Probing a theoretical framework for a Photonic Extreme Learning Machine

TL;DR

The paper addresses how to quantify and expand the expressive power of photonic extreme learning machines (PELM) by modeling the optical path with a transmission-matrix formalism. It develops a four-stage theoretical framework (input encoding, linear scattering propagation, nonlinear hidden-layer detection, and linear digital readout) and derives analytic bounds on the hidden-space dimensionality for amplitude- and phase-modulation schemes, linking these bounds to universal approximation capabilities. The authors validate the theory with experiments in diffusive media under low-dimensional inputs, using singular-value decomposition and Weyl-thresholds to separate signal from noise and to assess how detector nonlinearity (via exposure) enriches the hidden space. The results show that linear detection imposes fundamental limits on expressivity, while increasing detector nonlinearity can partially mitigate these limits, offering practical guidance for designing energy-efficient all-optical hardware and potential edge-sensing integrations.

Abstract

The development of computing paradigms alternative to von Neumann architectures has recently fueled significant progress in novel all-optical processing solutions. In this work, we investigate how the coherence properties can be exploited for computing by expanding information onto a higher-dimensional space in the photonic extreme learning machine framework. A theoretical framework is provided based on the transmission matrix formalism, mapping the input plane onto the output camera plane, resulting in the establishment of the connection with complex extreme learning machines and derivation of upper bounds for the hidden space dimensionality as well as the form of the activation functions. Experiments using free-space propagation through a diffusive medium, performed in low-dimensional input space regimes, validate the model and the proposed estimator for the dimensionality. Overall, the framework presented and the findings enclosed have the potential to foster further research in a multitude of directions, from the development of robust general-purpose all-optical hardware to a full-stack integration with optical sensing devices toward edge computing solutions.

Figures (5)

• Figure 1: Overview of the optoelectronic PELM experimental implementation and connection to the theoretical framework. A. The information of each input state $\boldsymbol{X}$ is encoded on the incident optical beam in phase or amplitude using an SLM. B. The beam is then focused on an optical media (here a diffusive media) inducing a random transmission matrix $\boldsymbol{\bar{M}}$ modeling the propagation before being collimated again resulting in a speckle patter then collected with a CMOS sensor, resulting in the hidden layer representation $\boldsymbol{Y}\left(\boldsymbol{X}\right)$. C. The imaged speckle pattern is then used as input to the linear model $\boldsymbol{O}(\boldsymbol{X}) = \boldsymbol{Y}\left(\boldsymbol{X}\right) \boldsymbol{\bar{\beta}}^*$, obtaining a prediction $\boldsymbol{O}\left( \boldsymbol{X}\right)$ by leveraging on a $\boldsymbol{\beta}^*$ trained with Ridge regression using the hidden layer output matrix $\boldsymbol{\bar{H}}$ built from $N_D$ training samples.
• Figure 2: Regression results. (a) Train and validation RMSE versus ridge parameter $\lambda$ for amplitude (top row) and phase (bottom row) encoding under low (left) and high (right) exposure settings. Gray bars above each plot show the singular value spectrum of the hidden layer such that $\sigma^2 = \lambda$, the red band marks the Weyl noise threshold. The green marker denotes the $\lambda$ value chosen for testing. (b) Single fold predictions for the low and high exposure settings compared with the ground truth sinc function (dashed black) for amplitude (top row) and phase (bottom row) modulations. A green edge denotes the test points while the training points have no edge. For both modulation settings, the low exposure configuration showcases the limited expressivity of the model to the underlying encoding functions while the high exposure nonlinearity enriches the effective hidden space, resulting in higher expressivity, as reflected in an improved fit between prediction and target function.
• Figure 3: Classification results. Low and high exposure versus amplitude and phase encoding table with each entry showing an accuracy versus $\lambda$ plot. The green marker denotes the optimal $\lambda$ used for evaluation. Alongside each plot is an example of the predictions of the train (no edges) and test (green edge) data points from a single fold. Correct predictions are marked with a dot, while incorrect predictions have a triangular shape. The background of these figures is the prediction map over a $32\times 32$ grid showcasing the prediction field used to analyse the generalisation capabilities of the model.
• Figure 4: Singular value spectra and Weyl threshold for the regression configuration. In the left amplitude encoding spectra for low and high exposure, identifying $3$ singular values above the Weyl threshold (highlighted region). Similar setup on the right for phase encoding we identify $5$ and $8$ singular values above the threshold.
• Figure 5: Singular value spectra and Weyl threshold for the classification configuration. In the left amplitude encoding spectra for low and high exposure, identifying $5$ and $6$ singular values above the Weyl threshold (highlighted region). Similar setup on the right for phase encoding we identify $7$ and $22$ singular values above the threshold.