Sparsity for isotropic spherical random fields

Abstract

We introduce a simple representation for isotropic spherical random fields and we discuss how it allows to discuss different notions of sparsity under isotropy. We also show how a suitable construction of sparse fields can mimic well the angular power spectrum and the polyspectra of some popular non-Gaussian fields, at the same time allowing for computationally efficient simulation algorithms. Using related ideas we also show how it is possible to obtain sparse approximations of spherical random fields which preserve isotropy, thus addressing an issue which has been raised in the Cosmological literature.

Figures (5)

• Figure 1: Simulation of a Whittle-Matérn field with $\beta=1.01$ (bottom image) with resolution $L_{\max}=128$ and $n_{\mathrm{side}}=64$. In the first three rows, we have plotted the sparse random fields written as superposition of $K=4,\,24,\,40$ random waves. On the left column, the random weights $\{\eta_{\ell k}\}$ are taken to be Gaussian random variables with variance $4\pi C_\ell/K$. On the right column, the random weights $\{\eta_{\ell k}\}$ are centered Bernoulli random variables normalized with $4\pi C_\ell/K$.
• Figure 2: Simulation of a Whittle-Matérn field with $\beta=1.5$ (bottom image) with resolution $L_{\max}=128$ and $n_{\mathrm{side}}=64$. In the first three rows, we have plotted the sparse random fields written as superposition of $K=4,\,24,\,40$ random waves. On the left column, the random weights $\{\eta_{\ell k}\}$ are taken to be Gaussian random variables with variance $4\pi C_\ell/K$. On the right column, the random weights $\{\eta_{\ell k}\}$ are centered Bernoulli random variables normalized with $4\pi C_\ell/K$.
• Figure 3: Simulation of a Whittle-Matérn field with $\beta=2$ (bottom image) with resolution $L_{\max}=128$ and $n_{\mathrm{side}}=64$. In the first three rows, we have plotted the sparse random fields written as superposition of $K=4,\,24,\,40$ random waves. On the left column, the random weights $\{\eta_{\ell k}\}$ are taken to be Gaussian random variables with variance $4\pi C_\ell/K$. On the right column, the random weights $\{\eta_{\ell k}\}$ are centered Bernoulli random variables normalized with $4\pi C_\ell/K$.
• Figure 4: Simulation of a Whittle-Matérn field with $\beta=1.01$ (bottom image) with resolution $L_{\max}=256$ and $n_{\mathrm{side}}=128$. In the first three rows, we have plotted the sparse random fields written as superposition of $K=4,\,24,\,40$ random waves. On the left column, the random weights $\{\eta_{\ell k}\}$ are taken to be Gaussian random variables with variance $4\pi C_\ell/K$. On the right column, the random weights $\{\eta_{\ell k}\}$ are centered Bernoulli random variables normalized with $4\pi C_\ell/K$.
• Figure 5: Simulation of a Whittle-Matérn field with $\beta=1.5$ (bottom image) with resolution $L_{\max}=256$ and $n_{\mathrm{side}}=128$. In the first three rows, we have plotted the sparse random fields written as superposition of $K=4,\,24,\,40$ random waves. On the left column, the random weights $\{\eta_{\ell k}\}$ are taken to be Gaussian random variables with variance $4\pi C_\ell/K$. On the right column, the random weights $\{\eta_{\ell k}\}$ are centered Bernoulli random variables normalized with $4\pi C_\ell/K$.

Theorems & Definitions (16)

• remark 1
• remark 2
• definition 1
• definition 2
• remark 3: Power spectrum
• remark 4
• theorem 1
• proof : Proof of \ref{['thm:superposition:representation']}
• corollary 1
• proof