Using Inherent Structures to design Lean 2layer RBMs
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Proceedings of the 35th International Conference on Machine Learning, PMLR 80:443451, 2018.
Abstract
Understanding the representational power of Restricted Boltzmann Machines (RBMs) with multiple layers is an illunderstood problem and is an area of active research. Motivated from the approach of Inherent Structure formalism (Stillinger & Weber, 1982), extensively used in analysing Spin Glasses, we propose a novel measure called Inherent Structure Capacity (ISC), which characterizes the representation capacity of a fixed architecture RBM by the expected number of modes of distributions emanating from the RBM with parameters drawn from a prior distribution. Though ISC is intractable, we show that for a single layer RBM architecture ISC approaches a finite constant as number of hidden units are increased and to further improve the ISC, one needs to add a second layer. Furthermore, we introduce Lean RBMs, which are multilayer RBMs where each layer can have atmost O(n) units with the number of visible units being n. We show that for every single layer RBM with Omega(n^{2+r}), r >= 0, hidden units there exists a twolayered lean RBM with Theta(n^2) parameters with the same ISC, establishing that 2 layer RBMs can achieve the same representational power as singlelayer RBMs but using far fewer number of parameters. To the best of our knowledge, this is the first result which quantitatively establishes the need for layering.
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