arXiv subjects

Jochen Trumpf

Publications and source records attributed to Jochen Trumpf.

22 records · Page 2Linked to original sources

arXiv2011

Degrees of Freedom of a Communication Channel and Kolmogorov numbers

In this note, we show that the operator theoretic concept of Kolmogorov numbers and the number of degrees of freedom at level $ε$ of a communication channel are closely related. Linear communication channels may be modeled using linear compact operators on Banach or Hilbert spaces and the number of degrees of freedom of such channels is defined to be the number of linearly independent signals that may be communicated over this channel, where the channel is restricted by a threshold noise level. Kolmogorov numbers are a particular example of $s$-numbers, which are defined over the class of bounded operators between Banach spaces. We demonstrate that these two concepts are closely related, namely that the Kolmogorov numbers correspond to the "jump points" in the function relating numbers of degrees of freedom with the noise level $ε$. We also establish a useful numerical computation result for evaluating Kolmogorov numbers of compact operators.

cs.IT↗

arXiv2009

Variable Metric Stochastic Approximation Theory

We provide a variable metric stochastic approximation theory. In doing so, we provide a convergence theory for a large class of online variable metric methods including the recently introduced online versions of the BFGS algorithm and its limited-memory LBFGS variant. We also discuss the implications of our results for learning from expert advice.

physics.data-an↗

arXiv2009

Degrees of Freedom of a Communication Channel: Using Generalised Singular Values

A fundamental problem in any communication system is: given a communication channel between a transmitter and a receiver, how many "independent" signals can be exchanged between them? Arbitrary communication channels that can be described by linear compact channel operators mapping between normed spaces are examined in this paper. The (well-known) notions of degrees of freedom at level $ε$ and essential dimension of such channels are developed in this general setting. We argue that the degrees of freedom at level $ε$ and the essential dimension fundamentally limit the number of independent signals that can be exchanged between the transmitter and the receiver. We also generalise the concept of singular values of compact operators to be applicable to compact operators defined on arbitrary normed spaces which do not necessarily carry a Hilbert space structure. We show how these generalised singular values can be used to calculate the degrees of freedom at level $ε$ and the essential dimension of compact operators that describe communication channels. We describe physically realistic channels that require such general channel models.

cs.IT↗

arXiv2007

Newton's method on Graßmann manifolds

A general class of Newton algorithms on Graßmann and Lagrange-Graßmann manifolds is introduced, that depends on an arbitrary pair of local coordinates. Local quadratic convergence of the algorithm is shown under a suitable condition on the choice of coordinate systems. Our result extends and unifies previous convergence results for Newton's method on a manifold. Using special choices of the coordinates, new numerical algorithms are derived for principal component analysis and invariant subspace computations with improved computational complexity properties.

math.OC↗