Information Geometry - Introduction - Alpha Connection

Alpha Connection

A non-metric connection is not determined by a metric tensor ; instead, it is and restricted by the requirement that the parallel transport between points and must be a linear combination of the base vectors in . Here,

expresses the parallel transport of as linear combination of the base vectors in, i.e. the new minus the change. Note that it is not a tensor (does not transform as a tensor).

For such a metric, one can construct a dual connection to make

,

for parallel transport using and .

For the mentioned -families the affine connection is called the -connection and can also be expressed in more ways.

For :

  • is a metric connection and with .
  • ,

    i.e. is dual to with respect to the Fisher metric.

  • If this is called -affine. Its dual is then -affine.

    , i.e. 0-affine, and hence, i.e. 1-affine.

Read more about this topic:  Information Geometry, Introduction

Famous quotes containing the words connection and/or alpha:

    One must always maintain one’s connection to the past and yet ceaselessly pull away from it. To remain in touch with the past requires a love of memory. To remain in touch with the past requires a constant imaginative effort.
    Gaston Bachelard (1884–1962)

    Imagination is a valuable asset in business and she has a sister, Understanding, who also serves. Together they make a splendid team and business problems dissolve and the impossible is accomplished by their ministrations.... Imagination concerning the world’s wants and the individual’s needs should be the Alpha and Omega of self-education.
    Alice Foote MacDougall (1867–1945)