Using some existing math libraries, though. The point is, that there is literally no research in this field in my language (Georgian). Is that possible? How long would I take? I know that this also depends on the skills, but still?
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Also answered at
Speech to text conversion for non-english language
You do not need to write engine from scratch, there are many engines already available, you can just pick one like CMUSphinx
http://cmusphinx.sourceforge.net
If you just interested in supporting Georgian, it's just a task to train Georgian model, you do not need to implement engine itself. Speech recognition engines do not depend on langauge.
Luckily there is a lot in English.
It takes about a month of single man work to create a database for a new language for CMUSphinx. For more details see the documentation:
http://cmusphinx.sourceforge.net/wiki/tutorial (Tutorial)
http://cmusphinx.sourceforge.net/wiki/tutorialam (Acoustic model training tutorial chapter)