Primitivity, freeness, norm and trace

Primitivity, freeness, norm and trace

Trace and Norm

Let L/K be a finite extension of fields, with n = [L : K]. We will associate to this extension two important functions L→ K, called the trace and the norm. For each α ∈ L, let mα : L → L be multiplication by α: mα(x) = αx for x ∈ L. Each mα is a K-linear map from L to L, so choosing a K-basis of L lets us write mα as an n×n matrix, which will be denoted [mα], or [mα]L/K if we need to emphasize ...

Rank, Trace-Norm and Max-Norm

We study the rank, trace-norm and max-norm as complexity measures of matrices, focusing on the problem of fitting a matrix with matrices having low complexity. We present generalization error bounds for predicting unobserved entries that are based on these measures. We also consider the possible relations between these measures. We show gaps between them, and bounds on the extent of such gaps.

Trace and Norm, Ii

where σ1(α), . . . , σd(α) are all the distinct values of σ(α) as σ runs over the Galois group. For each σ ∈ G, σ(α) = σi(α) for a unique i from 1 to d. Moreover, σ(α) = σi(α) if and only if σ ∈ σiH, where H = {τ ∈ G : τ(α) = α} = Gal(L/K(α)). Therefore as σ runs over G, the number σi(α) appears as σ(α) whenever σ is in the left coset σiH, so σi(α) occurs |H| times, and |H| = [L : K(α)] = [L : ...

Sparse Trace Norm Regularization

We study the problem of estimating multiple predictive functions from a dictionary of basis functions in the nonparametric regression setting. Our estimation scheme assumes that each predictive function can be estimated in the form of a linear combination of the basis functions. By assuming that the coefficient matrix admits a sparse low-rank structure, we formulate the function estimation prob...