Randomized Trigonometric Transforms

• Randomized trigonometric transforms are linear techniques that integrate random sampling and coefficient selection into classical trigonometric methods to enhance sparse recovery and stability.
• They employ random matrix constructions and probabilistic algorithms, offering improved recovery guarantees and efficiency through methods like FFT on randomized multiple rank-1 lattices.
• These transforms are pivotal in applications such as compressed sensing, spectral analysis, and random matrix theory, providing robust frameworks for high-dimensional signal processing.

Randomized trigonometric transforms are a family of linear transformations and computational techniques that introduce stochastic or pseudorandom structure into traditional trigonometric transforms (such as the discrete Fourier, cosine, or sine transforms) or utilize randomness for efficient sampling, sparse recovery, basis construction, or stability analysis. This topic sits at the intersection of harmonic analysis, signal processing, random matrix theory, and probabilistic algorithms, and is fundamental for understanding compressive sensing, randomized numerical linear algebra, and the behavior of random fields built from trigonometric functions.

1. Foundational Principles and Definitions

Randomized trigonometric transforms can take several forms, unified by their application of random elements in either the sampling, coefficient selection, construction of transformation matrices, or phase/randomness injection in the trigonometric kernel. The principal variants are:

• Random sampling of inputs: Sample points for the transform are chosen uniformly at random or via a probabilistic scheme rather than regularly spaced.
• Random coefficients: The coefficients in a trigonometric polynomial are randomly assigned (often as independent Gaussian or other random variables), yielding random fields.
• Random matrix transforms: Transformation matrices are built from random orthogonal/unitary bases (often via Givens rotations or randomization of angles).
• Randomized algorithmic constructions: Probabilistic methods for constructing sampling schemes or frames for sparse representation (e.g., randomly chosen multiple rank-1 lattices for multivariate polynomials).

These forms are motivated by the need for stable recovery (compressed sensing), dimensionality reduction, numerical efficiency, and robustness in noisy or high-dimensional settings.

2. Random Sampling and Sparse Recovery

A canonical application of randomized trigonometric transforms is in sparse recovery of trigonometric polynomials, as studied in (Xu, 2010) and (Kämmerer, 2017). In high dimensions or large ambient spaces, exact recovery of an $M$-sparse trigonometric polynomial typically requires $O(M \log D)$ or $O(M \log T)$ measurements, where $D$ is the ambient dimension and $T$ the sparsity.

Random sampling of the input points ($x_j$ drawn uniformly at random from $[0,1]^d$) yields measurement matrices whose columns correspond to randomly sampled trigonometric functions. These matrices typically satisfy incoherence or restricted isometry properties with high probability, allowing signal recovery via greedy or convex algorithms (Orthogonal Matching Pursuit, Basis Pursuit) with theoretical bounds:

• The randomized Fourier measurement matrix is $\mathcal{F}_X = [\exp(2\pi i k \cdot x_j)]_{j,k}$ for random $x_j$.
• With suitable oversampling (typically $N \geq C M \log D$), recovery of the coefficients is guaranteed w.h.p.

Probabilistic methods also enable efficient randomized lattice constructions (multiple rank-1 lattices) (Kämmerer, 2017), achieving nearly optimal oversampling $M/T \leq C \log T$ independently of ambient dimension, and supporting fast algorithms via FFTs for high-dimensional polynomials.

3. Randomized Matrix and Basis Construction

Random paraunitary projections (Queiroz, 2021) detail the construction of large random unitary matrices via hierarchical cascades of Givens rotations:

$U = S \prod_{i=0}^{M-2} \prod_{j=i+1}^{M-1} R_{ij}$

Random angles $\theta_{ij}$ are drawn from well-defined distributions to ensure near-Gaussian statistics and decorrelated outputs. Such random bases are essential for randomized projections in compressive sensing and iterative reconstruction (e.g., COSAMP), delivering rapid inverses through simple reversal of rotation order and angles.

Extensions include randomized paraunitary filter banks and adaptive under-decimated systems, where the compression ratio or number of projections may be tuned locally based on signal sparsity, with the transform remaining efficiently invertible via seed-based angle generation.

4. Spectral Properties and Universality

Randomized trigonometric transforms give rise to stochastic fields whose real zeros, local structure, and global fluctuations are robust to coefficient distribution and dependencies. Universality results, such as those in (Iksanov et al., 2016) and (Angst et al., 23 Sep 2024), establish that for trigonometric polynomials with independent or weakly dependent coefficients, the distribution of zero crossings and, critically, the asymptotic density of zeros remains unchanged under broad conditions:

• For polynomials $$X_n(t) = \sum_{k=1}^n (\xi_k \sin kt + \eta_k \cos kt)$$ with i.i.d. random coefficients, local zero statistics (when rescaled appropriately) converge to those of stationary Gaussian processes with $$\operatorname{Cov}(Z(t),Z(s)) = \frac{\sin(t-s)}{t-s}$$.
• Even with arbitrary dependence (given bounded moments and spectral densities), universality of the expected global zero count persists: $$\lim_{n\to\infty} \frac{E \mathcal{N}(f_n, [0,2\pi])}{n} = 2/\sqrt{3}$$ (Angst et al., 23 Sep 2024).

These findings underscore the robustness of nodal patterns in randomized spectra and inform the stability of algorithms based on random trigonometric transforms.

5. Multiplicative Chaos and Random Measures

When randomness in coefficients is sufficiently pronounced (e.g., $\sum_n \rho_n^2 = +\infty$ in $\sum_n \rho_n \cos (nt+\omega_n)$), the resulting trigonometric series defines a multifractal random measure through trigonometric multiplicative chaos (Fan et al., 2021). In such scenarios:

• The partial sums diverge almost surely, failing to represent Fourier–Stieltjes measures, and instead form pseudofunctions.
• The exponential martingale $Q_n(t) = \prod_{k=1}^n P_k(t)$ converges weakly to a nontrivial random measure $Q$, whose multifractal properties and Hausdorff dimension drop are explicitly linked to the correlation structure, e.g.:

$$H_\alpha(t) = -\frac{\alpha^2}{2} \log|2\sin(t/2)| + O(1)$$

• In higher dimensions ($\mathbb{T}^d$), the dimension formula is $d - (\tau(d)\alpha^2)/4$.

This theory interfaces ergodic theory, potential analysis, and quantum gravity, and offers a framework for randomizing trigonometric decompositions in analytical and physical models.

6. Fast Algorithms and Applications

Randomized trigonometric transforms undergird fast algorithms for high-dimensional Fourier analysis, compressed sensing, and sparse recovery. By leveraging randomization in sampling or basis construction:

• Nearly optimal sampling and recovery with minimal measurements are achieved—often with logarithmic oversampling independent of ambient dimension.
• Fast discrete transforms exploiting the structure of randomized multiple-rank-1 lattices enable computational complexity $O(T\log^2 T)$ (Kämmerer, 2017).
• Adaptive filter banks and paraunitary systems are realized with highly efficient inversion schemes (Queiroz, 2021).

These algorithms have direct applicability to uncertainty quantification, spectral analysis in physics, and fast signal processing in large-scale engineering problems.

7. Positivity and Stability Criteria

The introduction of randomness to the trigonometric kernel or envelope function is compatible with classical positivity and stability criteria for oscillatory transforms, provided monotonicity and controlled kernel zero patterns are preserved (Cho et al., 2023). For randomized transforms where the kernel is, for example, $u_R(xt) = \sin(xt + \theta(\omega))$, positivity is maintained almost surely under suitable conditions. This property is essential in signal reconstruction and harmonic analysis, ensuring algorithmic robustness against noise and perturbations.

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In summary, randomized trigonometric transforms bring together probabilistic sampling, random matrix theory, multiplicative chaos, and universality principles to yield efficient, robust, and theoretically sound frameworks for high-dimensional harmonic analysis, signal processing, and the paper of random fields and measures. The development of these methods is tightly linked to advances in sampling theory, spectral analysis of random matrices and polynomials, and the deep interplay between number theory and randomness in computational harmonic analysis.