Improved Cleanup and Decoding of Fractional Power Encodings

TL;DR

The paper tackles robust decoding and cleanup of continuous-valued SSPs encoded with Fourier Holographic Reduced Representation (FHRR) within VSAs. It introduces a phase-coupled least circular distance regression framework that combines $CLE$ and $MLE$, manifested through a two-stage iterative optimization that blends $E_D(\vec{x})$ and $E_C(\vec{x})$ with a controllable parameter $\lambda$. Key contributions include the coupling-based objective $E_C(\vec{x})$, its integration into a stable optimization scheme, and extensive experiments showing improved cleanup under bundling and noise without requiring training. The proposed method is fast, scalable, and amenable to neural-inspired implementations, offering practical gains for neuromorphic VSA processing and robust continuous-value decoding.

Abstract

High-dimensional vectors have been proposed as a neural method for representing information in the brain using Vector Symbolic Algebras (VSAs). While previous work has explored decoding and cleaning up these vectors under the noise that arises during computation, existing methods are limited. Cleanup methods are essential for robust computation within a VSA. However, cleanup methods for continuous-value encodings are not as effective. In this paper, we present an iterative optimization method to decode and clean up Fourier Holographic Reduced Representation (FHRR) vectors that are encoding continuous values. We combine composite likelihood estimation (CLE) and maximum likelihood estimation (MLE) to ensure convergence to the global optimum. We also demonstrate that this method can effectively decode FHRR vectors under different noise conditions, and show that it outperforms existing methods.

Figures (10)

• Figure 1: The uncorrupted (top), corrupted (middle), and twice corrupted (bottom) SSP phases (left) and similarity for different encoded $x$ values (right). The left plot displays the ideal phases in red and observed phases in blue. The right plot displays the similarity (blue) to SSPs encoding other $x$ values, and the true $x$-value (red).
• Figure 2: Failure of linear least-squares. For $x=2$, the left plot shows the unwrapped phases (blue dots) and the linear model $Ax$ (red line). The middle plot shows the same, but with wrapping to $(-\pi,\pi]$. The plot on the right shows the linear least-squares fit (purple) to the wrapped phases. That fit is not close to the true model (red).
• Figure 3: The objective function from coupling phases $i,j$ where the absolute value of the base phase combination $C_{j}A$ is in the range (from left to right) [0,0.05], [0.35, 0.4] and [0.7, 0.75]. The red line is at $x=4.5$ (the true $x$ value). The SSPs were $1000$-dimensional.
• Figure 4: Histograms of the base phase combinations, and the corresponding similarity functions. The distributions plotted are the uniform, Gaussian, Laplace, and triangular. In the plots of the objective function, the red line is at $x=2$ (the true $x$ value). The SSPs were $1000$-dimensional.
• Figure 5: Left: 15 random terms from the direct least circular distance objective function (top), and 15 random terms from the coupling least circular distance objective function (bottom). Right: The two objective functions, with the true $\vec{x}$-value plotted as a red line.