Explanation
Principal component analysis (PCA) involves a mathematical procedure that transforms a number of (possibly) correlated variables into a (smaller) number of uncorrelated variables called principal components. The first principal component accounts for as much of the variability in the data as possible: and each succeeding component accounts for as much of the remaining variability as possible.
Objectives of principal component analysis
1. To discover or to reduce the dimensionality of the data set.
2. To identify new meaningful underlying variables.

Explanation
Regularization is a very important technique in machine learning to prevent overfitting. Mathematically speaking, it adds a regularization term in order to prevent the coefficients to fit so perfectly to overfit. The difference between the L1 and L2 is just that L2 is the sum of the square of the weights, while L1 is just the sum of the weights.
Much of optimization theory has historically focused on convex loss functions because they're much easier to optimize than non-convex functions: a convex function over a bounded domain is guaranteed to have a minimum, and it's easy to find that minimum by following the gradient of the function at each point no matter where you start. For non-convex functions, on the other hand, where you start matters a great deal; if you start in a bad position and follow the gradient, you're likely to end up in a local minimum that is not necessarily equal to the global minimum.
You can think of convex functions as cereal bowls: anywhere you start in the cereal bowl, you're likely to roll down to the bottom. A non-convex function is more like a skate park: lots of ramps, dips, ups and downs. It's a lot harder to find the lowest point in a skate park than it is a cereal bowl.

Explanation
One scenario of collaborative filtering application is to recommend interesting or popular information as judged by the community. As a typical example, stories appear in the front page of Digg as they are "voted up" (rated positively) by the community. As the community becomes larger and more diverse, the promoted stories can better reflect the average interest of the community members.

Explanation
SGD-based classifiers avoid the need to predetermine vector size by simply picking a reasonable size and shoehorning the training data into vectors of that size. This approach is known as feature hashing. The shoehorning is done by picking one or more locations by using a hash of the name of the variable for continuous variables or a hash of the variable name and the category name or word for categorical, text*like, or word-like data.
This hashed feature approach has the distinct advantage of requiring less memory and one less pass through the training data, but it can make it much harder to reverse engineer vectors to determine which original feature mapped to a vector location. This is because multiple features may hash to the same location. With large vectors or with multiple locations per feature, this isn't a problem for accuracy but it can make it hard to understand what a classifier is doing.
An additional benefit of feature hashing is that the unknown and unbounded vocabularies typical of word-like variables aren't a problem.