Low-rank Approximation

Understanding low-rank approximation

Low-rank approximation finds a simplified representation of a matrix by decomposing it into the product of smaller matrices. For a matrix $M$ of rank $r$, the approximation $\hat{M}$ of rank $k < r$ minimizes the approximation error while using fewer dimensions.

Mathematically, for an $m \times n$ matrix $M$, the rank-$k$ approximation can be expressed as:

$\hat{M} = UV^T$

Where:

• $U$ is an $m \times k$ matrix
• $V$ is an $n \times k$ matrix
• $k$ is the target rank (smaller than the original rank)

Applications in financial markets

Portfolio optimization

Low-rank approximation helps simplify large covariance matrices in portfolio optimization:

1. Reduces noise in empirical correlation matrices
2. Identifies dominant risk factors
3. Improves stability of optimization results

Market microstructure analysis

In market microstructure research, low-rank approximation helps:

1. Extract common trading patterns from order book data
2. Identify market regimes from high-frequency data
3. Reduce dimensionality of feature spaces for machine learning models

Implementation techniques

Singular Value Decomposition (SVD)

The optimal low-rank approximation is achieved through singular value decomposition (SVD), which decomposes a matrix $M$ as:

$M = U\Sigma V^T$

Where:

• $\Sigma$ contains singular values in descending order
• The rank-$k$ approximation uses the top $k$ singular values

Truncated SVD

For large matrices, truncated SVD computes only the top $k$ singular values and vectors:

1. More computationally efficient
2. Requires less memory
3. Directly produces the desired rank-$k$ approximation

Error bounds and optimization

The Eckart-Young-Mirsky theorem provides error bounds for low-rank approximations:

$\|\|M - \hat{M}\|\|_F \geq \sqrt{\sum_{i=k+1}^r \sigma_i^2}$

Where:

• $\|\|\cdot\|\|_F$ is the Frobenius norm
• $\sigma_i$ are singular values
• $k$ is the approximation rank
• $r$ is the original rank

This helps practitioners:

1. Choose appropriate rank for approximation
2. Balance accuracy vs. computational efficiency
3. Quantify information loss

Best practices

When implementing low-rank approximation:

1. Rank selection

   • Use scree plots to identify significant components
   • Consider cross-validation for optimal rank selection
   • Balance complexity reduction vs. information preservation

2. Data preprocessing

   • Center and scale data appropriately
   • Handle missing values
   • Consider robust scaling for outlier-resistant approximations

3. Validation

   • Monitor approximation error
   • Validate results on out-of-sample data
   • Compare against domain-specific benchmarks