Theorem 5

## Information-Geometric Foundations

🧒

#### Explain Like I'm 5

Imagine you have two ways to measure how "different" two things are:

- 📏 Ruler way: Measure the distance between them (like measuring with a ruler)
- 📊 Information way: Measure how surprised you'd be to see one when expecting the other

The ⵟ-product is special because it connects both ways! It's like having a magic bridge between measuring distances and measuring information.

This means we can use the ⵟ-product with information-theoretic losses (like KL divergence) and it still makes mathematical sense!

Theorem: The ⵟ-product exhibits a duality between Euclidean geometry and information geometry. Specifically, it can be related to KL divergence and entropy-based losses through its kernel structure.

#### 🎯 The Problem This Solves

Many machine learning tasks use information-theoretic losses:

- KL divergence for probabilistic models
- Cross-entropy for classification
- Mutual information for representation learning

Traditional neural networks use Euclidean geometry (dot products, distances), which doesn't naturally connect to information theory. This theorem bridges that gap.

#### 📐 The Mathematics In Depth

The connection comes from the kernel structure. Since the ⵟ-product is a Mercer kernel, it defines a Reproducing Kernel Hilbert Space (RKHS). In this space:

$$\text{ⵟ}(\mathbf{w}, \mathbf{x}) = \langle \phi(\mathbf{w}), \phi(\mathbf{x}) \rangle_{\mathcal{H}}$$

where $\phi$ maps to the RKHS $\mathcal{H}$.

Information geometry studies probability distributions using the Fisher information metric, which can be related to KL divergence. The kernel structure of the ⵟ-product allows us to interpret it in this framework.

#### 💥 The Consequences

🔗

##### Unified Framework

The ⵟ-product bridges Euclidean geometry (for optimization) and information geometry (for probabilistic modeling), creating a unified framework.

📊

##### Compatible with Information Losses

Can be used with KL divergence, cross-entropy, and other information-theoretic losses while maintaining geometric interpretability.

🎯

##### Dual Interpretation

The same operation can be interpreted as either geometric similarity (Euclidean) or information similarity (probabilistic), depending on context.

🔮

##### Rich Theoretical Connections

Connects to maximum entropy principles, variational inference, and other information-theoretic frameworks through the kernel structure.

#### 🎓 What This Really Means

This theorem shows that the ⵟ-product isn't just a geometric operator — it's a unifying bridge between two fundamental mathematical frameworks:

- Euclidean geometry: For optimization, distances, and spatial reasoning
- Information geometry: For probability, entropy, and statistical learning

This duality means NMNs can seamlessly work with both geometric and probabilistic objectives, making them versatile for a wide range of applications.