Fig. 4: Input multiplicity allows overcoming reduced degrees of freedom and increases multi-class capacity.
From: Limits on the computational expressivity of non-equilibrium biophysical processes
A Plot of the learned classification functions π1(F), π2(F), π3(F), and π4(F) for the network shown in Fig. 1E, with only the solid arrows used for inputs (M = 1). B Schematic illustration of how the learnable parameters Wij (the edge rates on the left) are first multiplied within directed spanning trees into products called directed tree weights, which are then summed together to yield the polynomial functions \({\zeta }_{\mu }^{i}({{\boldsymbol{\theta }}})\) appearing in Equation (2). Although each function \({\zeta }_{\mu }^{i}({{\boldsymbol{\theta }}})\) is uniquely defined, there exist equality constraints owing to the physics of Markov networks which reduce the effective number of degrees of freedom below the number needed to solve the four-class classification task in A. C By including driving along the dashed arrows in Fig. 1E (setting M = 2), there are sufficiently many degrees of freedom to solve the four-class classification task.
