Estimating the Frequency of a Clustered Signal

@article{Chen2019EstimatingTF,
  title={Estimating the Frequency of a Clustered Signal},
  author={Xue Chen and Eric Price},
  journal={ArXiv},
  year={2019},
  volume={abs/1904.13043}
}
We consider the problem of locating a signal whose frequencies are "off grid" and clustered in a narrow band. Given noisy sample access to a function $g(t)$ with Fourier spectrum in a narrow range $[f_0 - \Delta, f_0 + \Delta]$, how accurately is it possible to identify $f_0$? We present generic conditions on $g$ that allow for efficient, accurate estimates of the frequency. We then show bounds on these conditions for $k$-Fourier-sparse signals that imply recovery of $f_0$ to within $\Delta… 
4 Citations

Sample Efficient Toeplitz Covariance Estimation

A new covariance estimation strategy is developed that further improves on all existing methods in the low-rank case and achieves sample complexity depending polynomially on $k$ and only logarithmically on $d$.

Fourier Sparse Leverage Scores and Approximate Kernel Learning

New explicit upper bounds on the leverage scores of Fourier sparse functions under both the Gaussian and Laplace measures are proved, which generalize existing work that only applies to uniformly distributed data.

(Nearly) Sample-Optimal Sparse Fourier Transform in Any Dimension; RIPless and Filterless

This algorithm uses O(k log k log n) samples, is dimension-free, operates for any universe size, and achieves the strongest ℓ_∞/ℓ-2 guarantee, while running in a time comparable to the Fast Fourier Transform.

The Statistical Cost of Robust Kernel Hyperparameter Tuning

This paper provides finite-sample guarantees for the problem of finding the best interpolant from a class of kernels with unknown hyperparameters, assuming only that the noise is square-integrable, and shows that hyperparameter optimization increases sample complexity by just a logarithmic factor in comparison to the setting where optimal parameters are known in advance.

References

SHOWING 1-10 OF 14 REFERENCES

Near-optimal sparse fourier representations via sampling

An algorithm for finding a Fourier representation of B for a given discrete signal signal A, such that A is within the factor (1 +ε) of best possible $\|\signal-\repn_\opt\|_2^2$.

Fourier-Sparse Interpolation without a Frequency Gap

It is shown how to estimate the signal with a constant factor growth of the noise and sample complexity polynomial in k and logarithmic in the bandwidth and signal-to-noise ratio for arbitrary k-Fourier-sparse signals under l2 bounded noise.

A universal sampling method for reconstructing signals with simple Fourier transforms

This work extends tools based on statistical leverage score sampling and column-based matrix reconstruction to the approximation of continuous linear operators that arise in the signal reconstruction problem, based on a novel connection between randomized linear algebra and the problem of reconstructing signals with constrained Fourier structure.

A Robust Sparse Fourier Transform in the Continuous Setting

  • Eric PriceZhao Song
  • Computer Science
    2015 IEEE 56th Annual Symposium on Foundations of Computer Science
  • 2015
An algorithm for robustly computing sparse Fourier transforms in the continuous setting and new results for how precisely the individual frequencies of x* can be recovered are given.

Simple and practical algorithm for sparse Fourier transform

This work considers the sparse Fourier transform problem, and proposes a new algorithm, which leverages techniques from digital signal processing, notably Gaussian and Dolph-Chebyshev filters, and is faster than FFT, both in theory and practice.

Improved time bounds for near-optimal sparse Fourier representations

A significantly improved algorithm for the problem of finding a Fourier representation R of m terms for a given discrete signal A of length N and a quadratic-in-m algorithm that works for any values of Ni's is given.

Sample-Optimal Fourier Sampling in Any Constant Dimension

  • P. IndykM. Kapralov
  • Computer Science
    2014 IEEE 55th Annual Symposium on Foundations of Computer Science
  • 2014
An algorithm for sparse recovery from Fourier measurements using O(k log N) samples, matching the lower bound of Do Ba-Indyk-Price-Woodruff'10 for non-adaptive algorithms up to constant factors for any k ≤ N<sup>1-δ</sup>.

The Threshold for Super-resolution via Extremal Functions

This work exactly resolve the threshold at which noisy super-resolution is possible and establishes a sharp phase transition for the relationship between the cutoff frequency (m) and the separation (∆).

Exact maximum likelihood parameter estimation of superimposed exponential signals in noise

A unified framework for the exact maximum likelihood estimation of the parameters of superimposed exponential signals in noise, encompassing both the time series and the array problems, is presented and the present formulation is used to interpret previous methods.

Randomized Interpolation and Approximation of Sparse Polynomials

  • Y. Mansour
  • Computer Science, Mathematics
    SIAM J. Comput.
  • 1992
A randomized algorithm is presented that interpolates a sparse polynomial in polynometric time in the bit complexity model and can be applied to approximate polynomials that can be approximated by sparse poynomials.