Hadamard transform

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

Communication Efficient, Sample Optimal, Linear Time Locally Private Discrete Distribution Estimation.

Abstract: We consider discrete distribution estimation over $k$ elements under $\varepsilon$-local differential privacy from $n$ samples The samples are distributed across users who send privatized versions of their sample to the server All previously known sample optimal algorithms require linear (in $k$) communication complexity in the high privacy regime $(\varepsilon<1)$, and have a running time that grows as $n\cdot k$, which can be prohibitive for large domain size $k$ We study the task simultaneously under four resource constraints, privacy, sample complexity, computational complexity, and communication complexity We propose \emph{Hadamard Response (HR)}, a local non-interactive privatization mechanism with order optimal sample complexity (for all privacy regimes), a communication complexity of $\log k+2$ bits, and runs in nearly linear time Our encoding and decoding mechanisms are based on Hadamard matrices, and are simple to implement The gain in sample complexity comes from the large Hamming distance between rows of Hadamard matrices, and the gain in time complexity is achieved by using the Fast Walsh-Hadamard transform We compare our approach with Randomized Response (RR), RAPPOR, and subset-selection mechanisms (SS), theoretically, and experimentally For $k=10000$, our algorithm runs about 100x faster than SS, and RAPPOR

Low complexity discrete Hartley transform precoded OFDM for peak power reduction

Abstract: A discrete Hartley transform (DHT) precoded OFDM system to counteract the peak power problem is proposed. By utilising the intrinsic relationship between DHT and discrete Fourier transform (DFT), the transmitter can be simplified to generate the precoded signals by only linear combination instead of inverse DFT and DHT, and the transceiver complexity can be reduced to simplify the system design. Simulations have confirmed that the peak-to-average power ratio of DHT precoded OFDM systems is substantially lower than that of both the discrete cosine transform and the discrete Walsh transform precoded OFDM system.

Hadamard Transform Photothermal Deflection Imaging

Abstract: A 69-element Hadamard mask is used to encode a line-focused excimer laser beam to obtain spatially multiplexed photothermal deflection signals. The spatial distribution and relative quantities of potassium chromate samples on silica are recovered by Hadamard transformation.

Generalized Mersenne matrices and Balonin’s conjecture

Abstract: Properties of generalized Hadamard matrices and a conjecture on the existence of Mersenne matrices included into the former are discussed. A new classification of few-level quasi-orthogonal matrices, including matrices of even and odd orders, is presented. Matrices of Euler, Mersenne, and Hadamard are considered. The fundamental differences between the matrices of Mersenne and Fermat, which explain the failure of the proof of the Hadamard conjecture, are shown.

A sinc quadrature rule for Hadamard finite-part integrals

Abstract: A Sinc quadrature rule is presented for the evaluation of Hadamard finite-part integrals of analytic functions. Integration over a general are in the complex plane is considered. Special treatment is given to integrals over the interval (?1,1). Theoretical error estimates are derived and numerical examples are included.