On the Performance of Manhattan Nonnegative Matrix Factorization.

Area of Science:

• Machine Learning
• Signal Processing
• Statistical Learning Theory

Background:

• Matrix factorization techniques are crucial for extracting low-rank and sparse structures.
• Conventional Nonnegative Matrix Factorization (NMF) has limitations with heavy-tailed noise.
• Manhattan Nonnegative Matrix Factorization (MahNMF) extends NMF to handle Laplacian noise.

Purpose of the Study:

• To statistically analyze the performance of MahNMF.
• To decompose the expected reconstruction error into estimation and approximation components.
• To provide a framework for understanding MahNMF's generalization error bounds.

Main Methods:

• Decomposition of expected reconstruction error in MahNMF.
• Bounding estimation error using generalization error bounds.
• Analyzing approximation error via asymptotic results of vector quantization.

Main Results:

• The generalization error bound is established, aiding in determining necessary training sample sizes.
• Statistical performance analysis quantifies the effect of dimensionality reduction on estimation and approximation errors.
• The developed framework is applicable to analyzing the performance of standard NMF.

Conclusions:

• MahNMF provides a statistically sound approach for matrix decomposition with heavy-tailed noise.
• Understanding the interplay between estimation and approximation errors is key to optimizing MahNMF.
• The theoretical framework enhances the interpretability and applicability of MahNMF in various domains.