A Hamiltonian driven Geometric Construction of Neural Networks via the Lognormal family, Application to Financial Fraud Detection and to Network Security

Technical Summary: A Hamiltonian Driven Geometric Construction of Neural Networks via the Lognormal Family

Problem Statement
Traditional artificial neural networks (ANNs) are typically constructed within Euclidean spaces, a limitation that restricts their ability to capture the intrinsic complexity of structured data residing on statistical manifolds. While information geometry, pioneered by Amari, has established the utility of Riemannian structures for synaptic parameters, few studies have explicitly constructed neural network architectures on specific statistical manifolds, particularly the lognormal manifold. The lognormal distribution is ubiquitous in finance, biology, and signal processing, yet its parameter space has not been systematically utilized to derive neural components (inputs, weights, activation functions) from first geometric principles. The paper addresses the question of how to structure a lognormal statistical manifold as a neural manifold and deduce its components using Hamiltonian geometry and Lie group formalism.

Methodology
The authors propose a systematic method to construct a neural network intrinsically on the lognormal statistical manifold by exploiting its Hamiltonian and geometric structure. The methodology proceeds through the following steps:

1. Statistical Formulation: The lognormal distribution is treated as an exponential family. The natural parameters $\theta = (\theta_1, \theta_2)$ are derived from the mean ($\mu$) and standard deviation ($\sigma$) of the log-data.
2. Hamiltonian Dynamics: The gradient flow on the lognormal manifold is shown to be equivalent to a Hamiltonian system. By defining phase-space coordinates $P$ and $Q$ based on the natural parameters, the system is mapped to the Poincaré disk ($D$), a model of hyperbolic geometry.
3. Lie Group Action: The dynamics of the system are described by the action of the Lie group $SU(1, 1)$ on the Poincaré disk. The group action represents the evolution of the system in phase space.
4. Neural Component Derivation:
   • Input: The input vector is defined by the coordinates of the Poincaré disk, derived from the Hamiltonian variables.
   • Weight Matrix: The synaptic weight matrix is constructed from the rotation component of the $SU(1, 1)$ action (represented by a rotation matrix $\Omega \in SO(2)$) and a translation vector $\vec{t}$. This matrix belongs to the Special Euclidean group $SE(2)$.
   • Activation Function: The activation function emerges from the symplectic structure of the system, defined as a group morphism involving exponential and trigonometric functions of the angular variable $\beta$.
5. Architecture: The resulting architecture is a fully connected single-layer network with a bias, where the linear combination and nonlinearity are rooted in the underlying geometry of the data rather than simple algebraic abstractions.

Key Contributions

• Geometric Construction: The paper provides a rigorous derivation of a neural network architecture directly from the lognormal statistical manifold, moving beyond Euclidean approximations.
• Hamiltonian-Driven Dynamics: It demonstrates that the gradient system on the lognormal manifold is integrable and equivalent to a Hamiltonian system with closed elliptical trajectories in phase space, which are compactified onto the Poincaré disk.
• Explicit Component Definition: The authors explicitly define the input functions, the weight matrix (incorporating rotation and translation), and the activation function based on the Lie group $SU(1, 1)$ and the symplectic structure.
• Theoretical Framework: The work establishes a bridge between information geometry, Lie group theory, and machine learning, showing how information processing systems can be "naturalized" on non-Euclidean spaces.

Results and Applications
The paper validates the proposed framework through two specific application scenarios, utilizing 10-day datasets to demonstrate the detection of anomalies:

1. Financial Fraud Detection:

   • Context: Transaction amounts, which follow a lognormal distribution, are monitored.
   • Mechanism: Fraudulent activity alters the baseline statistical distribution ($\mu, \sigma$). The system maps these parameters to the Poincaré disk.
   • Outcome: In the case study, a fraudulent day (Day 6) with high volatility caused a radical shift in the statistical properties. In the hyperbolic feature space of the Poincaré disk, this resulted in a significant "geodesic acceleration," separating the fraudulent state from the normal operational envelope much more distinctly than Euclidean distance metrics would.

2. Network Security (DDoS Intrusion Detection):

   • Context: Network packet rates, which are positive and skewed, are modeled as lognormal.
   • Mechanism: A DDoS attack drastically increases the mean packet rate and volatility.
   • Outcome: The geometric transformation pipeline showed that during normal traffic routines, the coordinates form a highly stable cluster on the extreme left segment of the internal manifold envelope ($x_D \approx -0.499$). When the DDoS attack triggers, the massive shift in distribution parameters causes an "explosive, near-orthogonal leap" towards the vertical boundary axis, forcing $x_D$ to travel from $-0.4996$ to $-0.0062$ and $y_D$ to climb to $0.4999$. This shift reflects a massive variation in the Rao-Fisher distance, providing an immediate geometric alert that is robust against adaptive attack profiles that might evade traditional threshold-based systems.

Significance and Claims
The paper claims to offer a new perspective for geometric machine learning by demonstrating that neural information processing systems based on the lognormal family can be modeled as dynamical systems on the Poincaré disk. The significance lies in the fact that the fundamental operations of the network (linear combination, nonlinearity) are not arbitrary algebraic choices but are intrinsically derived from the geometry of the data's parameter space.

The authors assert that this approach allows for the design of architectures that are naturally suited for data following lognormal distributions, where the parameter space is inherently a manifold. By utilizing the Hamiltonian formalism and the action of $SU(1, 1)$, the model preserves statistical identity during forward propagation within isolated information trajectories (orbits), rather than globally across the entire disk. This offers a computationally light geometric method for detecting anomalies in systems where data naturally follows a lognormal distribution. The work is presented as a foundational step in applying information geometry to the explicit construction of neural networks on specific statistical manifolds.