Why does suicide risk increase just after starting an antidepressant?
This comes from talking to a psychiatrist about the issue:

One of the effects of depression is a loss of motivation.  In some suicidal individuals, this has the result that they want to die, but can't work up the motivation to do anything about it.  Once they start taking antidepressants, however, the lack of motivation goes away before the depression does, creating a short period of increased suicide risk.