# Parallel Extreme Learning Machines Based on Frequency Multiplexing
## 4 Results
### Single Operation Mode  
To test the capability of the system to employ all the available combs for a single computation, the experiment was executed using \(N_{S}=3\) input light sources at wavelengths \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\). We tested the performances of the system on the mushroom classification task [30] that consists in selecting the correct class between two possible ones, given a set of 21 different input features (note that the dataset originally contained 22 features, but one of them assumes a constant value across all the entries and has thus been discarded). According to the description in Section 3.3, we encoded nine input features in the most central part of each of the three input combs. The feature order was selected at random, making sure that each feature appears at least once, and that six features appears twice, in two different combs. Results are reported in Figure 4. The accuracy of the classification is \(95.1\pm 0.4\%\), where the score is the average measured across 100 different random repartitions of the training and the testing set, and the error is the standard deviation of such distribution of scores. A software ELM composed of 90 hidden neurons with quadratic nonlinearity reached an accuracy of \(99.2\pm 0.4\%\), where the score was evaluated as described above. A simulation of this multi-comb scheme reached an accuracy of \(98.6\pm 0.6\%\), where the score was the average measured across 100 possible redistributions of the 21 features in the three combs, and the error was the standard deviation of such distribution of scores. Figure 3: Results of parallel operations mode tests. \(d\) indicates the number of consecutive comb lines on which each feature is encoded. The same task types with different \(d\) values are executed in parallel on two different combs. Each result is compared with the numerical simulation and previous experiments of a single-comb ELM [19]. Error bars represent the standard deviation of the score over 100 random repartitions of training and testing data (more information in Section 4.1). A photonic ELM exploiting free-space propagation in a scattering medium [8] scored an accuracy of 95.4% on this same task. To better investigate this single operation scheme, we conducted a test via numerical simulation on the MNIST handwritten digit classification task [31], which could not be implemented on a single-comb system since it required too many neurons. The only preprocessing of the dataset consisted in a resampling of the input images: we submitted to the network images composed of \(17\times 17\) pixels, corresponding to 289 input features. We simulated a system similar to the current one, with the only difference that it contained \(N_{S}=10\) light sources, each one generating a comb, resulting in approximately 300 input neurons and 300 hidden ones. The simulation reached an accuracy of \(90.0\pm 0.2\%\). When we simulated the same system with \(N_{S}=15\), which corresponds to 450 input neurons and 450 hidden ones, the accuracy increased to \(92.1\pm 0.2\%\). Note that our present system with \(N_{S}=3\) light sources has \(31\times 3=93\) comb lines, which is not enough to encode the 289 input features; however, the scenarios with \(N_{S}=10\) or \(N_{S}=15\) light sources are potentially realisable experimentally (see Section 5). A software ELM reported in [32] was tested on the same classification task, with no image resampling. When executed with 784 input neurons, 784 hidden neurons, and a tanh nonlinearity, this ELM reached an accuracy of 92.8% (the accuracy could be increased up to approximately 98%, increasing the number of hidden neurons by 20 times). These results clearly show that merging multiple independent combs in a single ELM is a viable way of tackling tasks that require a high number of neurons.