research-article

A Fast Hadamard Transform for Signals With Sublinear Sparsity in the Transform Domain

Authors:

Robin Scheibler,

Saeid Haghighatshoar,

Martin VetterliAuthors Info & Claims

IEEE Transactions on Information Theory, Volume 61, Issue 4

Pages 2115 - 2132

https://doi.org/10.1109/TIT.2015.2404441

Published: 01 April 2015 Publication History

Abstract

In this paper, we design a new iterative low-complexity algorithm for computing the Walsh-Hadamard transform (WHT) of an N dimensional signal with a K-sparse WHT. We suppose that N is a power of two and K = O(N<sup>α</sup>), scales sublinearly in N for some α ∈ (0, 1). Assuming a random support model for the nonzero transform-domain components, our algorithm reconstructs the WHT of the signal with a sample complexity O(K log<sub>2</sub>(N/K)) and a computational complexity O(K log<sub>2</sub>(K) log<sub>2</sub>(N/K)). Moreover, the algorithm succeeds with a high probability approaching 1 for large dimension N. Our approach is mainly based on the subsampling (aliasing) property of the WHT, where by a carefully designed subsampling of the time-domain signal, a suitable aliasing pattern is induced in the transform domain. We treat the resulting aliasing patterns as parity-check constraints and represent them by a bipartite graph. We analyze the properties of the resulting bipartite graphs and borrow ideas from codes defined over sparse bipartite graphs to formulate the recovery of the nonzero spectral values as a peeling decoding algorithm for a specific sparse-graph code transmitted over a binary erasure channel. This enables us to use tools from coding theory (belief-propagation analysis) to characterize the asymptotic performance of our algorithm in the very sparse (α ∈ (0, 1/3]) and the less sparse (α ∈ (1/3, 1)) regime. Comprehensive simulation results are provided to assess the empirical performance of the proposed algorithm.

References

[1]

W. Pratt, J. Kane, and H. C. Andrews, “Hadamard transform image coding,” Proc. IEEE, vol. 57, no. 1, pp. 58–68, Jan. 1969.

[2]

3GPP, Spreading and Modulation (FDD), 3rd Generation Partnership Project (3GPP), document Rec. TS 25.213, Sep. 2014.

[3]

E. D. Nelson and M. L. Fredman, “Hadamard spectroscopy,” J. Opt. Soc. Amer., vol. 60, no. 12, pp. 1664–1669, Dec. 1970.

[4]

S. Haghighatshoar and E. Abbe. (2013). “Polarization of the Rényi information dimension for single and multi terminal analog compression.” [Online]. Available: http://arxiv.org/abs/1301.6388

[5]

A. Hedayat and W. D. Wallis, “Hadamard matrices and their applications,” Ann. Statist., vol. 6, no. 6, pp. 1184–1238, 1978.

[6]

M. H. Lee and M. Kaveh, “Fast Hadamard transform based on a simple matrix factorization,” IEEE Trans. Acoust., Speech, Signal Process., vol. 34, no. 6, pp. 1666–1667, Dec. 1986.

[7]

J. R. Johnson and M. Püschel, “In search of the optimal Walsh–Hadamard transform,” in Proc. IEEE ICASSP, vol. 6. Istanbul, Turkey, Jun. 2000, pp. 3347–3350.

[8]

A. C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, and M. Strauss, “Near-optimal sparse Fourier representations via sampling,” in Proc. 34th Annu. ACM STOC, 2002, pp. 152–161.

[9]

A. C. Gilbert, M. J. Strauss, and J. A. Tropp, “A tutorial on fast Fourier sampling,” IEEE Signal Process. Mag., vol. 25, no. 2, pp. 57–66, Mar. 2008.

[10]

D. Lawlor, Y. Wang, and A. Christlieb, “Adaptive sub-linear Fourier algorithms,” Adv. Adapt. Data Anal., vol. 5, no. 1, p. 1350003, 2013.

[11]

H. Hassanieh, P. Indyk, D. Katabi, and E. Price, “Simple and practical algorithm for sparse Fourier transform,” in Proc. 23rd Annu. ACM-SIAM SODA, 2012, pp. 1183–1194.

[12]

H. Hassanieh, P. Indyk, D. Katabi, and E. Price, “Nearly optimal sparse Fourier transform,” in Proc. 44th Annu. ACM STOC, 2012, pp. 563–578.

[13]

B. Ghazi, H. Hassanieh, P. Indyk, D. Katabi, E. Price, and L. Shi, “Sample-optimal average-case sparse Fourier Transform in two dimensions,” in Proc. 51st Allerton, Oct. 2013, pp. 1258–1265.

[14]

S. Pawar and K. Ramchandran, “A hybrid DFT-LDPC framework for fast, efficient and robust compressive sensing,” in Proc. 50th Allerton, 2012, pp. 1943–1950.

[15]

S. Pawar and K. Ramchandran, “Computing a k-sparse n-length discrete Fourier transform using at most 4k samples and $O(k \log k)$ complexity,” in Proc. IEEE ISIT, Istanbul, Turkey, Jul. 2013, pp. 464–468.

[16]

T. Richardson and R. Urbanke, Modern Coding Theory. Cambridge, U.K.: Cambridge Univ. Press, 2008.

[17]

O. Goldreich and L. A. Levin, “A hard-core predicate for all one-way functions,” in Proc. 21st Annu. ACM STOC, Feb. 1989, pp. 25–32.

[18]

O. Goldreich, Modern Cryptography, Probabilistic Proofs and Pseudorandomness. New York, NY, USA: Springer-Verlag, 1999.

Digital Library

[19]

N. C. Wormald, “Differential equations for random processes and random graphs,” Ann. Appl. Probab., vol. 5, no. 4, pp. 1217–1235, Nov. 1995.

[20]

M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi, and D. A. Spielman, “Efficient erasure correcting codes,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 569–584, Feb. 2001.

Digital Library

[21]

X. Li, J. K. Bradley, S. Pawar, and K. Ramchandran, “The SPRIGHT algorithm for robust sparse Hadamard transforms,” in Proc. IEEE ISIT, Honolulu, HI, Jun./Jul. 2014, pp. 1857–1861.

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A Fast Hadamard Transform for Signals With Sublinear Sparsity in the Transform Domain

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cover image IEEE Transactions on Information Theory

IEEE Transactions on Information Theory Volume 61, Issue 4

April 2015

701 pages

ISSN:0018-9448

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Copyright © 2015.

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IEEE Press

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Published: 01 April 2015

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Cited By

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Liu XVenkataramanan R(2023)Sketching Sparse Low-Rank Matrices With Near-Optimal Sample- and Time-Complexity Using Message PassingIEEE Transactions on Information Theory10.1109/TIT.2023.327318169:9(6071-6097)Online publication date: 1-Sep-2023
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