The following is written on the Wikipedia entry of Taxicab Geometry:
The [taxicab] geometry has been used in regression analysis since the 18th century, and is often referred to as LASSO.
Unfortunately, no references are given on this point. I would appreciate if someone could explain this connection or point me to a reference.
“Taxicab geometry” refers to the $\ell_1$ norm in linear algebra or functional analysis, where, loosely speaking, you only travel left/right or up/down (never diagonal), kind of like how a taxicab drives on a the roads of a city with a grid layout (think of the dominant pattern in Manhattan).
LASSO regression uses the $\ell_1$ norm of the parameter vector (often without the intercept), giving the connection to taxicab geometry.
Taxicab geometry is also commonly referred to as Manhattan distance as Manhattan (a subdivision of New York, USA) has an exceptional number of roads which meet at right angles, and uses a numerical based indexing system (see https://www.etsy.com/listing/224325979/manhattan-map-prints-and-manhattan-wall for an example of how angular the roads are).
The reason it's called Taxicab distance is this is the distance a taxi would need to drive and since it's all right angles and easily indexed, taxis could usually pre-compute the approximate cost of taking you places in Manhattan.
LASSO is used to refer to regularizing using Manhattan distance. Personally, I believe this is a backronym because a Lasso is a rope knot which shrinks to hold things tight (usually livestock) making it an easy mnemonic to remember that it shrinks the parameter magnitude (and often number of parameters)
