Fast physical repetitive patterns generation for masking in time-delay reservoir computing

Albeit the conceptual simplicity of hardware reservoir computing, the various implementation schemes that have been proposed so far still face versatile challenges. The conceptually simplest implementation uses a time delay approach, where one replaces the ensemble of nonlinear nodes with a unique nonlinear node connected to a delayed feedback loop. This simplification comes at a price in other parts of the implementation; repetitive temporal masking sequences are required to map the input information onto the diverse states of the time delay reservoir. These sequences are commonly introduced by arbitrary waveform generators which is an expensive approach when exploring ultra-fast processing speeds. Here we propose the physical generation of clock-free, sub-nanosecond repetitive patterns, with increased intra-pattern diversity and their use as masking sequences. To that end, we investigate numerically a semiconductor laser with a short optical feedback cavity, a well-studied dynamical system that provides a wide diversity of emitted signals. We focus on those operating conditions that lead to a periodic signal generation, with multiple harmonic frequency tones and sub-nanosecond limit cycle dynamics. By tuning the strength of the different frequency tones in the microwave domain, we access a variety of repetitive patterns and sample them in order to obtain the desired masking sequences. Eventually, we apply them in a time delay reservoir computing approach and test them in a nonlinear time-series prediction task. In a performance comparison with masking sequences that originate from random values, we find that only minor compromises are made while significantly reducing the instrumentation requirements of the time delay reservoir computing system.


Discussion on noise effects of the SL-OF system
The presence of laser noise, as well as other noise sources and instabilities that may be present in possible physical implementations of the SL-OF system, can make it impossible to observe some of the dynamical responses of this system. For example, period-4 (P4) or even period-2 (P2) dynamics that are found numerically to exist in this system, might not be possible to observe experimentally. In this study we have considered a low-noise laser emission, which is introduced by a value of D = 3ns -1 in the Lang-Kobayashi model (Methods). At this laser noise level, we can observe clearly the gradual increase of the number of frequency tones (e.g. Figure 2, points (b), (c) and (d)). Moreover the fully-deployed frequency tones appear always in a wide {rc,φc} parameter space. When increasing the laser noise parameter D by one order of magnitude (30ns -1 ) the gradual increase of the number of frequency tones (points (b), (c) and (d) in Figure 2) can be hardly observed. However, the parameter space {rc,φc} for which we observe the fully-deployed 11 frequency tones is still very wide. Regarding the conditions for which we observe the integer relation between the high order and the first frequency tones (as shown in Figure 3b), this is also significantly reduced. In this case, the feedback parameters of the SL-OF system must be defined with higher precision in order to obtain repetitive patterns without any periodicity drift.

Pattern repetitions for large reservoirs
In the presented analysis we select a feedback delay time of τ = 200ps and a given set of parameters for describing the laser operation. These result in the generation of repetitive patterns of τe = 226ps duration for the pre-selected feedback conditions. When sampling this pattern with 50 sample values, this physically defines a sampling distance of 4.52ps. Thus, for a reservoir with Vn = 50 virtual nodes, these 50 sampled values from the analogue patterns will serve as the masking values (Supplementary Figure 1). For reservoirs with larger number of virtual nodes, we use multiple patterns to reach the appropriate masking sequence length. For example, for Vn = 250, we use χ = 5 repetitions of the selected sampled pattern to obtain the desired 250 masking values.

Increment entropy for ordinal patterns with m>3
In permutation entropy calculations, longer ordinal pattern lengths consider more distant neighbouring samples. In this case, the generated multi-dimensional vectors are mapped into a larger number of unique permutations. In Figure 5 we presented the PEinc metric computation with m=3 and R=4. In this case, there are (2R+1) m = 729 possible unique ordinal patterns that are considered. This number increases exponentially when increasing the value of m. Even though the highest value of the PEinc metric increases slightly with the ordinal pattern length, the qualitative interpretation of our results remains the same (Supplementary Figure 2). The conclusion that high PEinc values can be obtained from the repetitive patterns of the SL-OF system when at least one of the 2 nd and 3 rd amplification stages has significant gain is still valid for m>3.
Supplementary Figure 2. Increment entropy PEinc of the repetitive patterns obtained for different amplification gain conditions at the second and the third spectral regime, when considering an increased length of ordinal patterns, from m=4 to m=7.

Masked input and time delay reservoir response
While the response of the time delay reservoir is computed for every time step t, as given by equation (8), we only use the specific samples which are allocated to the virtual nodes of the reservoir and create a matrix representation for implementing the input layer of the TDRC. The schematic of Supplementary Figure 3 shows how the masking sequence and the input are applied in the operation of the TDRC. The masking sequence M obtained from the repetitive patterns of the SL-OF system -consists of Vn values, with temporal distance θ and has a repetitive temporal duration of T. A change of the masking value is applied only after a temporal duration of θ, under a sample and hold operation. The input Y with dimension n x 1, with n being the total number of samples, changes its value only after every temporal duration of T. From equation (8) we obtain the temporal nonlinear transformation of the masked input, from which we retain the reservoir's responses X, in a form of matrix with dimension n x Vn, to the input samples Y. Thus, in the representation showed in Supplementary Figure 3, the first input sample Y(1) is expanded -after the masking and the reservoir transformation -into the vector X(1) with Vn values. From the trained classifier and the defined weights, a prediction value Ỹ(2) is calculated from the X(1) and compared to the initial input value Y(2). Eventually, the first n-1 responses of X are used to make the prediction Ỹ, shifted by one sample.
Supplementary Figure 3. Masking methodology and reservoir operation. The input sequence Y is masked by the repetitive pattern M and it goes through a nonlinear transformation by the time delay reservoir, resulting in the response X. These responses are used to calculate the classifier's one-step-ahead prediction value of Ỹ.

NMSE performance versus PEinc
Another approach to visualize the NMSE performance of the computational task for the different masking sequences, as presented in Figure 7, is versus the increment entropy (PEinc) of the evaluated pattern. The connection between the patterns' profile and their corresponding PEinc value can be easily extracted from Figure 5. Here we associate the pattern ID, with the calculated PEinc value, as extracted from Figure 5:

Pattern ID PEinc value
Initial pattern from SL-OF system 2 Pattern A 6 Pattern B 5.9 Pattern C 5.8 Pattern D 5.9 Pattern E 4 In a new visualization of the presented results in Figure 7, we show in Supplementary Figure 4 the NMSE performance of the different patterns A-E, including also the initial pattern from the SL-OF system, versus their PEinc value. In order to keep a clear visualization, we only show here the NMSE performance for Vn=50 and Vn=400, for all the investigated NF conditions (Supplementary Figure 4, a -d) of the amplification stages.
Focusing first on the case where no amplification noise is considered (Supplementary Figure 4, a), the use of the initial pattern from the SL-OF system as masking sequence with a much lower PEinc=2, results in much higher NMSE for the computing task. Especially for large reservoirs (Vn=400) where lower error values are obtained, the masking patterns with high PEinc (≥5.8) always provide lower errors compared to the pattern with PEinc=4. However, this relation of NMSE versus PEinc is not always linear when we zoom at the region of high entropy patterns (≥5.8); for example, there are cases, such as in Supplementary Figure 4, a and d, that the pattern C exhibits lower NMSE error than the pattern A. The discussion on the impact of amplification noise, versus the reservoir size and the masking pattern properties is included in the manuscript.
Supplementary Figure 4. NMSE performance of the Santa-Fe timeseries one-step-ahead prediction benchmark task with TDRC versus the increment entropy PEinc of the different evaluated patterns, as they appear in Figure 5, after their sampling to masking sequences. The NMSE performance is shown here versus two reservoir sizes (Vn=50 and Vn=400) and under different amplification noise conditions: (a) NF=0dB, (b) NF=1dB, (c) NF=2dB and (d) NF=3dB.