# Effective Rank

Roy *et al*^{1} provides the following definition for the effective rank:

Consider a matrix $A$ with a singular value decomposition (SVD)

where D is a $M \times N$ diagonal matrix with the singular values

where $Q = \min\{M, N\}$.

**Definition 1 (Nuclear norm)** *The nuclear norm of the matrix $A$ is defined as*

Then the sequence of singular values gives rise to a nominal distribution

**Definition 2 (Effective rank)** *The effective rank of a matrix $A$ is defined
as the volume*

*where $H$ is the Shannon entropy of the nominal distribution given by the matrix spectrum.*

# Entropy of A Continuous Spectrum

The effective rank given by Roy *et al* is only defined on discrete spectrum. We know that
computing the entropy of a continuous distribution require special care^{2}.

# Spectral Entropy in Finite Sampling Window

An additional consideration when computing the spectral entropy on a discrete spectrum, is that
the windowing function produced by applying discrete Fourier transform in a finite data
window imprints a spectral floor that contributes to the spectral distribution. We can subtract
the spectral entropy of this windowing function from the spectral entropy of the discrete Fourier
spectrum, *s.t.* the entropy of the discrete spectrum is $0$ when there is just a single component.

**Acknowledgements:**

Leave the Acknowledgements here.

# References

- Roy, O. and Vetterli, M. (2007) ‘The effective rank: A measure of effective dimensionality’, in 2007 15th European Signal Processing Conference, pp. 606–610. Available at: http://ieeexplore.ieee.org/document/7098875/↩
- Ge Yang, 'Differential Entropy and Invariant Entropy Measure', Available at: https://www.episodeyang.com/blog/2022/10-23/entropy_continuous/↩