All of those use fundamentally use back propagation though, they just approximate the gradients. Pretraining with autoencoders even used back propagation to train the autoencoders. And researchers have entirely moved away from that strategy because it just doesn't work as well as pure supervised learning.
I don't think backpropagation will ever go away because it is so simple and powerful. Instead it will be tweaked and approximated like in your examples.
The paper presents these and other approaches as being instances of a classical, general form, regularized regression but with the "stacked" property involving each layer iterating however many times and then the next layer changing parameters (or features) and further iterating.
From my barely-informed viewpoint, this sounds like a fascinating way to unify the earlier efforts and one which could yield a variety of other approaches - even if the particular variation they use doesn't work out. But I assume lots of more-informed people are going to be looking at this.
But if this works as advertised, these guys must be doing something new.
If the layers are learned locally, can it take into account interactions between layers?
If the claim holds, the implication is that layers can quickly learn much of what they need to learn locally, that is, without requiring backpropagation of gradients from potentially very distant layers. I can't help but wonder if this opens the door for more efficient asynchronous/parallel/distributed training of layers, potentially leading to models that update themselves continuously (i.e., "online" instead of in a batch process).
I wouldn't be surprised if the claim holds. There is mounting evidence that standard end-to-end backpropagation is a rather inefficient learning mechanism. For example, we now know that deep neural nets can be trained with approximate gradients obtained by shifting bits to get the sign and order of magnitude of the gradient roughly right.[1] In some cases it's even possible to restrict learning to use binary weights.[2] More recently, we have learned that it's possible to use "helper" linear models during training to predict what the gradients will be for each layer, in-between true-gradient updates, allowing layers to update their parameters locally during backpropagation.[3] Finally, don't forget that in the late 2000's, AI researchers were doing a lot of interesting work with unsupervised layer-wise training (e.g., DBNs composed of RBMs, stacked autoencoders).[4]
This is a fascinating area of research with potentially huge payoffs. For example, it would be really neat if we find there's a "general" algorithm via which layers can learn locally from inputs continuously ("online"), allowing us to combine layers into deep neural nets for specific tasks as needed.
[1] https://arxiv.org/abs/1510.03009
[2] https://arxiv.org/abs/1602.02830
[3] https://deepmind.com/blog#decoupled-neural-interfaces-using-...
[4] https://www.iro.umontreal.ca/~lisa/pointeurs/TR1312.pdf
EDITS: Expanded the original comment so it conveys better what I actually meant to write, while keeping language as casual and informal as possible. Also, I softened the tone of my more speculative observations.