A Hamiltonian driven Geometric Construction of Neural Networks on the Lognormal Statistical Manifold

TL;DR

The paper addresses neural network construction on non-Euclidean parameter spaces by focusing on the lognormal statistical manifold. It reframes gradient flow as an integrable Hamiltonian system on the Poincaré disk and derives a geometric neural network whose weights live in the SE(2) group and whose activation structure arises from the underlying symplectic geometry via SU(1,1) actions. Key contributions include an explicit Hamiltonian form $\mathcal{H}=-(P^2+Q^2)\cos(\beta)\sin(\beta)$, a disk-based input mapping $Z$, explicit $SU(1,1)$-generated transforms $g_1,g_2$ on $D$, and a fully connected single-layer network on the lognormal manifold with a group-theoretic weight representation. This framework provides a principled, interpretable approach to geometry-aware neural computation on manifold-valued parameters, with potential relevance for domains where lognormal statistics are natural.

Abstract

Bridging information geometry with machine learning, this paper presents a method for constructing neural networks intrinsically on statistical manifolds. We demonstrate this approach by formulating a neural network architecture directly on the lognormal statistical manifold. The construction is driven by the Hamiltonian system that is equivalent to the gradient flow on this manifold. First, we define the network's input values using the coordinate system of this Hamiltonian dynamics, naturally embedded in the Poincare disk. The core of our contribution lies in the derivation of the network's components from geometric principles: the rotation component of the synaptic weight matrix is determined by the Lie group action of SU(1,1) on the disk, while the activation function emerges from the symplectic structure of the system. We subsequently obtain the complete weight matrix, including its translation vector, and the resulting output values. This work shows that the lognormal manifold can be seamlessly viewed as a neural manifold, with its geometric properties dictating a unique and interpretable neural network structure. The proposed method offers a new paradigm for building learning systems grounded in the differential geometry of their underlying parameter spaces.

Figures (1)

• Figure 1: Diagram of the neural network architecture on lognormal statistical manifold

Theorems & Definitions (3)

• Theorem 1
• Lemma 1
• Theorem 2