we'd want to match "Microsoft,Excel,Bar Graph" to "Visio,Bar Graph" while for "Similar but different images" we'd want to match "Microsoft,Excel,Bar Graph" to "Excel,Spreadsheet".
My best guess at the moment is to treat the tags like text and throw them into Solr. On the other hand, maybe a different kind of database, like Neo4j, would be the way to go.
Any suggestions on how to take a few steps forward? I'm not expecting a full solution, but suggestions for a general approach would be appreciated.
Extra Credit:
To make things more difficult, when tags are assigned to images, they are designated as "primary" or "secondary" and of course we want to take that into account.

Answer 1

Update
Let's repeat the problem.
The input data consists of sets of tags = set of strings (and pointers to the associated image resources)
The strings are just character sequences, there is no additional semantic information available
However there is a weighting of strings into 'primary' (higher weight) and 'secondary' (lower weight)
This means that the search has to rely solely on some similarity measure of sets (and strings).
Examples for such measures are:
Jaccard Similarity
Dice Similarity
Tverski Similarity
Cosine Similarity
This paper from 2010: A weighted tag similarity measure based on a collaborative weight model applies several of those (and others) on the tag problem and shows how to include weighting. That should be helpful, IMHO.
Another (simpler) application can be seen in this paper from 2013: Using of Jaccard Coefficient for Keywords Similarity.
About the examples from the question
For "Similar images" we'd want to match "Microsoft, Excel, Bar Graph" to "Visio, Bar Graph"
It would have some similarity due to that one tag ("Bar Graph") is common to both sets of tags.
while for "Similar but different images" we'd want to match "Microsoft, Excel, Bar Graph" to "Excel, Spreadsheet".
Again one tag in common ("Excel"). But how should the system know that "Visio" is more similiar to the set "Microsoft, Excel, Bar Graph" than "Spreadsheet"?
That would require semantic information. I don't see how to solve this otherwise.
Old Part
I found not much which would help you with your chosen approach (you restricted it quite a bit), except for a discussion of various metrics in the 2009 paper below.
But I would like to keep the steps of my little searching on this topic online here, because it puts your problem into context.
Where others go
The research community seems to go in these directions:
using the additional information about the users who provided the tags (folksonomies, social tagging)
making use of semantic meta data (ontologies, semantic similarity)
making use of visual image content (content-based visual information retrieval)
Folksonomnies, Social Tagging
See this paper from 2009: Evaluating Similarity Measures for Emergent Semantics of Social Tagging.
Instead of the traditional approach to define similarity by comparing the graphical data of the images
I = { (x, y, colour) }
by some measure (content-based image retrieval, query by image content, content-based-visual information retrieval), those authors use information (harvest semantics) from the tags, like you intend.
Their basic model consists of user assigned tags for a resource, comparing tuples of a so-called folksonomy
F = { (user, resource, tag) }
which can be scaled down to your case of (resource, tag) tuples by different approaches to aggregate over the users resulting in different similarity measures.
Semantic Similarity
Interesting is the use of semantic similarity, e.g. Jiang-Conrad, but alas, you have no semantic meta data (ontology e.g.) for your tags, which leaves you sticking to the similarity of the string representation of the words.
Again this paper The Use of Ontologies for Improving Image Retrieval and Annotation from 2008 favours the use of ontologies, but I think it gives a nice discussion of the various approaches.
Folksonomies, social tagging systems that relies on the idea of the
wisdom of the people. One representative example of this is Flickr.
com. This approach overcomes the so much time consuming of
manual annotation but the inconsistency in tag use can difficult the
search through the entire collection of data.
Combination with Content-Based Visual Information Retrieval
Both papers above cite this paper Augmenting Navigation for Collaborative Tagging with Emergent Semantics from 2006.
However, using tags alone for searching and browsing databases clearly has
its limitations. First, people make mistakes while tagging, such as spelling mistakes,
or accidental tagging with the wrong tag. Second, there is no solution to
cope with homonymy, i.e. to distinguish different meanings of a word. Third,
synonymy or different languages can only be handled by tagging data explicitly
with all terms.
These authors combine social tagging with the initially mentioned content-based image retrieval.
Yet another link: collaborative tagging.