Parallel Bidirectional Reservoir Computing

• Parallel Bidirectional Reservoir Computing (PBRC) is a computing architecture that enriches reservoir states by combining bidirectional (forward and backward) temporal encoding with parallel processing modules.
• It leverages independent reservoir modules—implemented via ESN-based or all-optical memristive systems—to capture comprehensive context and perform diverse input fusion for tasks like sign language recognition.
• The design emphasizes computational efficiency, using fixed random recurrent dynamics and ridge regression-trained readouts, making it ideal for edge applications and real-time signal processing.

Searching arXiv for the specified PBRC papers and closely related reservoir-computing work. Parallel Bidirectional Reservoir Computing (PBRC) denotes reservoir-computing architectures that combine bidirectional state construction with parallel information pathways. In "Sign Language Recognition using Parallel Bidirectional Reservoir Computing" (Singh et al., 22 Dec 2025), PBRC is an echo state network (ESN)-based architecture in which two bidirectional reservoir computing (BRC) modules are arranged in parallel and driven by MediaPipe keypoints for word-level American Sign Language recognition. In "All-Optically Controlled Memristive Reservoir Computing Capable of Bipolar and Parallel Coding" (Hu et al., 13 Feb 2026), the same designation is associated with an all-optically controlled memristive reservoir in which bidirectional behavior is realized through bipolar optical coding and parallel behavior through simultaneous multi-source injection. This suggests that PBRC is best interpreted as a structural motif for enriching reservoir states, rather than as a single fixed algorithm.

1. Reservoir-computing basis

PBRC in its ESN-based form inherits the standard reservoir-computing separation between fixed recurrent dynamics and a trained linear readout. The leaky-integrator ESN state update is

$$x(t+1) = (1-\alpha)x(t) + \alpha f\!\left(W_{in}u(t) + W_r x(t)\right),$$

where $x(t) \in \mathbb{R}^{N_r}$ is the reservoir state, $u(t) \in \mathbb{R}^{N_{in}}$ is the input, $W_{in}$ is the input weight matrix, $W_r$ is the fixed internal recurrent weight matrix, $f(\cdot)$ is the nonlinear activation, and $\alpha \in (0,1]$ is the leak rate. For $\alpha = 1$, this simplifies to

$x(t+1) = f\!\left(W_{in}u(t) + W_r x(t)\right).$

The corresponding readout is linear,

$y(t) = W_{out}x(t),$

with only $x(t) \in \mathbb{R}^{N_r}$0 trained by ridge regression. The reservoir weights are randomly initialized; $x(t) \in \mathbb{R}^{N_r}$1 is normalized and rescaled by the spectral radius hyperparameter $x(t) \in \mathbb{R}^{N_r}$2, with

$x(t) \in \mathbb{R}^{N_r}$3

and

$x(t) \in \mathbb{R}^{N_r}$4

In the reported PBRC sign-language experiments, $x(t) \in \mathbb{R}^{N_r}$5 for each bidirectional reservoir, $x(t) \in \mathbb{R}^{N_r}$6 for every BRC, and the activation is $x(t) \in \mathbb{R}^{N_r}$7 (Singh et al., 22 Dec 2025).

The all-optical memristive realization also adopts a leaky nonlinear state evolution, but its state variable is physical rather than algorithmic. A Pt/ZnO/Pt memristor node is biased at a constant read voltage $x(t) \in \mathbb{R}^{N_r}$8 and receives dual-wavelength optical powers at $x(t) \in \mathbb{R}^{N_r}$9 and $u(t) \in \mathbb{R}^{N_{in}}$0. The normalized fraction of ionized oxygen vacancies, $u(t) \in \mathbb{R}^{N_{in}}$1, governs conductance; blue photons favor ionization and red photons favor neutralization. The state dynamics are modeled as

$u(t) \in \mathbb{R}^{N_{in}}$2

with the measured current

$u(t) \in \mathbb{R}^{N_{in}}$3

A compact reservoir map is then written as

$u(t) \in \mathbb{R}^{N_{in}}$4

which makes explicit the signed optical drive induced by the dual-wavelength competition (Hu et al., 13 Feb 2026).

2. Bidirectionality and parallelism in the ESN architecture

In the sign-language PBRC system, bidirectionality is temporal. Each BRC processes the input sequence both in the original order and in reversed temporal order. Within a single BRC, one internal recurrent matrix is shared by the forward and backward directions, while the input matrices are direction-specific. The updates are

$u(t) \in \mathbb{R}^{N_{in}}$5

$u(t) \in \mathbb{R}^{N_{in}}$6

where $u(t) \in \mathbb{R}^{N_{in}}$7 is the reversed input sequence, implemented by reversing the time axis while preserving the feature dimension. The forward and backward states are concatenated at each time step,

$u(t) \in \mathbb{R}^{N_{in}}$8

so that the readout has simultaneous access to past and future context (Singh et al., 22 Dec 2025).

Parallelism is introduced by instantiating two independent BRC modules, Reservoir A and Reservoir B. Each module has its own internal recurrent matrix, $u(t) \in \mathbb{R}^{N_{in}}$9 or $W_{in}$0; within a given module, that matrix is shared across forward and backward directions, but the two modules are independent of each other. The full PBRC state is

$W_{in}$1

followed by the readout

$W_{in}$2

Fusion is simple concatenation, with no dimensionality reduction or PCA. With $W_{in}$3 nodes per BRC, forward-plus-backward concatenation yields $W_{in}$4 features per BRC, and the two parallel BRCs produce a $W_{in}$5-dimensional concatenated state at each time step (Singh et al., 22 Dec 2025).

A plausible implication is that the architecture uses parallel reservoirs not for depth in the deep-learning sense, but for diversity of fixed dynamical embeddings under the same input stream.

3. Readout training, classification, and computational profile

The readout in PBRC is trained by ridge regression. For design matrix $W_{in}$6 and targets $W_{in}$7 in vector form, the objective is

$W_{in}$8

with closed-form solution

$W_{in}$9

For reservoir computing, if $W_r$0 is the time-stacked reservoir-state matrix and $W_r$1 is the target matrix, then

$W_r$2

In the reported WLASL100 setting, $W_r$3 and $W_r$4. The readout produces class scores $W_r$5; top-$W_r$6 accuracy is computed by ranking these scores. If probabilities are required, a softmax can be applied post hoc, but the system treats ridge-regression outputs as scores for classification by $W_r$7 or ranking. For inference, scores are computed at each time step and then aggregated across time; the paper notes average or last-step aggregation as examples, while not prescribing a single aggregation rule (Singh et al., 22 Dec 2025).

The computational efficiency follows directly from readout-only training. Training cost is dominated by forming $W_r$8, with complexity $W_r$9 if $f(\cdot)$0 is the total number of accumulated time steps, and by solving the $f(\cdot)$1 system, with complexity $f(\cdot)$2. With $f(\cdot)$3, the parameter count of $f(\cdot)$4 is $f(\cdot)$5 parameters, plus any bias if used. Reservoir weights remain fixed after random initialization. The paper reports CPU training time for PBRC of $f(\cdot)$6 on an Intel Core i7 system with $f(\cdot)$7 RAM, compared with $f(\cdot)$8 for Bi-GRU in the same setup. No GPU is required, and the reported memory footprint is low because the architecture uses small concatenated states and readout-only training (Singh et al., 22 Dec 2025).

4. Sign-language recognition instantiation

The sign-language instantiation combines MediaPipe with PBRC. MediaPipe provides real-time hand tracking and precise extraction of hand joint coordinates, which serve as input features for the reservoir architecture. The system is trained on WLASL100, a 100-class subset of the Word-Level American Sign Language dataset, with splits of $f(\cdot)$9 for train/validation/test. The paper states that MediaPipe makes available $\alpha \in (0,1]$0 landmarks per hand and $\alpha \in (0,1]$1 pose landmarks, but does not specify a fixed feature subset count per frame in the experiments beyond that availability (Singh et al., 22 Dec 2025).

The reported PBRC configuration uses two BRC modules, $\alpha \in (0,1]$2, each with $\alpha \in (0,1]$3 nodes, spectral radius $\alpha \in (0,1]$4, leak rate $\alpha \in (0,1]$5, $\alpha \in (0,1]$6 activation, random initialization of $\alpha \in (0,1]$7 and $\alpha \in (0,1]$8, and ridge regularization with empirically tuned $\alpha \in (0,1]$9. Baselines include a single BRC with $\alpha = 1$0 nodes total, a single unidirectional ESN with $\alpha = 1$1 nodes, and a Bi-GRU trained for $\alpha = 1$2 epochs with $\alpha = 1$3 hidden layers (Singh et al., 22 Dec 2025).

On WLASL100, the proposed PBRC achieves top-1, top-5, and top-10 accuracies of $\alpha = 1$4, $\alpha = 1$5, and $\alpha = 1$6, respectively, with training time $\alpha = 1$7. Additional comparisons on the same dataset report Pose-TGCN at top-1 $\alpha = 1$8, top-5 $\alpha = 1$9, top-10 $x(t+1) = f\!\left(W_{in}u(t) + W_r x(t)\right).$0, and $x(t+1) = f\!\left(W_{in}u(t) + W_r x(t)\right).$1 on GPU; I3D at top-1 $x(t+1) = f\!\left(W_{in}u(t) + W_r x(t)\right).$2, top-5 $x(t+1) = f\!\left(W_{in}u(t) + W_r x(t)\right).$3, top-10 $x(t+1) = f\!\left(W_{in}u(t) + W_r x(t)\right).$4, and $x(t+1) = f\!\left(W_{in}u(t) + W_r x(t)\right).$5 on GPU; and MRC at top-1 $x(t+1) = f\!\left(W_{in}u(t) + W_r x(t)\right).$6, top-5 $x(t+1) = f\!\left(W_{in}u(t) + W_r x(t)\right).$7, top-10 $x(t+1) = f\!\left(W_{in}u(t) + W_r x(t)\right).$8, and $x(t+1) = f\!\left(W_{in}u(t) + W_r x(t)\right).$9 on CPU. Within-paper baselines are reported as PBRC $y(t) = W_{out}x(t),$0 with $y(t) = W_{out}x(t),$1, BRC $y(t) = W_{out}x(t),$2 with $y(t) = W_{out}x(t),$3, standard ESN $y(t) = W_{out}x(t),$4 with $y(t) = W_{out}x(t),$5, and Bi-GRU $y(t) = W_{out}x(t),$6 with $y(t) = W_{out}x(t),$7 (Singh et al., 22 Dec 2025).

These figures establish the specific tradeoff claimed for the architecture: the system is lightweight, cost-effective, and oriented toward real-time SLR on edge devices, while remaining competitive with more computationally intensive methods. The data do not show PBRC exceeding the highest top-1 accuracy in the comparison set, but they do show a favorable accuracy-speed balance.

5. All-optical memristive PBRC

In the all-optically controlled memristive system, the meanings of bidirectionality and parallelism differ from those in the ESN-based SLR model. Bidirectionality is realized as bipolar coding. Logic “1” is encoded by a $y(t) = W_{out}x(t),$8 pulse with $y(t) = W_{out}x(t),$9, and logic “0” by a $x(t) \in \mathbb{R}^{N_r}$00 pulse with $x(t) \in \mathbb{R}^{N_r}$01, using pulse width $x(t) \in \mathbb{R}^{N_r}$02 and no inter-pulse gap. Under $x(t) \in \mathbb{R}^{N_r}$03 illumination, ionization dominates and yields positive persistent photoconductivity; under $x(t) \in \mathbb{R}^{N_r}$04 illumination, neutralization dominates and yields negative persistent photoconductivity. The resulting signed discrete update is

$x(t) \in \mathbb{R}^{N_r}$05

which reproduces the experimentally observed spread of reservoir states across positive and negative regions (Hu et al., 13 Feb 2026).

Parallel coding is realized by simultaneous injection of multiple streams through different wavelength channels into the same device. For $x(t) \in \mathbb{R}^{N_r}$06 synchronous inputs, per-wavelength injected powers are written as

$x(t) \in \mathbb{R}^{N_r}$07

leading to the signed state update

$x(t) \in \mathbb{R}^{N_r}$08

In the dual-authentication demonstration, $x(t) \in \mathbb{R}^{N_r}$09: face information is mapped to the $x(t) \in \mathbb{R}^{N_r}$10 channel and fingerprint information to the $x(t) \in \mathbb{R}^{N_r}$11 channel. Pairs of 4-digit codes are applied simultaneously, with $x(t) \in \mathbb{R}^{N_r}$12 and state sampling $x(t) \in \mathbb{R}^{N_r}$13 after coding. Because both wavelength terms act on a single physical state variable, the device performs nonlinear fusion in hardware rather than simple post hoc concatenation (Hu et al., 13 Feb 2026).

At array level, with $x(t) \in \mathbb{R}^{N_r}$14 nodes, the state-vector formulation is

$x(t) \in \mathbb{R}^{N_r}$15

with linear readout

$x(t) \in \mathbb{R}^{N_r}$16

The paper also gives the full PBRC vector form

$x(t) \in \mathbb{R}^{N_r}$17

where $x(t) \in \mathbb{R}^{N_r}$18 makes the signed input map explicit (Hu et al., 13 Feb 2026).

Quantitatively, on word recognition with a $x(t) \in \mathbb{R}^{N_r}$19 array and $x(t) \in \mathbb{R}^{N_r}$20 four-letter words, bipolar coding achieves approximately $x(t) \in \mathbb{R}^{N_r}$21 accuracy after $x(t) \in \mathbb{R}^{N_r}$22 epochs, whereas unipolar coding achieves approximately $x(t) \in \mathbb{R}^{N_r}$23; for the pair “LOCK” and “LUCK,” confusion drops from $x(t) \in \mathbb{R}^{N_r}$24 in the unipolar case to $x(t) \in \mathbb{R}^{N_r}$25 in the bipolar case. On chaotic time-series prediction of the Lorenz system, the average NRMSE over $x(t) \in \mathbb{R}^{N_r}$26, $x(t) \in \mathbb{R}^{N_r}$27, and $x(t) \in \mathbb{R}^{N_r}$28 is approximately $x(t) \in \mathbb{R}^{N_r}$29 for bipolar coding and approximately $x(t) \in \mathbb{R}^{N_r}$30 for unipolar coding; under fixed total reservoir size $x(t) \in \mathbb{R}^{N_r}$31, the best performance occurs around mask length $x(t) \in \mathbb{R}^{N_r}$32. In dual-factor authentication with $x(t) \in \mathbb{R}^{N_r}$33 paired identities, parallel coding achieves approximately $x(t) \in \mathbb{R}^{N_r}$34 accuracy with one reservoir and $x(t) \in \mathbb{R}^{N_r}$35 features per pair, while bipolar serial coding achieves approximately $x(t) \in \mathbb{R}^{N_r}$36 accuracy but requires two reservoirs and $x(t) \in \mathbb{R}^{N_r}$37 features, and unipolar coding achieves approximately $x(t) \in \mathbb{R}^{N_r}$38 accuracy (Hu et al., 13 Feb 2026).

6. Conceptual distinctions and significance

A common source of confusion is that PBRC does not denote identical operations across the two reported implementations. The term unifies parallelism and bidirectionality, but the mechanisms are architecture-specific.

Realization	“Bidirectional”	“Parallel”
ESN-based SLR	Forward and backward temporal processing of the same sequence	Two independent BRC modules, concatenated at readout
All-optical memristive RC	Signed bipolar coding via opposite photoresponses	Simultaneous multi-source injection and in-materia fusion

In the ESN system, bidirectionality means access to both past and future context through forward and reversed sequence processing, and parallelism means the coexistence of two independent reservoirs whose states are concatenated. In the memristive system, bidirectionality means positive and negative state excursions induced by $x(t) \in \mathbb{R}^{N_r}$39 and $x(t) \in \mathbb{R}^{N_r}$40 excitation, while parallelism means spectrally multiplexed co-injection of multiple streams into a single physical reservoir. The first implementation therefore increases representational richness through multiple fixed temporal embeddings, whereas the second increases it through physically coupled dual-wavelength dynamics (Singh et al., 22 Dec 2025, Hu et al., 13 Feb 2026).

Another potential misconception is that PBRC is synonymous with generic multi-reservoir computing. The SLR paper explicitly compares PBRC with MRC and reports that MRC trains much faster on CPU, at $x(t) \in \mathbb{R}^{N_r}$41, with top-1 $x(t) \in \mathbb{R}^{N_r}$42, top-5 $x(t) \in \mathbb{R}^{N_r}$43, and top-10 $x(t) \in \mathbb{R}^{N_r}$44. PBRC is instead defined there by the specific composition of two bidirectional ESN reservoirs in parallel. In the memristive context, PBRC is not a multi-reservoir ensemble at all; it is a single-reservoir realization of bipolar and parallel coding. This suggests that the shared conceptual core is not the number of reservoirs, but the deliberate expansion of reservoir state diversity through bidirectional encoding and parallel pathways.

Across both realizations, the reported significance is computational efficiency under constrained resources. In the SLR setting, efficiency derives from fixed random reservoirs and closed-form ridge-regression training of a small readout. In the memristive setting, efficiency derives from exploiting intrinsic device dynamics, wavelength-dependent bipolar photoresponse, and hardware-level fusion to reduce hardware consumption while maintaining high computational accuracy. A plausible implication is that PBRC, as presently represented in the arXiv literature, is most relevant where temporal context, heterogeneous input streams, and edge-oriented efficiency must be balanced without full backpropagation through time.