Global stability of a Hebbian/anti-Hebbian network for principal subspace learning

TL;DR

This work establishes global stability for a Hebbian/anti-Hebbian PSA network in the continuum limit at equal learning-rate timescales (τ=1/2), proving convergence from almost all initializations to equilibria whose neural filters form an orthonormal basis spanning the input's principal subspace. The analysis reveals a two-phase convergence: first, rapid contraction to an invariant manifold with orthonormal neural filters, and second, slow gradient-flow dynamics along this manifold toward the PSA subspace minima of a non-convex potential. The results bridge local synaptic updates with stable, high-level computations, providing a rigorous link between biologically plausible plasticity and principal subspace learning, and suggest practical initialization and extensions to general timescales. The work also discusses extensions to general 0<τ≤1/2 and outlines empirical evidence that the online algorithm inherits the two-phase behavior, with implications for neuromorphic and PSA-related algorithms.

Abstract

Biological neural networks self-organize according to local synaptic modifications to produce stable computations. How modifications at the synaptic level give rise to such computations at the network level remains an open question. Pehlevan et al. [Neur. Comp. 27 (2015), 1461--1495] proposed a model of a self-organizing neural network with Hebbian and anti-Hebbian synaptic updates that implements an algorithm for principal subspace analysis; however, global stability of the nonlinear synaptic dynamics has not been established. Here, for the case that the feedforward and recurrent weights evolve at the same timescale, we prove global stability of the continuum limit of the synaptic dynamics and show that the dynamics evolve in two phases. In the first phase, the synaptic weights converge to an invariant manifold where the `neural filters' are orthonormal. In the second phase, the synaptic dynamics follow the gradient flow of a non-convex potential function whose minima correspond to neural filters that span the principal subspace of the input data.

Figures (6)

• Figure 1: Hebbian/anti-Hebbian network for PSA. Single layer network with $k$ neurons that receives $n$ inputs. Feedforward Hebbian synapses ${\bf W}$ connect the $n$ inputs to the $k$ neurons and recurrent anti-Hebbian synapses $-{\bf M}$ connect the $k$ neurons.
• Figure 2: Dependency diagram for our results. Local stability of equilibrium points (orange region) is established in section \ref{['sec:local']}. Convergence of solutions to the invariant manifold $\mathcal{O}$ (blue region) is shown in section \ref{['sec:Oconvergence']}. Convergence starting in or near the invariant manifold to the equilibrium points $\mathcal{E}$ (green region) is shown in section \ref{['sec:gradflow']}. Finally, global stability of the ODE (red box) is shown in section \ref{['sec:proof']}. Technical results (gray region) are proved in appendices \ref{['apdx:exist']} and \ref{['apdx:setN']}.
• Figure 3: Plot of the vector field ${\bf G}(W,M)$ in the case $k=n=1$ and $\lambda_1=2$. The grayscale indicates the logarithm of the vector magnitude. The blue lines denote the set $\mathcal{O}$, the orange vertical line denotes the set $\mathcal{N}$, the 2 red dots denote the set $\mathcal{E}_0$ (which is equal to $\mathcal{E}$ in this case), and the green line indicates that the line $M=0$ does not belong to $\mathcal{D}=\mathbb{R}\times(0,\infty)$.
• Figure 4: Plot of the vector field $-\nabla V({\bf W})$ in the case $n=2$, $k=1$ and ${\bf A}=\mathop{\mathrm{diag}}\nolimits(2,\frac{1}{2})$. The grayscale indicates the vector magnitude. The red dots denote the global minima of $V$, the purple dots denote the saddle points of $V$, and the cyan dot at the origin denotes the set $\{{\bf W}:\det({\bf W}{\bf W}^\top)=0\}$.
• Figure 5: Convergence of the ODE and the online algorithm in a network with $d=4$ inputs and $k=2$ neurons. (a) Convergence to the invariant manifold $\mathcal{O}$, measured using the Lyapunov function $L({\bf W},{\bf M})$. (b) Convergence to the principal subspace, measured using $V_\ast({\bf W})=V({\bf W})-V({\bf W}_\ast)$, where $({\bf W}_\ast,{\bf M}_\ast)$ is any stable equilibrium point of the ODE. In panels (a) and (b), shaded regions indicate the middle 80th percentile over 100 random initializations; solid/dotted lines denote the median values. Simulation details are in the main text.

Theorems & Definitions (18)

• Theorem 1
• Lemma 1
• Theorem 2
• Lemma 2
• proof
• Lemma 3
• Corollary 1
• proof
• Lemma 4
• proof