Geometric Theory of Information by Frank Nielsen

By Frank Nielsen

This publication brings jointly geometric instruments and their purposes for info research. It collects present and plenty of makes use of of within the interdisciplinary fields of data Geometry Manifolds in complicated sign, snapshot & Video Processing, complicated information Modeling and research, info rating and Retrieval, Coding, Cognitive platforms, optimum keep watch over, information on Manifolds, desktop studying, Speech/sound acceptance and common language remedy that are additionally considerably suitable for the industry.

Show description

Read or Download Geometric Theory of Information PDF

Similar machine theory books

Theory And Practice Of Uncertain Programming

Real-life judgements tend to be made within the kingdom of uncertainty equivalent to randomness and fuzziness. How can we version optimization difficulties in doubtful environments? How will we remedy those types? so one can resolution those questions, this ebook offers a self-contained, entire and updated presentation of doubtful programming conception, together with quite a few modeling principles, hybrid clever algorithms, and purposes in method reliability layout, undertaking scheduling challenge, car routing challenge, facility situation challenge, and computing device scheduling challenge.

Algebras in Genetics

The aim of those notes is to provide a slightly entire presentation of the mathematical thought of algebras in genetics and to debate intimately many functions to concrete genetic occasions. traditionally, the topic has its foundation in different papers of Etherington in 1939- 1941. basic contributions were given through Schafer, Gonshor, Holgate, Reiers¢l, Heuch, and Abraham.

Augmented Marked Graphs

Petri nets are a proper and theoretically wealthy version for the modelling and research of structures. A subclass of Petri nets, augmented marked graphs own a constitution that's specially fascinating for the modelling and research of structures with concurrent strategies and shared assets. This monograph comprises 3 components: half I presents the conceptual history for readers who've no previous wisdom on Petri nets; half II elaborates the idea of augmented marked graphs; ultimately, half III discusses the applying to method integration.

Large-Scale Scientific Computing: 9th International Conference, LSSC 2013, Sozopol, Bulgaria, June 3-7, 2013. Revised Selected Papers

This ebook constitutes the completely refereed post-conference lawsuits of the ninth overseas convention on Large-Scale medical Computations, LSSC 2013, held in Sozopol, Bulgaria, in June 2013. The seventy four revised complete papers offered including five plenary and invited papers have been conscientiously reviewed and chosen from a number of submissions.

Extra resources for Geometric Theory of Information

Sample text

The proof can be found in the Appendix C. 2 Geometry of Submanifolds ˜ Next, we further investigate the geometry of the submanifolds Ls . Let (˜g(V ) , ⊂, ∗⊂ ˜ (V ) ) be the dualistic structure on Ls induced from (g(V ) , ⊂, ∗ ⊂ (V ) ) on PD(n, R). Since each transformation φG where G ∈ SL(n, R) is an isometry of g˜ (V ) and an element of automorphism group acting on Ls transitively, the Riemannian manifold (Ls , g˜ (V ) ) is a homogeneous space. The isotropy group at κI consists of φG where G ∈ SO(n).

If t = 0, then this equation implies that β3 (s)tr(P−1 X)tr(P−1 Y )P−1 − β2 (s)tr(P−1 XP−1 Y )P−1 ⎝ ⎬ ⎛ ⎥ d∂t (Y ) P−1 − β2 (s)tr(P−1 Y )P−1 XP−1 + β2 (s)tr P−1 dt t=0 50 A. Ohara and S. Eguchi − β2 (s)tr P−1 X P−1 YP−1 + β1 (s)P−1 XP−1 YP−1 ⎛ ⎝ −1 d∂t (Y ) − β1 (s)P P−1 + β1 (s)P−1 YP−1 XP−1 = 0, dt t=0 where s = det P. 25) + β1 (s)(P−1 XP−1 Y + P−1 YP−1 X) − β2 (s) tr P−1 Y P−1 X + tr P−1 X P−1 Y + β3 (s)tr P−1 X tr P−1 Y I − β2 (s)tr P−1 YP−1 X I. 26) dt t=0 ⎞ ⎠ ⎞ ⎠ ⎞ ⎠ =(2β1 (s) − nβ2 (s))tr P−1 XP−1 Y + (nβ3 (s) − 2β2 (s))tr P−1 X tr P−1 Y .

V ) ψ ψx i ψ ψx j ⎝ ⎛ = P d∂−t ((ψ/ψx j )Pt ) dt ⎝ , Pt ∈ PD(n, R), P = P0 . t=0 This completes the proof. , ⎞ ⎠ φG∗ (⊂X Y )P = ⊂X Y P , φG∗ ∗ (V ) ⊂X Y P = ∗ (V ) ⊂X Y P holds for any G ∈ SL(n, R), where P = φG P, X = φG∗ X and Y = φG∗ Y . Particularly, both of the connections induced from the power potential α(V ) , defined via V (s) = c1 + c2 sγ with real constants c1 , c2 and γ, are GL(n, R)-invariant. In addition, so is the orthogonality with respect to g(V ) . Hence, we conclude that both ⊂and ∗ ⊂ (V ) -projections [1, 2] are GL(n, R)-invariant for the power potentials, while so is not g(V ) .

Download PDF sample

Rated 4.12 of 5 – based on 44 votes