Why Representation Learning Matters

The mathematical and practical advantages of representation learning in financial markets.

The Power of Learned Representations

Representation learning addresses a fundamental challenge in market analysis: how to transform raw market data into meaningful, actionable features. Unlike predetermined features or LLM embeddings, learned representations capture the inherent structure of market data:

$$\phi: \mathcal{X} \to \mathcal{H}$$

Where $\mathcal{X}$ is the space of raw market data and $\mathcal{H}$ is a learned representation space that captures meaningful market dynamics.

Mathematical Foundations

The power of representation learning comes from its ability to capture complex market structures. Consider a market with multiple assets and various types of relationships. We can model this as a heterogeneous graph:

$$\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{A}, \mathcal{R})$$

Where:

• $\mathcal{V}$ represents vertices (assets, traders, protocols)

• $\mathcal{E}$ represents edges (relationships)

• $\mathcal{A}$ represents vertex attributes

• $\mathcal{R}$ represents relationship types

Through representation learning, we can learn embeddings that preserve the essential structure of this market graph:

$$h_v = f_\phi(v, \mathcal{N}(v)) \\ \text{where }\mathcal{N}(v) = \text{ neighborhood of vertex }v$$

Temporal Dynamics

Market behavior is inherently temporal. Representation learning allows us to capture these dynamics through specialized architectures:

$$h_t = f_\phi(x_t, h_{t-1}) \\ \text{where }h_t\text{ captures market state at time }t$$

This allows for:

1. Multi-scale temporal patterns

2. Regime detection

3. Trend analysis

4. Volatility modeling

The Information Bottleneck

Representation learning operates on the principle of the information bottleneck:

$$\min_{p(h|x)} I(X; H) - \beta I(H; Y)$$

Where:

• $I(X;H)$ is the mutual information between input and representation
• $I(H;Y)$ is the mutual information between representation and target
• $β$ controls the trade-off between compression and prediction

This framework ensures that learned representations:

• Capture relevant market information

• Discard noise

• Maintain predictive power

• Generalize well to new conditions