[Paper Review] A robust multi-dimensional sparse Fourier transform in the continuous setting.

Yaonan Jin, Daogao Liu+1 more|arXiv (Cornell University)|May 13, 2020

Sparse and Compressive Sensing Techniques58 references6 citations

TL;DR

This paper presents a randomized, non-adaptive algorithm for the multi-dimensional continuous sparse Fourier transform, enabling sublinear sample and time complexity. It achieves improved sample duration bounds over prior one-dimensional continuous methods by introducing novel filter design, permutation-hashing schemes, and frequency localization techniques tailored for higher-dimensional continuous signals.

ABSTRACT

Sparse Fourier transform (Sparse FT) is the problem of learning an unknown signal, whose frequency spectrum is dominated by a small amount of $k$ individual frequencies, through fast algorithms that use as few samples as possible in the time domain. The last two decades have seen an extensive study on such problems, either in the one-/multi-dimensional discrete setting [Hassanieh, Indyk, Katabi, and Price STOC'12; Kapralov STOC'16] or in the one-dimensional continuous setting [Price and Song FOCS'15]. Despite this rich literature, the most general multi-dimensional continuous case remains mysterious. This paper initiates the study on the Sparse FT problem in the multi-dimensional continuous setting. Our main result is a randomized non-adaptive algorithm that uses sublinear samples and runs in sublinear time. In particular, the sample duration bound required by our algorithm gives a non-trivial improvement over [Price and Song FOCS'15], which studies the same problem in the one-dimensional continuous setting. The dimensionality in the continuous setting, different from both the discrete cases and the one-dimensional continuous case, turns out to incur many new challenges. To overcome these issues, we develop a number of new techniques for constructing the filter functions, designing the permutation-then-hashing schemes, sampling the Fourier measurements, and locating the frequencies. We believe these techniques can find their applications in the future studies on the Sparse FT problem.

Motivation & Objective

• To address the lack of efficient algorithms for the multi-dimensional continuous sparse Fourier transform problem.
• To develop a non-adaptive, sublinear-time algorithm that uses minimal time-domain samples.
• To overcome new challenges introduced by dimensionality in the continuous setting, distinct from discrete or one-dimensional cases.
• To improve upon existing sample duration bounds from one-dimensional continuous Sparse FT methods.
• To establish foundational techniques applicable to future research in continuous multi-dimensional Sparse FT.

Proposed method

• Designs novel filter functions to isolate dominant frequencies in multi-dimensional continuous signals.
• Develops a permutation-then-hashing scheme to efficiently map and detect high-energy frequency components.
• Samples Fourier measurements using a structured, randomized sampling strategy to reduce time-domain sample count.
• Employs a frequency localization procedure to accurately identify the k dominant frequencies from noisy or sparse measurements.
• Combines these components into a non-adaptive algorithm that operates in sublinear time and sample complexity.
• Leverages mathematical analysis of Fourier concentration and spectral sparsity to ensure robustness and correctness.

Experimental results

Research questions

• RQ1 Can a sublinear-time, non-adaptive algorithm be designed for the multi-dimensional continuous sparse Fourier transform?
• RQ2 How does dimensionality in the continuous setting introduce unique challenges not present in discrete or one-dimensional cases?
• RQ3 What sampling and filtering strategies can achieve improved sample duration bounds compared to prior one-dimensional continuous methods?
• RQ4 Can robust frequency localization be achieved in high-dimensional continuous signals with minimal time-domain samples?
• RQ5 What new algorithmic techniques are required to handle the spectral structure of multi-dimensional continuous signals?

Key findings

• The proposed algorithm achieves sublinear time and sample complexity for the multi-dimensional continuous sparse Fourier transform.
• It provides a non-trivial improvement in sample duration bounds over the one-dimensional continuous method by Price and Song (FOCS'15).
• The algorithm is non-adaptive, meaning all sampling decisions are made in advance without feedback.
• New techniques in filter construction and permutation-hashing are introduced to handle the increased complexity of multi-dimensional continuous spectra.
• The method enables accurate recovery of k dominant frequencies with high probability using minimal time-domain samples.
• The framework establishes a foundation for future research in continuous multi-dimensional sparse Fourier analysis.