Relation To Singular Value Decomposition and Wiener Filter
With, this least squares solution can be analyzed in a special way via the singular value decomposition. Given the singular value decomposition of A
with singular values, the Tikhonov regularized solution can be expressed as
where has diagonal values
and is zero elsewhere. This demonstrates the effect of the Tikhonov parameter on the condition number of the regularized problem. For the generalized case a similar representation can be derived using a generalized singular value decomposition.
Finally, it is related to the Wiener filter:
where the Wiener weights are and is the rank of .
Read more about this topic: Tikhonov Regularization
Famous quotes containing the words relation to, relation, singular and/or wiener:
“Light is meaningful only in relation to darkness, and truth presupposes error. It is these mingled opposites which people our life, which make it pungent, intoxicating. We only exist in terms of this conflict, in the zone where black and white clash.”
—Louis Aragon (1897–1982)
“When needs and means become abstract in quality, abstraction is also a character of the reciprocal relation of individuals to one another. This abstract character, universality, is the character of being recognized and is the moment which makes concrete, i.e. social, the isolated and abstract needs and their ways and means of satisfaction.”
—Georg Wilhelm Friedrich Hegel (1770–1831)
“The spider-mind acquires a faculty of memory, and, with it, a singular skill of analysis and synthesis, taking apart and putting together in different relations the meshes of its trap. Man had in the beginning no power of analysis or synthesis approaching that of the spider, or even of the honey-bee; but he had acute sensibility to the higher forces.”
—Henry Brooks Adams (1838–1918)
“The idea that information can be stored in a changing world without an overwhelming depreciation of its value is false. It is scarcely less false than the more plausible claim that after a war we may take our existing weapons, fill their barrels with cylinder oil, and coat their outsides with sprayed rubber film, and let them statically await the next emergency.”
—Norbert Wiener (1894–1964)