Bidirectional Reservoir Computing (BRC)

• BRC is a neural computation paradigm that processes sequential data in both forward and reverse directions using coupled reservoirs.
• It utilizes a fixed, sparse random reservoir with a controlled spectral radius, enabling swift, closed-form training via ridge regression.
• BRC is effectively applied in sign language recognition, offering improved accuracy and real-time performance on resource-constrained devices.

Bidirectional Reservoir Computing (BRC) is a neural computation paradigm that extends the classical echo state network (ESN) architecture by coupling two reservoirs to jointly process sequential data in both forward and reverse temporal directions. This approach enables efficient exploitation of temporal dependencies in sequence modeling tasks while maintaining computational efficiency and suitability for deployment on resource-constrained devices (Singh et al., 30 Nov 2025).

1. Mathematical Framework of Bidirectional Reservoirs

Let $u(t)\in\mathbb{R}^D$ denote the input vector at time $t$. The BRC model instantiates two echo-state reservoirs of size $N$:

• the forward reservoir with state $x_f(t)\in\mathbb{R}^N$,
• and the backward reservoir with state $x_b(t)\in\mathbb{R}^N$.

Both reservoirs share a recurrent internal weight matrix $W\in\mathbb{R}^{N\times N}$ and an input-to-reservoir weight matrix $W_{\text{in}}\in\mathbb{R}^{N\times D}$ but process the input sequence in opposite temporal orders.

The state update equations, incorporating a leak rate parameter $\alpha\in[0,1]$, are: $$x_f(t) = (1-\alpha)x_f(t-1) + \alpha f(W x_f(t-1) + W_{\text{in}} u(t))$$

$$x_b(t) = (1-\alpha)x_b(t+1) + \alpha f(W x_b(t+1) + W_{\text{in}} u(t))$$

where $f(\cdot)=\tanh(\cdot)$ is applied element-wise. The leak rate $\alpha$, typically in the range $0.2$–$0.5$, modulates the integration of new input versus retention of past state. To guarantee the echo state property, the spectral radius $\rho(W)$ is constrained to satisfy $\rho(W)<1$ (typically $\rho(W)\approx0.9$).

2. Initialization and Echo-State Properties

Initialization of the BRC’s forward and backward reservoirs involves a single setup per model instance:

• Input weights: $W_{\text{in}}$ entries are sampled uniformly from $[-s_{\text{in}}, +s_{\text{in}}]$; a typical input scaling constant is $s_{\text{in}}=0.1$.
• Recurrent weights: $W$ is formed by first generating a sparse random matrix $W_{\text{raw}}$ with preset density (e.g., $10\%$ nonzero entries), entries drawn uniformly in $[-1, +1]$. Its largest-modulus eigenvalue $\lambda_{\text{max}}$ is computed, and $W$ is scaled as $$W = (\rho_{\text{desired}}/|\lambda_{\text{max}}|)W_{\text{raw}}$$ with $\rho_{\text{desired}}<1$.

In practical implementations, the forward and backward reservoirs share $W$ and $W_{\text{in}}$. The fixed, non-trainable nature of reservoir weights ensures the echo state property and stability of the network dynamics.

3. State Concatenation and Closed-form Readout Training

After processing an input sequence, the BRC architecture concatenates the states of the forward and backward reservoirs at each time step: $$x(t) = \begin{bmatrix} x_f(t) \ x_b(t) \end{bmatrix} \in \mathbb{R}^{2N}$$ A state matrix $X\in\mathbb{R}^{2N\times T}$ is formed by stacking these vectors, where $T$ is the sequence length. For each sequence with one-hot target label vectors $Y_{\text{target}}\in\mathbb{R}^{C\times T}$, the only trainable parameter is the readout matrix $W_{\text{out}}\in\mathbb{R}^{C\times 2N}$, obtained by ridge regression: $$W_{\text{out}} = Y_{\text{target}} X^\top \, \bigl(X X^\top + \lambda I_{2N}\bigr)^{-1}$$ with regularization parameter $\lambda>0$. The objective function minimized is: $$\mathcal{L}(W_{\text{out}}) = \|Y_{\text{target}} - W_{\text{out}} X\|_F^2 + \lambda \|W_{\text{out}}\|_F^2$$ This procedure eschews backpropagation through time, allowing for extremely rapid, closed-form training.

4. Spatiotemporal Feature Extraction via MediaPipe

The implementation of BRC for sign language recognition leverages MediaPipe’s Hand-Landmarker to extract input features:

• Each video frame yields 21 landmark coordinates $(x, y, z)$ per hand (up to 42 landmarks for both hands).
• These coordinates are concatenated and flattened to form $u(t)\in\mathbb{R}^{D}$, with $D$ as large as $126$.
• Features are normalized per coordinate (mean zero, unit variance across the training set). Optional linear interpolation addresses frames with missed detections.

The resultant sequence $\{u(1),\ldots,u(T)\}$ forms the input to the bidirectional ESN, enabling frame-level fusion of low-dimensional, semantically meaningful hand pose descriptors.

5. Computational Properties and Empirical Evaluation

The computational complexity and empirical attributes of BRC are as follows:

• Training: Dominated by the inversion of a $(2N\times 2N)$ matrix ($O((2N)^3)$), plus the covariance operation $O((2N)^2T)$. With $N=100$, $T\approx150$, training completes in under $9$ seconds on an Intel i7-11700 CPU.
• Inference: For a sequence, the joint forward+backward pass costs $O(2N\cdot T \cdot \text{density}(W))$; classification requires a single matrix-vector product. Per-sequence time is under $10$ ms on the same hardware.
• Accuracy: On WLASL 100, BRC achieves $57.71\%\pm1.35$, outperforming a Bi-GRU baseline ($49.90\%\pm2.56$) trained for $55$ minutes, $38$ seconds. A unidirectional ESN with $200$ nodes attains $54.31\%\pm1.45$ ($7$ s train time).
• Device Suitability: Training involves only the linear readout; no backpropagation or reservoir updates occur. The fixed sparse reservoir affords low memory usage and stable runtime. Real-time deployment on embedded CPUs and lightweight NPUs is feasible.

Method	Accuracy (%)	Training Time
BRC (N=100)	57.71±1.35	9 seconds
Bi-GRU	49.90±2.56	55 min, 38 seconds
Unidirectional ESN (N=200)	54.31±1.45	7 seconds

6. Context, Applications, and Implications

The BRC paradigm is demonstrated in sign language recognition, where temporal dependencies are bidirectional due to coarticulation and non-causal gestures. By capturing both past and future context, BRC provides a richer temporal representation than unidirectional ESNs while maintaining the training and resource efficiency characteristic of fixed-weight reservoir architectures (Singh et al., 30 Nov 2025).

A plausible implication is that such architectures are broadly applicable to sequence classification tasks where both prior and subsequent information improve context disambiguation, and where deployment constraints preclude the resource demands of deep recurrent architectures. The approach is particularly salient for edge inference scenarios where power and memory are at a premium.

7. Comparative Assessment and Limitations

BRC offers a trade-off profile favoring rapid training, minimal tuning, and real-time inference over deep recurrent networks. Its empirically demonstrated gains over traditional ESNs and Bi-GRU baselines in sign language recognition indicate potential for edge applications. However, the reliance on fixed random recurrent dynamics and a linear readout may limit expressivity for tasks demanding higher representational capacity or long-term temporal reasoning beyond the spectral radius-constrained reservoir.

The significance of bidirectional processing for SLR highlights the limitations of strictly causal models in domains with temporally distributed dependencies, suggesting that further research on hybrid or adaptive reservoir frameworks could yield performance improvements, particularly as input feature extraction pipelines (e.g., MediaPipe descriptors) become more robust and informative (Singh et al., 30 Nov 2025).