Atomic norm minimization for decomposition into complex exponentials and optimal transport in Fourier domain

Abstract

This paper is devoted to the decomposition of vectors into sampled complex exponentials; or, equivalently, to the information over discrete measures captured in a finite sequence of their Fourier coefficients. We study existence, uniqueness, and cardinality properties, as well as computational aspects of estimation using convex semidefinite programs. We then explore optimal transport between measures, of which only a finite sequence of Fourier coefficients is known.

An Epigraphical Approach to the Representer Theorem

“…, t fc−1 }. Since that set is equispaced in T, we notice as in [Con19] that, to recover the amplitudes a j , we may invert the system Since the function cotan is (strictly) decreasing on ]0, π[, we see that a ℓ = 0 with sign(a ℓ ) = sign(η * (t ℓ )), which is the desired optimality condition. It remains to check that (33) also holds for k = f c .…”

“…where {ϕ k } 2fc k=0 is again the trigonometric system. In [Con19], Laurent Condat has observed that when y is the Fourier coefficient vector of two opposite close spikes, a solution to (28) is a Dirac comb. More precisely, if y = Φm 0 def.…”

“…While this observation in [Con19] seems to rely on numerical experiments, we provide in Appendix A a proof relying on a duality argument.…”

“…As a consequence of Proposition 1, the number of Dirac masses predicted by Corollary 1 is almost optimal (2f c + 1 Dirac masses are predicted whereas 2f c actually appear). In fact, one cannot do "better": it is proved in [Con19] that for every y ∈ R 2fc+1 , there is a solution to (28) which is a sum of at most 2f c Dirac masses.…”

Tikhonov Regularization of Circle-Valued Signals

“…In this work, we use trigonometric moments, or Fourier coefficients. The characterization of a measure on the circle from a subset of its Fourier coefficients has a long history, rooted in Carathéodory's work [24]; in short, the latter constraint is satisfied if the Toeplitz matrix formed by the c m is positive semidefinite [25], [26].…”