By Gary F. Marcus
An try to combine theories approximately how the brain works, one who says that the brain is a computer-like manipulator of symbols, and one other that claims that the brain is a huge community of neurons operating jointly in parallel.
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Extra resources for Algebraic Mind: Integrating Connectionism and Cognitive Science
5x, f(x) = 2x, and so on. Such models cannot learn arbitrary mappings. ) As such they provide a candidate hypothesis for how operations over variables can be implemented in a neural substrate and not for a mental architecture that eliminates the representation of abstract relationships between variables. 4). When such a network represents identity or some other UQOTOM, it represents an abstract relationship between variables—which is to say that such a network implements an algebraic rule. Advocates of multilayer perceptrons might resist the claim that I am making here, for I am claiming that some multilayer perceptrons (such as the one in the left panel) implement—rather than eliminate—algebraic rules.
With respect to such an encoding, all cats are represented equivalently. There has been some confusion in the literature on this point. For example, people have talked about Elman’s sentence-prediction model as if it had context-dependent representations of its input words. But in fact, the input nodes are context-independent (the word cat always turns on the same node regardless of where in a sentence it appears), and the hidden nodes do not truly represent individual words; instead, the hidden units represent sentence fragments.
To sum them up, models that allocate a single node to each variable have (putting aside the worries about nonlinear activation functions and arbitrary representational schemes) no choice but to represent abstract relationships between variables, whereas models that allocate multiple nodes to each variable sometimes represent abstract relationships between variables and sometimes do not: what they represent is a function of what their connection weights are. In multiplenodes-per-variable multilayer perceptrons, some connection weights represent UQOTOM, others represent many-to-one mappings, and still others can represent purely arbitrary mappings.
Algebraic Mind: Integrating Connectionism and Cognitive Science by Gary F. Marcus