Receptron

The receptron (short for "reservoir perceptron") is a neuromorphic data processing model — specifically neuromorphic computing — that generalizes the traditional perceptron, by incorporating non-linear interactions between inputs.¹²³ Unlike classical perceptron, which rely on linearly independent weights, the receptron leverages complexity in physical substrates,⁴ such as the electric conduction properties of nanostructured materials or optical speckle fields, to perform classification tasks.⁵⁶ The receptron bridges unconventional computing and neural network principles,⁷ enabling solutions that do not require the training approaches typical of artificial neural networks based on the perceptron model.⁸

The receptron is an algorithm for supervised learning of binary classifiers, so a classification algorithm that makes its predictions based on a predictor function, combining a set of weights with the feature vector.⁹ The mathematical model is based on the sum of inputs with non-linear interactions:

${\displaystyle S=\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})|S\in R}$      (1)

where ${\displaystyle j\in [1,n]}$ and ${\displaystyle {\widetilde {w}}_{j}}$ are non-linear weight functions depending on the inputs, ${\displaystyle {\vec {x}}}$. Nonlinearity will typically make the system extremely complex, and allowing for the solution of problems not solvable through the simpler rules of a linear system, such as the perceptron or McCulloch Pitts neurons, which is based on the sum of linearly independent weights:¹⁰

${\displaystyle S=\sum _{k=1}^{n}x_{j}w_{j}^{p}}$     (2)

where ${\displaystyle w_{j}}$are constant real values. A consequence of this simplicity is the limitation to linearly separable functions, which necessitates multi-layer architectures and training algorithms like backpropagation¹¹

As in the perceptron case,¹² the summation in Eq. 1 origins the activation of the receptron output through the thresholding process,

${\displaystyle Y(x_{1},...,x_{n})={\begin{cases}1&{\text{if }}S>{\text{th}}\\0&{\text{if }}S\leq {\text{th}}\end{cases}}}$ (3)

where th is a constant threshold parameter. Equation 3 can be written by using the Heaviside step function.

The weight functions ${\displaystyle {\widetilde {w}}({\vec {x}})}$ can be written with a finite number of parameters ${\displaystyle w_{j_{1}...j_{n}}}$, simplifying the model representation. One can Taylor-expand ${\displaystyle {\widetilde {w}}({\vec {x}})}$ and use the idempotency of Boolean variables ${\displaystyle (x_{j})^{q}=x_{j}\forall q\geq 1}$ such that ${\displaystyle S'=b+\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})}$ can be written as

${\displaystyle S'({\vec {x}})=b+\sum _{j}w_{j}x_{j}+\sum _{j<k}w_{jk}x_{j}x_{k}+\sum _{j<k<l}w_{jkl}x_{j}x_{k}x_{l}+...}$ (4)

where ${\displaystyle w_{j_{1}...j_{n}}}$ are independent parameters that can be seen as the components of a tensor ${\displaystyle W}$ (“weight tensor”) of rank ${\displaystyle n}$ and type ${\displaystyle (n,0)}$.

The sum in Eq. [3] reduces to the perceptron case when off-diagonal terms of ${\displaystyle W}$ vanish. If one considers ${\displaystyle n=2}$, one gets:

${\displaystyle S'({\vec {x}})=b+x_{1}w_{11}+x_{2}w_{22}+x_{1}x_{2}w_{12}}$ (5)

in the perceptron case, the vanishing of ${\displaystyle w_{12}}$ implies linearity ${\displaystyle S(1,1)=S(0,1)+S(1,0)}$. In the receptron case ${\displaystyle S(1,1)\neq S(0,1)+S(1,0)}$, meaning that the superposition principle is no longer valid, the latter terms being responsible of the more complex non-linear interaction between the inputs.

Substrate: Nanostructured and nanocomposite films (Au, Pt, Zr Au/Zr). These films form disordered networks of nanoparticles with resistive switching and non-linear electrical conduction.

Substrate: Optical speckle fields generated by random interference of light emerging from a disordered medium illuminated by a laser or coherent radiation.¹³

Physical Substrate Computing: The receptron does not require digital training; instead, it exploits the natural complexity of materials (e.g., nanowire networks, diffractive media) to perform computations.

Non-Linear Separability: Unlike traditional perceptrons, which fail on problems like the XOR function, the receptron can solve such tasks due to its inherent non-linearity.

Training-Free Operation: Classification is achieved through the physical system's response rather than iterative weight adjustments, reducing computational overhead.