MC has pointed out that we have convenient transportation to my house is ambiguous. He has shown another possible meaning. (only to the house but not from the house). Ambiguity is one major reason we will change an English sentence, rephrase it to something more clear.
On the other hand when you say, "public transportation is convenient to my house", it is an idiomatic way of saying it is close to my house. The substitution of "is close to" for "is convenient to" happens and there is no confusion.
Thanks, Avig, for the clear explanation.
By the way, if it's daily speech, is it acceptible to say "we have convenient transportation to my house?" It seems to me that "from my house" is understood in the expression.