Using Statistical Mechanics to understand depth in Deep Networks

Understanding the representational power of Restricted Boltzmann Machines (RBMs) with multiple layers is an ill-understood 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.

We introduce Lean RBMs, which are multi-layer RBMs where each layer can have at-most 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 two-layered lean RBM with Theta(n^2) parameters with the same ISC, establishing that 2 layer RBMs can achieve the same representational power as single-layer 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.