Latent FxLMS: Accelerating Active Noise Control with Neural Adaptive Filters

1. Problem Statement

Active Noise Control (ANC) aims to minimize unwanted noise at a specific location (error microphone) by generating an anti-noise signal via control speakers. The standard algorithm for this is Filtered-x Least Mean Squares (FxLMS).

• Limitations of Standard FxLMS: While effective, FxLMS adapts filter weights directly in a high-dimensional space (dimension $L$, the filter length). When the noise source moves within a spatial region, the optimal filter coefficients change. However, these optimal coefficients do not fill the entire $L$-dimensional space; rather, they lie on a low-dimensional manifold determined by the geometry of the room and the source locations.
• The Challenge: Standard FxLMS does not exploit this geometric constraint, leading to slower convergence when the noise source position changes or when adapting to new conditions. Deriving a closed-form mathematical representation of this manifold is difficult.
• Goal: To accelerate the convergence of ANC systems by constraining the adaptive filter weights to lie on a learned low-dimensional manifold using neural networks, specifically when the noise source is spatially constrained.

2. Methodology: Latent FxLMS (LFxLMS)

The proposed Latent FxLMS (LFxLMS) algorithm replaces the direct adaptation of filter weights with adaptation in a latent space.

Core Architecture

1. Offline Training (Auto-Encoder):

   • An auto-encoder is trained on a dataset of converged FxLMS filter coefficients ($W$), where each coefficient set corresponds to a different noise source location within a known spatial region.
   • Encoder ($E$): Maps high-dimensional filter weights ($w \in \mathbb{R}^L$) to a low-dimensional latent vector ($z \in \mathbb{R}^k$, where $k \ll L$). The encoder first transforms time-domain weights into the frequency domain (Real Fast Fourier Transform) before passing them through fully connected layers.
   • Decoder ($D$): Maps the latent vector $z$ back to the filter weights ($w = D(z)$). The LFxLMS algorithm constrains the adaptive filter to always be the output of this decoder.

2. Online Adaptation (Latent Space Update):

   • Instead of updating $w$ directly, the algorithm updates the latent variable $z$.
   • Using the chain rule, the gradient of the error with respect to $w$ is mapped to the gradient with respect to $z$ using the Jacobian matrix ($\Upsilon$) of the decoder:
     $$z[n] = z[n-1] - \mu_z \Upsilon(z[n-1]) (e_n \hat{x}_n)$$
   • The new filter weights are generated by $w[n] = D(z[n])$.

Neural Constraints & Normalization

The paper investigates how different neural constraints and normalization schemes impact performance:

• Variational Auto-Encoders (VAE) & InfoVAE: Enforce a disentangled latent space (Gaussian distribution) to prevent correlations between latent variables from hindering gradient updates. InfoVAE is used to prevent "posterior mode collapse."
• Convex Mixup Constraint: The auto-encoder is trained not just on raw data, but also on convex combinations of filter pairs ($\gamma w_1 + (1-\gamma)w_2$). This forces the network to learn a smooth manifold, ensuring that intermediate adaptation steps (which are convex combinations of start and end states) remain valid filter configurations.
• Normalization Schemes:
  • Data-normalized: Normalizes the gradient based on the input signal energy (standard FxLMS approach) before mapping to latent space.
  • Latent-normalized: Normalizes the gradient after mapping to the latent space (dividing by the norm of the latent gradient vector).

3. Key Contributions

1. Novel Integration: First application of a neural-network-based system identification (NNSI) framework specifically to constrain the adaptive filter of an ANC system to a learned manifold, rather than replacing the filter entirely.
2. Manifold Exploitation: Demonstrates that constraining updates to a low-dimensional manifold of converged filters significantly accelerates convergence compared to standard FxLMS.
3. Normalization Analysis: Introduces and evaluates "Latent-normalized" updates, showing they often outperform data-normalized approaches in the latent space.
4. Mixup for Manifold Learning: Proposes using convex mixup training to ensure the decoder can accurately reconstruct intermediate filter states during the adaptation process.

4. Experimental Results

The authors simulated an ANC scenario in a reverberant room ($6 \times 6.2 \times 3$ m) using the Image Source Model. The noise source moved along a line segment, and the system had to adapt to these changes.

• Convergence Speed:
  • All LFxLMS models converged faster than the baseline FxLMS.
  • The best performing model was the Latent-normalized LFxLMS using an InfoVAE decoder trained with the Mixup constraint.
  • Standard VAE models (without InfoVAE or Mixup) performed poorly, often worse than FxLMS, highlighting the importance of the specific constraints.
• Steady-State Performance:
  • Despite faster convergence, the steady-state Mean Squared Error (MSE) of the best LFxLMS models was comparable to standard FxLMS (within 3 dB).
  • Standard FxLMS achieved a ~37 dB noise reduction; the neural models achieved similar reduction without the traditional trade-off where faster convergence usually implies higher steady-state error.
• Adaptation to Change: When the noise source position switched abruptly (at block 100), the optimized LFxLMS models re-converged significantly faster than the standard algorithm.

5. Significance and Conclusion

• Hardware Potential: The use of shallow neural networks (2 layers) suggests that LFxLMS is computationally feasible for real-time hardware integration in ANC devices (e.g., headphones, active car cabins).
• Spatial Constraints: The method is particularly effective when the noise source is known to be within a specific spatial region. It leverages prior knowledge of the environment to "shortcut" the adaptation process.
• Future Work: The authors note that further research is needed to evaluate the interpolation capabilities of the algorithm when the noise source moves to locations outside the training region.

Summary: Latent FxLMS successfully accelerates Active Noise Control by shifting the adaptation process from a high-dimensional weight space to a low-dimensional, learned manifold. By combining auto-encoders with specific constraints (InfoVAE, Mixup) and normalization techniques, the system achieves faster convergence without sacrificing steady-state noise cancellation performance.