Hadamard transform

About: Hadamard transform is a research topic. Over the lifetime, 7262 publications have been published within this topic receiving 94328 citations.

Approximate message-passing decoder and capacity-achieving sparse superposition codes

Abstract: We study the approximate message-passing decoder for sparse superposition coding on the additive white Gaussian noise channel and extend our preliminary work [1]. We use heuristic statistical-physics-based tools such as the cavity and the replica methods for the statistical analysis of the scheme. While superposition codes asymptotically reach the Shannon capacity, we show that our iterative decoder is limited by a phase transition similar to the one that happens in Low Density Parity check codes. We consider two solutions to this problem, that both allow to reach the Shannon capacity: i) a power allocation strategy and ii) the use of spatial coupling, a novelty for these codes that appears to be promising. We present in particular simulations suggesting that spatial coupling is more robust and allows for better reconstruction at finite code lengths. Finally, we show empirically that the use of a fast Hadamard-based operator allows for an efficient reconstruction, both in terms of computational time and memory, and the ability to deal with very large messages.

New Subsampling Algorithms for Fast Least Squares Regression

Abstract: We address the problem of fast estimation of ordinary least squares (OLS) from large amounts of data (n ≫ p). We propose three methods which solve the big data problem by subsampling the covariance matrix using either a single or two stage estimation. All three run in the order of size of input i.e. O(np) and our best method, Uluru, gives an error bound of O(√p/n) which is independent of the amount of subsampling as long as it is above a threshold. We provide theoretical bounds for our algorithms in the fixed design (with Randomized Hadamard preconditioning) as well as sub-Gaussian random design setting. We also compare the performance of our methods on synthetic and real-world datasets and show that if observations are i.i.d., sub-Gaussian then one can directly subsample without the expensive Randomized Hadamard preconditioning without loss of accuracy.

Fast Hadamard transform based on a simple matrix factorization

Abstract: This correspondence presents a simple factorization of the Hadamard matrix which is used to develop a fast algorithm for the Hadamard transform. The matrix decomposition is of the form of the Kronecker products of identity matrices and successively lower order Hadamard matrices. This decomposition leads very clearly to a sparse-matrix factorization of the Hadamard matrix.

Z4-linear Hadamard and extended perfect codes

Abstract: If $N=2^k > 8$ then there exist exactly $[(k-1)/2]$ pairwise nonequivalent $Z_4$-linear Hadamard $(N,2N,N/2)$-codes and $[(k+1)/2]$ pairwise nonequivalent $Z_4$-linear extended perfect $(N,2^N/2N,4)$-codes A recurrent construction of $Z_4$-linear Hadamard codes is given

Hadamard transform capillary electrophoresis.

Abstract: This paper reports the first demonstration of a multiplex sample injection technique in capillary electrophoresis. The sample was injected into a capillary (effective length, 4 cm) as a pseudorandam Hadamard sequence by a photodegradation technique using a high-power gating laser, and the fluorescence signal, which was measured using a probe excitation beam, was decoded by an inverse Hadamard transformation. The signal-to-noise ratio was improved by a factor of 8, which was in good agreement with the theoretically predicted value of 8.02. This approach is potentially useful for the enhancement of the sensitivity by 3 orders of magnitude in high-resolution capillary electrophoresis, combined with fluorescence detection.