Computational Analysis — Neural Matter Networks
Description: FLOP counts, memory analysis, and computational trade-offs of NMN layers vs standard neural network components.

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# Computational Analysis

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#### Explain Like I'm 5

Building with LEGOs takes time and table space! ⏱️🧱

- ⏱️ FLOPs = How many "clicks" to snap pieces together (time)
- 📦 Memory = How big a table you need (space)
- 🤔 Trade-off: NMN takes slightly more clicks, but needs a smaller table!

It's like trading: do 2× the snapping, but your table can be 25% smaller!

#### 🧮 Interactive FLOP Calculator

##### ⚡ Layer Complexity Calculator

Batch Size (B)
Input Dim (d)
Output Dim (n)

#### 📐 Complexity Analysis

Key Identity: We reuse computations using: $$\|\mathbf{w} - \mathbf{x}\|^2 = \|\mathbf{w}\|^2 + \|\mathbf{x}\|^2 - 2\mathbf{w}^\top\mathbf{x}$$ Since $\mathbf{w}^\top\mathbf{x}$ is already computed for the numerator, the denominator comes "almost free"!

#### 📊 Detailed FLOP Breakdown

Operation | Linear + ReLU | NMN Layer
--- | --- | ---
Matrix multiply | $2Bnd$ | $2Bnd$ (same)
Weight norms | — | $nd$ (can cache)
Input norms | — | $Bd$
Distance computation | — | $3Bn$ (from identity)
Square + Division | — | $2Bn$
ReLU | $Bn$ | —
Total | $\approx 2Bnd$ | $\approx 4Bnd$

#### 💾 Memory Analysis

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##### Linear + ReLU

Must store $Bn$ activation values for ReLU backward pass (to compute gradient mask).

✨

##### NMN Layer

No activation storage needed — gradient flows through geometric computation directly.

⚖️
The Trade-off: NMN uses ~2× FLOPs but saves 15-25% peak memory. For large models (LLMs, vision transformers), memory is often the bottleneck — the extra compute is worth it if you can fit larger batches or longer sequences!

#### 🚀 Optimization Techniques

- Norm Caching: $\|\mathbf{w}\|^2$ can be computed once and cached per layer
- Fused Kernels: Custom CUDA kernels can combine operations for better memory bandwidth
- Mixed Precision: BF16 shows excellent stability due to bounded outputs
- Gradient Checkpointing: Less relevant since we already save activation memory

#### 📈 Scaling Behavior

High-Dimensional Scaling: In high dimensions ($d \to \infty$), for random unit vectors: $$\mathbb{E}[\text{ⵟ}(\mathbf{w}, \mathbf{x})] \approx \frac{1/d}{2 + \epsilon} = O(1/d)$$ The ⵟ-product naturally handles the "curse of dimensionality" through its self-regulating denominator.