Decoding the Black Box: Analogies Between Neural Network Embeddings, Modulation, and Spectral Decomposition in Large Language Models

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Abstract

This paper explores the interpretability of artificial neural networks (ANNs), particularly large language models (LLMs), through the lens of signal processing and spectral analysis. Drawing parallels to transmission carrier modulation and Fourier transform-like decomposition, we analyze how high-dimensional vector embeddings encode linguistic patterns and relationships. While acknowledging the non-linear, adaptive nature of ANNs, we propose frameworks inspired by classical signal theory to disentangle latent representations. The discussion addresses challenges in mechanistic interpretability, evaluates existing spectral methods, and outlines future research directions for bridging the gap between empirical observation and theoretical understanding.

Introduction
- 1.1 The Interpretability Challenge in Modern AI
- 1.2 Motivation: Lessons from Signal Processing
- 1.3 Scope and Objectives
Background
- 2.1 Neural Networks and Embedding Spaces
  - 2.1.1 From Word2Vec to Transformer-Based LLMs
  - 2.1.2 Emergent Properties of High-Dimensional Embeddings
- 2.2 Primer on Transmission Modulation and Fourier Analysis
  - 2.2.1 Carrier Signals and Information Encoding
  - 2.2.2 Frequency Domains and Basis Decomposition
Analogies Between ANN Processing and Modulation
- 3.1 Embeddings as Carrier Signals
  - 3.1.1 Token Vectors and Baseband Encoding
  - 3.1.2 Non-Linear Modulation via Attention Mechanisms
- 3.2 Multiplexing and Parallel Processing
  - 3.2.1 Attention Heads as Frequency-Division Channels
  - 3.2.2 Contextual Bandwidth: Local vs. Global Dependencies
- 3.3 Demodulation: Probing Latent Representations
  - 3.3.1 Linear Probes and Semantic Decoding
  - 3.3.2 Limitations of Deterministic Analogies
Spectral Methods for ANN Interpretability
- 4.1 Linear Algebra Approaches
  - 4.1.1 Singular Value Decomposition (SVD) of Weight Matrices
  - 4.1.2 Eigenvalue Analysis of Attention Layers
- 4.2 Frequency-Domain Insights in Deep Learning
  - 4.2.1 Fourier Transforms in Convolutional Networks
  - 4.2.2 Wavelet Transforms for Multi-Scale Activation Analysis
- 4.3 Disentangling Representations
  - 4.3.1 PCA, t-SNE, and Manifold Learning
  - 4.3.2 Sparse Coding and Adaptive Basis Functions
Case Study: Transformers as Signal Processors
- 5.1 Anatomy of a Transformer Layer
  - 5.1.1 Self-Attention as Adaptive Filtering
  - 5.1.2 Feed-Forward Networks as Non-Linear Mixers
- 5.2 Frequency Roles in Linguistic Tasks
  - 5.2.1 Low-Frequency Features: Semantic Coherence
  - 5.2.2 High-Frequency Features: Syntactic Structure
- 5.3 Experimental Validation
  - 5.3.1 Visualizing Attention Head “Bands”
  - 5.3.2 Perturbation Analysis and Signal Recovery
Challenges and Limitations
- 6.1 Non-Linearity and the Curse of Dimensionality
- 6.2 Emergent Behaviors and Scalability
- 6.3 The Myth of a Universal “ANN Fourier Transform”
Future Research Directions
- 7.1 Hybrid Spectral-Neural Architectures
- 7.2 Quantum-Inspired Tensor Decompositions
- 7.3 Toward a Unified Theory of Neural Modulation
Conclusion
References

Expanded Content Overview

1. Introduction

Modern LLMs like GPT-4 operate as “black boxes,” with their internal mechanics often inscrutable. This paper argues that insights from signal processing—particularly modulation and spectral decomposition—can illuminate how these models encode and transform information. By framing token embeddings as modulated carrier signals and attention mechanisms as adaptive filters, we propose a vocabulary to describe ANN behavior without relying on anthropomorphic metaphors. The goal is not to reduce ANNs to linear systems but to identify transferable principles for interpretability.

3. Analogies Between ANN Processing and Modulation

Carrier Signal Analogy: Token embeddings (e.g., 768-dimensional vectors in BERT) act as baseband carriers, initialized randomly and modulated during training to encode semantic relationships. Training adjusts their “phase” (contextual meaning) and “amplitude” (salience).
Attention as Frequency-Division Multiplexing: Each attention head processes distinct relational “bands,” such as subject-verb agreement (high-frequency) or narrative coherence (low-frequency).

4. Spectral Methods

SVD in Embedding Spaces: Decomposing the embedding matrix W∈RV×dW∈RV×d (where VV = vocabulary size) into UΣVTUΣVT reveals dominant semantic axes (e.g., gender, tense).
Wavelet Transforms: Applied to spatial activations in vision models, wavelet banks could isolate hierarchical features in transformers (e.g., n-gram patterns vs. discourse-level links).

5. Case Study

Sentence Analysis: For the input “The cat sat on the mat,” low-frequency attention heads link “cat” and “mat” across the sentence, while high-frequency heads resolve positional encoding and verb tense (“sat”).

7. Future Directions

Neural Tangent Kernel (NTK): This framework, which linearizes network training dynamics, could map ANN feature spaces to frequency-domain duals.
Cross-Disciplinary Tools: Combining information theory (e.g., entropy measures) with harmonic analysis to quantify “spectral leakage” in attention matrices.

Figures and Tables

Figure 1: Modulation vs. Transformer Layer (Block diagram comparing AM/FM modulation to embedding transformations).
Figure 2: Spectral Decomposition of Embedding Space (t-SNE plots vs. Fourier power spectra).
Table 1: Attention Head Roles Mapped to Frequency Bands (e.g., Head 12: Syntax, Head 34: Coreference).

References

Vaswani et al. (2017). “Attention Is All You Need.”
Olah et al. (2020). “Zoom In: An Introduction to Circuits.”
Goodfellow et al. (2016). Deep Learning (Chapter on Representation Learning).
Bruna & Mallat (2013). “Invariant Scattering Convolution Networks.”

Conclusion

While ANNs defy direct analogy to classical signal processing, their study benefits from adopting modular, frequency-inspired frameworks. By treating embeddings as modulated carriers and leveraging spectral tools, researchers can systematically reverse-engineer the “signals” governing LLM behavior. However, the non-linear, emergent nature of these systems demands hybrid approaches, blending empirical observation with theoretical innovation. The path forward lies not in forcing ANNs into classical molds but in developing a new language of interpretability—one as revolutionary as the Fourier transform was for engineering.