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Dry Cleaning Carpet
There are several variations of this method. It's worth noting that none of these involves rinsing your carpet when finished. One uses a foam that dries in your carpet and is then vacuumed up. It is debatable whether all the residue can be extracted.
Another variation uses a large cotton bonnet that spins. This is supposed to absorb the dirt in your carpet. Obviously, there are some questions as to how deeply this method can clean your carpet.
Finally, there is a compound that looks like wet sawdust. A machine brushes this into your carpet, where the dirt adheres to it. When it is dry, it is vacuumed out.
The primary reason these methods are used may be that the equipment needed costs thousands of dollars less than hot-water extraction equipment.