Bifurcation structure of equilibria of iterated softmax

Peter Tino

doi:10.1016/j.chaos.2008.07.026

Bifurcation structure of equilibria of iterated softmax

Peter Tino

Computer Science

Research output: Contribution to journal › Article

2 Citations (Scopus)

Abstract

We present a detailed bifurcation study of iterated renormalization process driven by softmax transformation parametrized by a temperature parameter. For each emerging equilibrium we give exact characterization of stable/unstable manifolds of the linearized dynamics. As the system cools down, new equilibria emerge in a strong structure until finally a complex skeleton of saddle type equilibria surrounding an unstable maximum entropy point, with decision enforcing "one-hot" stable equilibria emerges. (C) 2008 Elsevier Ltd. All rights reserved.

Original language	English
Pages (from-to)	1804-1816
Number of pages	13
Journal	Chaos, Solitons and Fractals
Volume	41
Issue number	4
Early online date	10 Sept 2008
DOIs	https://doi.org/10.1016/j.chaos.2008.07.026
Publication status	Published - 30 Aug 2009

Access to Document

10.1016/j.chaos.2008.07.026

Cite this

@article{f5519fe04c1e4f63bceea957b655fcca,

title = "Bifurcation structure of equilibria of iterated softmax",

abstract = "We present a detailed bifurcation study of iterated renormalization process driven by softmax transformation parametrized by a temperature parameter. For each emerging equilibrium we give exact characterization of stable/unstable manifolds of the linearized dynamics. As the system cools down, new equilibria emerge in a strong structure until finally a complex skeleton of saddle type equilibria surrounding an unstable maximum entropy point, with decision enforcing {"}one-hot{"} stable equilibria emerges. (C) 2008 Elsevier Ltd. All rights reserved.",

author = "Peter Tino",

year = "2009",

month = aug,

day = "30",

doi = "10.1016/j.chaos.2008.07.026",

language = "English",

volume = "41",

pages = "1804--1816",

journal = "Chaos, Solitons and Fractals",

publisher = "Elsevier",

number = "4",

}

TY - JOUR

T1 - Bifurcation structure of equilibria of iterated softmax

AU - Tino, Peter

PY - 2009/8/30

Y1 - 2009/8/30

N2 - We present a detailed bifurcation study of iterated renormalization process driven by softmax transformation parametrized by a temperature parameter. For each emerging equilibrium we give exact characterization of stable/unstable manifolds of the linearized dynamics. As the system cools down, new equilibria emerge in a strong structure until finally a complex skeleton of saddle type equilibria surrounding an unstable maximum entropy point, with decision enforcing "one-hot" stable equilibria emerges. (C) 2008 Elsevier Ltd. All rights reserved.

AB - We present a detailed bifurcation study of iterated renormalization process driven by softmax transformation parametrized by a temperature parameter. For each emerging equilibrium we give exact characterization of stable/unstable manifolds of the linearized dynamics. As the system cools down, new equilibria emerge in a strong structure until finally a complex skeleton of saddle type equilibria surrounding an unstable maximum entropy point, with decision enforcing "one-hot" stable equilibria emerges. (C) 2008 Elsevier Ltd. All rights reserved.

U2 - 10.1016/j.chaos.2008.07.026

DO - 10.1016/j.chaos.2008.07.026

M3 - Article

VL - 41

SP - 1804

EP - 1816

JO - Chaos, Solitons and Fractals

JF - Chaos, Solitons and Fractals

IS - 4

ER -

Bifurcation structure of equilibria of iterated softmax

Abstract

Access to Document

Fingerprint

Cite this