The Effective Rank: A Measure of Effective Dimensionality

Many signal processing algorithms include numerical problems where the solution is obtained by adjusting the value of parameters such that a specific matrix exhibits rank deficiency. Since rank minimization is generally not practicable owing to its integer nature, we propose a real-valued extension that we term effective rank. After proving some of its properties, the effective rank is provided with an operational meaning using a result on the coefficient rate of a stationary random process. Finally, the proposed measure is assessed in a practical scenario and other potential applications are suggested.