Non-negative Matrix Factorization (NMF)

Understanding NMF

NMF decomposes a matrix $V$ into two matrices $W$ and $H$ such that:

$V \approx WH$

where:

• $V$ is the original $m \times n$ non-negative matrix
• $W$ is an $m \times k$ non-negative matrix
• $H$ is a $k \times n$ non-negative matrix
• $k$ is the chosen rank (typically $k < \min(m,n)$)

The key constraint is that all elements must be non-negative:

$V_{ij}, W_{ij}, H_{ij} \geq 0$

Applications in financial markets

Portfolio analysis

NMF helps decompose market returns into interpretable factors, with the non-negativity constraint naturally aligning with long-only portfolio constraints. This makes it valuable for:

• Factor investing strategies
• Risk decomposition
• Portfolio allocation

Market microstructure

In market microstructure analysis, NMF can help identify:

• Trading patterns in order flow
• Market regimes
• Liquidity components

Optimization approach

The NMF problem is typically solved through iterative optimization. The most common objective function is the Frobenius norm:

$\min_{W,H \geq 0} \|V - WH\|_F^2$

Common update rules include:

1. Multiplicative updates: $H_{ij} \leftarrow H_{ij} \frac{(W^TV)_{ij}}{(W^TWH)_{ij}}$

   $W_{ij} \leftarrow W_{ij} \frac{(VH^T)_{ij}}{(WHH^T)_{ij}}$

2. Alternating least squares with non-negativity constraints

Advantages and limitations

Advantages

• Produces naturally interpretable components due to non-negativity
• Well-suited for additive data models
• Handles sparse data effectively
• Results often align with physical or economic constraints

Limitations

• Non-convex optimization problem
• Multiple local minima
• Sensitive to initialization
• Computationally more intensive than SVD

Implementation considerations

When implementing NMF for financial applications, key considerations include:

1. Rank selection: The choice of $k$ impacts:

   • Model complexity
   • Interpretability
   • Computational cost

2. Initialization strategies:

   • Random initialization
   • SVD-based initialization
   • Multiple restarts to avoid poor local minima

3. Convergence criteria:

   • Relative improvement threshold
   • Maximum iterations
   • Reconstruction error targets

Relationship to other techniques

NMF complements other matrix decomposition methods:

• Unlike singular value decomposition (SVD), NMF enforces non-negativity
• Compared to low-rank approximation, NMF provides more interpretable factors
• Often used alongside principal component analysis (PCA) for comprehensive analysis

Applications in time-series analysis

In time-series analysis, NMF can:

1. Decompose temporal patterns:

   • Identify recurring market regimes
   • Extract seasonal components
   • Detect anomalies

2. Feature extraction:

   • Create low-dimensional representations
   • Extract interpretable features for machine learning
   • Reduce noise while preserving signal

3. Pattern discovery:

   • Uncover hidden market dynamics
   • Identify trading opportunities
   • Detect structural breaks