The easiest way would probably be to make your own constructors and link the data by hand. You can use mapping if you really want to, but you'll basically have to do the same manual linking, only in a more verbose format.
This is the fiddle with the example implementation: http://jsfiddle.net/aKpS9/3/
The most important part of the code is the linking, you have to take care to create the activity objects only once, and use the same objects everywhere, as opposed to creating new activity objects for the parts.
var TrainingSession = function(rawData, actualActivities){
var self = this;
self.name = ko.observable(rawData.name);
self.parts = ko.observableArray(ko.utils.arrayMap(rawData.parts, function(rawPart){
return ko.utils.arrayFirst(actualActivities(), function(ac){
return ac.ID() == rawPart.ID;
})
}));
}
var Activity = function(rawData){
var self = this;
self.ID = ko.observable(rawData.ID);
self.name = ko.observable(rawData.name);
}
var MainVM = function(rawData){
var self = this;
//first create an array of all activities
self.activities = ko.observableArray(ko.utils.arrayMap(rawData.activities, function(rawAc){
return new Activity(rawAc);
}));
self.trainingSessions = ko.observableArray(ko.utils.arrayMap(rawData.trainingSessions, function(session){
return new TrainingSession(session, self.activities);
}));
}