Fig. 2: Classification with nonlinear mapping.

a, Training data from the Fashion MNIST datasets are used to train a one-layer neural network as a digital decoder for classification tasks. Additionally, the percentage of the modulated area on the DMD is changed among 6.25%, 25% and 100% to adjust the order of nonlinear mapping. With full (100%) modulation of DMD, the nonlinear order is further enhanced by covering the output port with a partial reflector (silicon wafer). b, Fashion MNIST classification results with a linear classifier are presented under different numbers of output modes (speckle grains) and varying nonlinear strengths. The optical linear features with quadratic detection are simulated by scattering from a single layer with intensity detection to create a quadratic nonlinear response. Note that a linear regression for binarized Fashion MNIST data cannot exceed 77.6% with the same number of modes. c,d, Violin plots representing the distributions of mutual information between the speckle grains and classification targets under varying numbers of output modes (c) and differing orders of nonlinear mapping by changing the modulated area on the DMD or partially closing the cavity (enhanced) (d). For n speckle mode (n on the x axis), 4n replicated measurements from the same input were performed in c and d. The dashed line plots depict the median values of the mutual information. Each violin’s width reflects the distribution of the mutual information values of the speckle grains and its probability density. Within each violin, the slim black vertical line represents the range of minimum and maximum values; the black box represents the first to third percentile; the white dot represents the median. c, Mutual information analysis when the number of output modes (speckle grains) varies under the highest-order nonlinear mapping. d, Mutual information analysis with low-dimensional speckle features (four output modes) for Fashion MNIST as a function of the nonlinear orders varied by modulated area on the DMD, showing the advantage of going to higher-order nonlinear mapping.