Supplement to “Higher-Order Total Variation Classes on Grids: Minimax Theory and Trend Filtering Methods”
نویسندگان
چکیده
the Kronecker sum of DD with itself, a total of d times. Using a standard fact about Kronecker sums, if ρ1, . . . , ρN denote the eigenvalues of DD then ρi1 + ρi2 + · · ·+ ρid , i1, . . . , id ∈ {1, . . . , N}, are the eigenvalues of (∆̃) ∆̃. By counting the multiplicity of the zero eigenvalue, we arrive at a nullity for ∆̃ of (k + 1). One can now directly check that each of the polynomials specified in the lemma is indeed in the null space (e.g., simply by inspection of the formula in (8)), and that these polynomial functions are linearly independent, which completes the proof.
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