Editing Singular value decomposition (section)

===Separable models===
The SVD can be thought of as decomposing a matrix into a weighted, ordered sum of separable matrices. By separable, we mean that a matrix {{tmath|\mathbf A}} can be written as an [[outer product]] of two vectors {{tmath|\mathbf A {{=}} \mathbf u \otimes \mathbf v,}} or, in coordinates, {{tmath|A_{ij} {{=}} u_i v_j.}} Specifically, the matrix {{tmath|\mathbf M}} can be decomposed as,

<math display=block>
\mathbf{M} = \sum_i \mathbf{A}_i
= \sum_i \sigma_i \mathbf U_i \otimes \mathbf V_i.
</math>

Here {{tmath|\mathbf U_i}} and {{tmath|\mathbf V_i}} are the {{tmath|i}}-th columns of the corresponding SVD matrices, {{tmath|\sigma_i}} are the ordered singular values, and each {{tmath|\mathbf A_i}} is separable. The SVD can be used to find the decomposition of an image processing filter into separable horizontal and vertical filters. Note that the number of non-zero {{tmath|\sigma_i}} is exactly the rank of the matrix.{{citation needed|date=November 2023}} Separable models often arise in biological systems, and the SVD factorization is useful to analyze such systems. For example, some visual area V1 simple cells' receptive fields can be well described<ref>{{cite journal |doi=10.1016/0166-2236(95)94496-R |last1=DeAngelis |first1=G. C. |last2=Ohzawa |first2=I. |last3=Freeman |first3=R. D. |title=Receptive-field dynamics in the central visual pathways |journal=Trends Neurosci. |volume=18 |issue=10 |pages=451–8 |date=October 1995 |pmid=8545912 |s2cid=12827601 }}</ref> by a [[Gabor filter]] in the space domain multiplied by a modulation function in the time domain. Thus, given a linear filter evaluated through, for example, [[Spike-triggered average|reverse correlation]], one can rearrange the two spatial dimensions into one dimension, thus yielding a two-dimensional filter (space, time) which can be decomposed through SVD. The first column of {{tmath|\mathbf U}} in the SVD factorization is then a Gabor while the first column of {{tmath|\mathbf V}} represents the time modulation (or vice versa). One may then define an index of separability

<math display=block>
\alpha = \frac{\sigma_1^2}{\sum_i \sigma_i^2},
</math>

which is the fraction of the power in the matrix M which is accounted for by the first separable matrix in the decomposition.<ref>{{cite journal |last1=Depireux |first1=D. A. |last2=Simon |first2=J. Z. |last3=Klein |first3=D. J. |last4=Shamma |first4=S. A. |title=Spectro-temporal response field characterization with dynamic ripples in ferret primary auditory cortex |journal=J. Neurophysiol. |volume=85 |issue=3 |pages=1220–34 |date=March 2001 |pmid=11247991 |doi=10.1152/jn.2001.85.3.1220}}</ref>