# An Adaptive Sublinear-Time Block Sparse Fourier Transform

## TL;DR

This paper introduces an adaptive, sublinear-time algorithm for efficiently computing the dominant Fourier coefficients of block sparse signals, reducing sample complexity below traditional bounds and demonstrating the necessity of adaptivity in such measurements.

## Contribution

It presents the first sublinear-time sparse FFT algorithm for block sparse signals with improved sample complexity, highlighting the importance of adaptivity in measurement strategies.

## Key findings

- Achieves $O^*(k_0k_1 + k_0 ext{log}(1+k_0) ext{log} n)$ sample complexity.
- First sublinear-time algorithm for model-based compressed sensing.
- Non-adaptive algorithms require $oldsymbol{ ext{Omega}(k_0k_1 ext{log}(n/k_0k_1)))}$ samples, showing adaptivity's necessity.

## Abstract

The problem of approximately computing the $k$ dominant Fourier coefficients of a vector $X$ quickly, and using few samples in time domain, is known as the Sparse Fourier Transform (sparse FFT) problem. A long line of work on the sparse FFT has resulted in algorithms with $O(k\log n\log (n/k))$ runtime [Hassanieh et al., STOC'12] and $O(k\log n)$ sample complexity [Indyk et al., FOCS'14]. These results are proved using non-adaptive algorithms, and the latter $O(k\log n)$ sample complexity result is essentially the best possible under the sparsity assumption alone. This paper revisits the sparse FFT problem with the added twist that the sparse coefficients approximately obey a $(k_0,k_1)$-block sparse model. In this model, signal frequencies are clustered in $k_0$ intervals with width $k_1$ in Fourier space, where $k= k_0k_1$ is the total sparsity. Signals arising in applications are often well approximated by this model with $k_0\ll k$. Our main result is the first sparse FFT algorithm for $(k_0, k_1)$-block sparse signals with the sample complexity of $O^*(k_0k_1 + k_0\log(1+ k_0)\log n)$ at constant signal-to-noise ratios, and sublinear runtime. A similar sample complexity was previously achieved in the works on model-based compressive sensing using random Gaussian measurements, but used $\Omega(n)$ runtime. To the best of our knowledge, our result is the first sublinear-time algorithm for model based compressed sensing, and the first sparse FFT result that goes below the $O(k\log n)$ sample complexity bound. Our algorithm crucially uses {\em adaptivity} to achieve the improved sample complexity bound, and we prove that adaptivity is in fact necessary if Fourier measurements are used: Any non-adaptive algorithm must use $\Omega(k_0k_1\log \frac{n}{k_0k_1})$ samples for the $(k_0,k_1$)-block sparse model, ruling out improvements over the vanilla sparsity assumption.