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Support Recovery of Periodic Mixtures with Nested Periodic Dictionaries

Pouria Saidi, George K. Atia

TL;DR

This work addresses exact support recovery for periodic mixtures represented in nested periodic dictionaries (NPDs), enabling reliable recovery of hidden periods. It introduces new coherence notions—nested periodic inter/intra-coherence (NPI/NPA) and their restricted/cumulative variants (CNPI/CNPA)—to exploit the Euler and LCM structures of Ramanujan subspaces, yielding tighter recovery guarantees than traditional RIP or coherence-based bounds. The authors derive exact recovery conditions in the noise-free setting (ζ_{k,m}+ν_{k,m}<1) and extend them to noisy scenarios (bounded and Gaussian) using OMP/BP, with refined bounds that account for actual sparsity s≤k and problem-specific sets Q_k(m). Numerical results on Farey and Ramanujan-type dictionaries (RPT and Farey) demonstrate significant improvements over generic bounds and illustrate practical phase transitions, informing dictionary choice for period estimation. Overall, the work advances theoretical guarantees for period estimation from NPD representations and guides robust recovery in real-world, noisy data contexts.

Abstract

Periodic signals composed of periodic mixtures admit sparse representations in nested periodic dictionaries (NPDs). Therefore, their underlying hidden periods can be estimated by recovering the exact support of said representations. In this paper, support recovery guarantees of such signals are derived both in noise-free and noisy settings. While exact recovery conditions have been studied in the theory of compressive sensing, existing conditions fall short of yielding meaningful achievability regions in the context of periodic signals with sparse representations in NPDs, in part since existing bounds do not capture structures intrinsic to these dictionaries. We leverage known properties of NPDs to derive several conditions for exact sparse recovery of periodic mixtures in the noise-free setting. These conditions rest on newly introduced notions of nested periodic coherence and restricted coherence, which can be efficiently computed. In the presence of noise, we obtain improved conditions for recovering the exact support set of the sparse representation of the periodic mixture via orthogonal matching pursuit based on the introduced notions of coherence. The theoretical findings are corroborated using numerical experiments for different families of NPDs. Our results show significant improvement over generic recovery bounds as the conditions hold over a larger range of sparsity levels.

Support Recovery of Periodic Mixtures with Nested Periodic Dictionaries

TL;DR

This work addresses exact support recovery for periodic mixtures represented in nested periodic dictionaries (NPDs), enabling reliable recovery of hidden periods. It introduces new coherence notions—nested periodic inter/intra-coherence (NPI/NPA) and their restricted/cumulative variants (CNPI/CNPA)—to exploit the Euler and LCM structures of Ramanujan subspaces, yielding tighter recovery guarantees than traditional RIP or coherence-based bounds. The authors derive exact recovery conditions in the noise-free setting (ζ_{k,m}+ν_{k,m}<1) and extend them to noisy scenarios (bounded and Gaussian) using OMP/BP, with refined bounds that account for actual sparsity s≤k and problem-specific sets Q_k(m). Numerical results on Farey and Ramanujan-type dictionaries (RPT and Farey) demonstrate significant improvements over generic bounds and illustrate practical phase transitions, informing dictionary choice for period estimation. Overall, the work advances theoretical guarantees for period estimation from NPD representations and guides robust recovery in real-world, noisy data contexts.

Abstract

Periodic signals composed of periodic mixtures admit sparse representations in nested periodic dictionaries (NPDs). Therefore, their underlying hidden periods can be estimated by recovering the exact support of said representations. In this paper, support recovery guarantees of such signals are derived both in noise-free and noisy settings. While exact recovery conditions have been studied in the theory of compressive sensing, existing conditions fall short of yielding meaningful achievability regions in the context of periodic signals with sparse representations in NPDs, in part since existing bounds do not capture structures intrinsic to these dictionaries. We leverage known properties of NPDs to derive several conditions for exact sparse recovery of periodic mixtures in the noise-free setting. These conditions rest on newly introduced notions of nested periodic coherence and restricted coherence, which can be efficiently computed. In the presence of noise, we obtain improved conditions for recovering the exact support set of the sparse representation of the periodic mixture via orthogonal matching pursuit based on the introduced notions of coherence. The theoretical findings are corroborated using numerical experiments for different families of NPDs. Our results show significant improvement over generic recovery bounds as the conditions hold over a larger range of sparsity levels.

Paper Structure

This paper contains 26 sections, 11 theorems, 62 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and paper organization
  3. Background and preliminaries
  4. Sparse recovery conditions
  5. Nested periodic matrices
  6. Ramanujan periodicity transform (RPT)
  7. Farey dictionary
  8. Nested periodic dictionaries (NPDs)
  9. Ramanujan subspaces
  10. Period estimation via support recovery
  11. Problem Setup and Main Results
  12. Exact sparse recovery guarantees in the noise-free case
  13. Support recovery guarantees in the presence of noise
  14. Bounded noise
  15. Gaussian noise
  16. ...and 11 more sections

Key Result

Theorem 1

Suppose that $\mathbf{K}$ is an NPD of size $L\times N$, $\mathcal{T} \in \mathbb{Q}_k\left(m\right)$ contains the hidden periods of a periodic mixture where $\mathbb{Q}_k\left(m\right)$ is as in Definition def:Qkm, and $\zeta_{k,m}$ and $\nu_{k,m}$ are the NPI and NPA in eq:npi_coherence and eq:npa then one can identify all the hidden periods of the periodic mixture by recovering the sparse vecto

Figures (10)

  • Figure 1: Sparse recovery of periodic signals in NPDs using basis pursuit (a sparse recovery framework) and its agreement with true sparse representation of the signal (a) L = 200 and (b) L = 80.
  • Figure 2: The nonzero coefficients of the sparse representations of the periodic signals with period 20 (a) L = 200 and (b) L = 80. Here, each group of nonzero coefficients illustrated with a distinct color corresponds to the same submatrix $\mathbf{R}_p$.
  • Figure 3: RPT dictionary with $P_{\max} = 6$ and $L = 8$. The periodically extended version of submatrix $\mathbf{C}_5$ and its corresponding indices in $I_5$ are shown in a distinct color. The columns of the dictionary are not normalized.
  • Figure 4: Calculation of the NPI in \ref{['eq:npi_coherence']}. (a) For $\mathcal{T} = \{3,4\} \in \mathbb{Q}_6\left(2\right)$, the atoms in $S^c_{\mathcal{T}}$ and $S_{\mathcal{T}}$ are shown in green and blue boxes, respectively. The inner summation in \ref{['eq:npi_coherence']} is over the absolute value of the inner products between one atom from $S^c_{\mathcal{T}}$ (e.g., $i = 7$ marked with a red rectangle) and all the atoms in $S_{\mathcal{T}}$. (b) The inner products of the atoms in $S_\mathcal{T}$ and $S_\mathcal{T}^c$ are shown in blue color. Each row corresponds to an atom from $S_\mathcal{T}^c$, e.g., the first row shows the summands (inner products) in the summation over $j\in S_{\mathcal{T}}$ for $i = 7$. The inner maximization in \ref{['eq:npi_coherence']} selects the maximizing row. (c) The difference between the NPI and CNPI, for $\mathcal{T} = \{3,4\}$ and $s = 3$: instead of summing over all $j \in S_{\mathcal{T}}$, we sum over the largest $s$ values.
  • Figure 5: Evaluation of the condition in (\ref{['eq:ERC_mixture_general']}) restricted to set $\mathbb{Q}_k\left(m\right)$ for (a) the Farey dictionary and (b) the RPT dictionary when $L = 1915$.
  • ...and 5 more figures

Theorems & Definitions (21)

  • Definition 1: Support set
  • Example 1
  • Definition 2: Union of support sets
  • Example 2
  • Definition 3
  • Definition 4
  • Theorem 1: Recovery condition for mixtures in $\mathbb{Q}_k\left(m\right)$
  • Definition 5
  • Theorem 2: Computationally efficient alternative bound
  • Corollary 3
  • ...and 11 more