Science Limelight

  Imagine a chain of linear mappings each selected by something. If the same mapping was selected each time that would be boring. Matrices such as A and B can be collapsed together to a single matrix M by matrix multiply, M=AB. And M is actually very different from its parents A and B. If you select A from a pool of 10 and likewise B you have 100 different possible M's from 20 matrices. They are like rabbits aren't they? If you have 10 of A and B and C you have 1000 different matrices from 30. This is exponentiation of expressiveness with depth similar to a neural network. The something that is doing the matrix selection at each layer you can call the context. And that can be liberal in meaning. That's nice but what if you now apply pressure from backpropagation on the matrix chain? For a data channel input x and the co-selected linear mapping (eg A₂B₇C₅...) backpropagation drifts each chosen linear mapping to be an expert for the slice of the training data it sees. Further...

  Linear Mappings Have a Memory I've been looking at something surprisingly simple: what happens when a linear mapping is treated explicitly as a linear associative memory . One particularly interesting observation is that when a weighted sum is trained beyond capacity, new examples don't simply erase old ones. Instead, the weight vector is continually perturbed, gradually adding noise to previously stored associations. Recent examples are recalled better, but older examples can persist statistically for a surprisingly long time. Even more interestingly, below capacity, removing a training example may produce no change whatsoever in the weight vector—until weight decay is introduced. Then the mapping can move to a different, lower-norm solution while preserving the remaining associations. These elementary observations have some interesting consequences for initialization, weight decay, SGD dynamics, and ultimately CCSLM, where the local experts can themselves be viewed as fact...

The core mathematics of concatenated context selected linear mappings https://archive.org/details/core-ccslm

One for the skeptics but it is not argued as well as possible. It still retains some old fashioned views of neural networks. https://docs.google.com/document/d/e/2PACX-1vRLqFOuoGC3jS9HqJebDW5PHscznODG5XaCrsVN7MmoMRVkDl2r5Iu8sEjLpoN6vuH-PoAer0M1BT1w/pub

Context-Only CCSLM sets the data input to a constant vector, such as (x=all 1's), so that all computational variation is controlled by context . y=Lₙ(Cₙ)... L₂(C₂)L₁(C₁)1 The finite set of context states visited during training becomes the primary organizing structure. Each context selects specialized linear mappings, making context effectively a computational address . This is particularly interesting for robotics and other stateful systems where useful extrinsic context—task, mode, gait phase, contact state, environment, time, etc.—may exist without any sensible analog data vector to supply. It also provides a clean experimental setting for studying context topology, state transitions, specialization, capacity, and computational structure largely independently of conventional input-driven learning.

CCSLM: Context as Parameter Assignment for Expertise Central idea The main role of context is not to determine the output . The main role of context is to assign regions of parameter space in which expertise can grow . Context determines which parameters participate in a particular portion of the learning problem. Repeated exposure to similar contexts causes those parameters to specialize. From context to expertise Context selects or modulates parameters. Training repeatedly adjusts the selected parameters. Over time, particular parameter regions become specialized. These regions can be regarded as future experts . Each expert is fundamentally a linear mapping and linear associative memory . Context is control, not computation Context provides a control pathway. It does not itself transform the data. The selected parameters perform the data transformation. Consequently, the final output is entirely the result of the selected linear mappings and their concatenation. There is no separate...