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Hadamard transform

The Hadamard transform, also known as the Walsh-Hadamard transform, is a discrete orthogonal transformation that decomposes a vector of length $2^m into coefficients using the Hadamard matrix H_m, a square matrix of order $2^m with entries \pm 1 whose rows (and columns) are mutually orthogonal, satisfying H_m H_m^T = 2^m I where I is the identity matrix.^[1] The transform is defined as \hat{x} = H_m x / \sqrt{2^m}, where x is the input vector, providing an orthonormal basis similar to the discrete Fourier transform but using square waves instead of sinusoids, and it is reversible since H_m is symmetric and its own inverse up to scaling.^[2] This transformation is computationally efficient, requiring only additions and subtractions in its fast implementation (fast Hadamard transform), which operates in O(2^m \log 2^m) time via a butterfly structure analogous to the fast Fourier transform, making it advantageous for hardware implementations without multipliers.^[3] In signal processing, it excels at analyzing signals with discontinuities or binary data, enabling applications such as data compression, filtering, and power spectrum estimation, particularly in electrocardiogram (ECG) analysis and speech processing where it outperforms the Fourier transform for certain non-sinusoidal components.^[3] Beyond classical signal processing, the Hadamard transform plays a key role in coding theory for constructing error-correcting codes like Reed-Muller codes and in quantum computing as the quantum Hadamard gate, which applies the transform to superposed states over Boolean domains, facilitating algorithms such as Simon's algorithm for period finding.^[2]^[4] Its basis functions, known as Walsh functions in sequency order, provide a complete representation for functions on the hypercube \{0,1\}^m, with the transform coefficients given by \hat{f}(s) = 2^{-m/2} \sum_x f(x) (-1)^{\langle x, s \rangle}, underscoring its utility in probabilistic analysis and machine learning tasks involving binary features.^[4]

Mathematical Foundations

Definition

The Hadamard transform is a linear transformation defined on real-valued vectors of length N = 2^n, where n is a non-negative integer, given by \mathbf{y} = H \mathbf{x}, with H denoting the Hadamard matrix of order N.^[5] The matrix H consists entirely of entries \pm 1 and is constructed recursively, starting from the base case of order 2. The unnormalized Hadamard matrix of order 2 is

H_2 = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix},

while the normalized version scales the entries by $1/\sqrt{2}:

H_2 = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}.

Higher-order matrices are obtained via the Kronecker product, such as H_{2^{k+1}} = H_2 \otimes H_{2^k}. For the unnormalized Hadamard matrix H of order N = 2^n, the rows (and columns) are pairwise orthogonal, satisfying H^T H = N I, where I is the identity matrix of order N. This property ensures that the transform is invertible, with the inverse given by H^{-1} = H / N. The normalized Hadamard matrix, with entries scaled by $1/\sqrt{N}, yields a unitary transformation where H^T H = I, preserving the Euclidean norm of the input vector \mathbf{x}. The Hadamard transform thus maps real-valued input sequences of length $2^n to output sequences that maintain the total energy (Parseval's relation holds for the normalized case) but are expressed in a basis of \pm 1 entries, facilitating applications in domains requiring orthogonal decompositions with simple arithmetic operations.^[5] The Hadamard matrix was introduced by Jacques Hadamard in 1893 in the context of maximizing determinants of \pm 1 matrices. Aspects of the transform relate to Walsh functions, a complete orthogonal system introduced by J. L. Walsh in 1923, where the Hadamard transform corresponds to a specific sequency ordering of the Walsh transform.^[6]

Hadamard Matrices

A Hadamard matrix of order m is a square matrix H_m with entries consisting solely of +1 and -1, such that the rows are pairwise orthogonal and similarly for the columns, satisfying the relation H_m H_m^T = m I_m, where I_m is the m \times m identity matrix.^[7] Such matrices exist only for orders m = 1, m = 2, or m a multiple of 4.^[7] From this orthogonality, each row (and column) has equal Euclidean norm \sqrt{m}.^[7] Among all m \times m matrices with entries \pm 1, Hadamard matrices achieve the maximum possible absolute determinant, given by m^{m/2}.^[8] In applications such as transforms, Hadamard matrices are often normalized by scaling each entry by $1/\sqrt{m}, yielding a unitary matrix \tilde{H}_m = H_m / \sqrt{m} that preserves the Euclidean norm of vectors under transformation.^[9] The Sylvester construction provides a recursive method to build Hadamard matrices of order $2^k for any nonnegative integer k, starting from the $1 \times 1 matrix

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and iteratively forming H_{2k} = \begin{bmatrix} H_k & H_k \\ H_k & -H_k \end{bmatrix}.^[7] This doubling procedure, equivalent to the Kronecker product H_2 \otimes H_k, guarantees existence for all powers of 2, including up to order $2^{20} = 1,048,576.^[9] The Paley construction generates Hadamard matrices for certain orders congruent to 0 modulo 4, specifically when the order is q + 1 where q is a prime power congruent to 3 modulo 4 (for skew matrices) or 1 modulo 4 (for symmetric matrices); it relies on the Jacobian symbol over finite fields to fill the matrix entries.^[9] The Hadamard conjecture posits that a Hadamard matrix exists for every order m = 4k where k is a positive integer.^[10] Hadamard matrices are known to exist for all multiples of 4 up to 1208 except 668, 716, 892, and 1132, and for all powers of 2 via Sylvester's method; the conjecture remains open, with the smallest unresolved order being 668 as of 2025.^[11] Walsh matrices represent a variant of Hadamard matrices obtained by reordering the rows (or columns) of H_{2^n} according to sequency, which counts the number of sign changes and aligns the basis functions in increasing order of frequency-like behavior.^[7]

Properties

Advantages

The Hadamard transform operates exclusively with real-valued arithmetic, circumventing the complex number operations inherent in transforms such as the discrete Fourier transform (DFT). This real-valued nature simplifies processing for applications involving real signals.^[12] A key computational benefit arises from the Hadamard matrix entries, which consist solely of +1 and -1, enabling the entire transform to be performed using only additions and subtractions without any multiplications. This property substantially lowers hardware requirements and boosts execution speed, offering up to an order of magnitude improvement over the DFT in image coding tasks.^[13] Hadamard matrices possess orthogonality, meaning their rows (and columns) are mutually orthogonal vectors. This orthogonality ensures the transform is invertible and shares energy-preserving qualities with other orthogonal transforms, such as the DFT. Specifically, for an n × n normalized Hadamard matrix H, Parseval's theorem applies:

\| H \mathbf{x} \|^2 = n \| \mathbf{x} \|^2

where \mathbf{x} is the input vector, thereby conserving the total signal energy across the transform domain.^[14] The transform basis consists of Walsh functions, which are piecewise constant square waves that provide a sequency-ordered decomposition analogous to frequency ordering in the Fourier transform. These functions yield localized representations for signals with abrupt changes or binary characteristics, enhancing interpretability in time-sequency analysis.^[12] Due to the bounded entries of the matrix, the Hadamard transform produces outputs with limited dynamic range, proportional to the input vector's norm, which mitigates overflow risks and supports efficient fixed-point implementations in resource-constrained systems.

Relation to Fourier Transform

The Hadamard transform, also known as the Walsh–Hadamard transform, serves as the Fourier transform over the finite abelian group (\mathbb{Z}/2\mathbb{Z})^n, which corresponds to the Boolean hypercube \{0,1\}^n. In this framework, the characters of the group are the Walsh functions \chi_S(x) = (-1)^{\sum_{i \in S} x_i} for subsets S \subseteq , which are products of the Rademacher functions \chi_{\{i\}}(x) = (-1)^{x_i}. These characters form an orthonormal basis for L^2(\{0,1\}^n, \mu), where \mu is the uniform measure, enabling the Fourier expansion f(x) = \sum_S \hat{f}(S) \chi_S(x) with coefficients \hat{f}(S) = \mathbb{E}[f \chi_S] = 2^{-n} \sum_x f(x) \chi_S(x). This structure parallels the standard discrete Fourier transform (DFT) on \mathbb{Z}/N\mathbb{Z}, but replaces complex exponentials with real-valued parity checks, making it particularly suited for binary signals on the hypercube graph.^[15] Both transforms share key properties: they are orthogonal and unitary (up to scaling), hence invertible via their adjoints, and they satisfy analogous convolution theorems. Specifically, the Hadamard convolution (f * g)(x) = 2^{-n} \sum_y f(y) g(x + y) (modulo 2) has Fourier transform \widehat{f * g}(S) = \hat{f}(S) \hat{g}(S), mirroring the circular convolution property of the DFT where \widehat{f \circledast g}(\omega) = \hat{f}(\omega) \hat{g}(\omega). For binary signals, the Hadamard transform approximates the DFT on the vertices of the hypercube, providing a real-valued spectral decomposition that captures correlations via parities rather than phases. However, unlike the sinusoidal basis of the DFT, which yields smooth frequency representations, the Hadamard basis consists of square-wave-like Rademacher products, resulting in blocky, discontinuous frequency components that emphasize abrupt changes over gradual oscillations.^[15] A precise connection arises in the one-dimensional case, where the Sylvester-Hadamard matrix H_2^n applied to a vector x computes the Fourier transform on the finite dyadic group, with explicit mappings to the standard DFT achievable via bit-reversal or Gray code reordering of the indices to align sequency with frequency. This equivalence highlights how the Hadamard transform diagonalizes convolution operators on binary domains, akin to circulant matrices in the DFT. In the limit as n \to \infty, the Hadamard transform extends to the Walsh–Fourier transform on the infinite dyadic group, offering sampling theorems for sequency-bandlimited signals analogous to Shannon sampling in the classical Fourier setting.^[16]

Algorithms and Computation

Fast Walsh–Hadamard Transform

The fast Walsh–Hadamard transform (FWHT) is an efficient divide-and-conquer algorithm for computing the Walsh–Hadamard transform of a vector of length N = 2^n, analogous to the Cooley–Tukey fast Fourier transform (FFT) in structure but using only additions and subtractions instead of complex multiplications. It exploits the recursive construction of the Hadamard matrix, where the transform matrix H_{2^n} is built as the Kronecker product H_2 \otimes H_{2^{n-1}}, with H_2 = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, allowing the computation to be broken into smaller subproblems by splitting the input into even and odd indexed parts. This recursive splitting reduces the operation count from O(N^2) for direct matrix multiplication to O(N \log N), performed through \log_2 N stages of butterfly operations.^[17] The algorithm operates in-place on the input vector x of length N, processing it through n = \log_2 N stages. In each stage s = 0, 1, \dots, n-1, the vector is divided into blocks of size $2^{s+1}, and within each block, pairs of elements separated by distance $2^s are combined using the basic Hadamard operation: for indices i and i + 2^s, let a = x, b = x[i + 2^s]; set x \leftarrow a + b and x[i + 2^s] \leftarrow a - b (unnormalized form). Each stage consists of N/2 such butterfly operations, totaling N \log_2 N additions and subtractions across all stages. The process starts with the input in natural order and produces the output in bit-reversed order for the Hadamard ordering of Walsh functions.^[18]^[19] To obtain the transform in sequency ordering (where functions are ordered by the number of zero crossings, similar to frequency in the Fourier transform), a post-computation reordering of the output indices via bit-reversal permutation is required, which can be done via bit manipulation or a permutation table. For the unitary version of the transform, which preserves the L^2 norm, the final output must be scaled by dividing each element by \sqrt{N}. The algorithm's simplicity stems from the absence of twiddle factors, making it particularly suitable for hardware implementations and real-valued computations.^[17]^[20] Historically, significant algorithmic advancements for fast Walsh transforms occurred in the 1960s and 1970s building on Walsh's foundational 1923 theory of dyadic groups. A unified matrix framework for various orderings, including efficient computation procedures, was formalized by Fino and Algazi in 1976.^[17]

Complexity

The fast Walsh–Hadamard transform (FWHT), the efficient algorithm for computing the Hadamard transform on inputs of length N = 2^n, achieves a time complexity of O(N \log N), requiring exactly N \log_2 N additions and subtractions through its butterfly-structured stages.^[21]^[22] In contrast, the naive approach via direct matrix multiplication yields O(N^2) complexity, making it impractical for large N. Space complexity for the FWHT is O(N) overall for in-place computation, utilizing O(1) additional space beyond the input and output arrays. The algorithm's structure enables high parallelization, with a computational depth of \log_2 N and independent butterfly operations within each stage, rendering it well-suited for SIMD instructions, multi-core CPUs, and GPUs.^[21]^[23] On hardware, the FWHT demonstrates efficiency by avoiding floating-point multiplications entirely, as the \pm 1 entries in Hadamard matrices allow integer implementations using only bit shifts and additions for scaling.^[21]^[24] Compared to the fast Fourier transform (FFT), which also runs in O(N \log N) time but requires multiplications by twiddle factors, the FWHT uses only real additions and subtractions, making it simpler for certain hardware implementations despite its restriction to lengths that are powers of 2; generalizations to arbitrary-order Hadamard transforms revert to O(N^2) complexity without specialized fast algorithms.^[22]

Applications

Quantum Computing

In quantum computing, the Hadamard gate, denoted as H, is a fundamental single-qubit unitary operation that applies the $2 \times 2 Hadamard matrix to create superposition states. Specifically, it transforms the computational basis states as H|0\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}} and H|1\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}}, thereby mapping a definite state into an equal superposition of the two basis states with a relative phase. This gate is essential for initializing qubits into superposition, enabling quantum parallelism by allowing computations to explore multiple possibilities simultaneously. For multi-qubit systems, the Hadamard transform generalizes via the tensor product H^{\otimes n}, which applies the gate independently to each of n qubits. This operation maps the all-zero state |00\dots0\rangle to a uniform superposition \frac{1}{\sqrt{2^n}} \sum_{x=0}^{2^n-1} |x\rangle over all $2^n basis states, facilitating the exploration of the entire Hilbert space in algorithms requiring global superposition. The Hadamard gate exhibits key properties that make it versatile in quantum circuits: it is self-inverse, satisfying H^2 = I, Hermitian (H = H^\dagger), and a member of the Clifford group, the normalizer of the Pauli group under conjugation. In circuit implementations, each Hadamard gate has constant depth O(1) and acts locally on a single qubit, but realizing the full n-qubit transform requires n such gates in parallel, resulting in overall circuit depth O(1) for the superposition layer, though total gate count scales as O(n). It is also commonly used to change the measurement basis, such as converting from the computational to the Hadamard (or X) basis. The Hadamard gate builds on foundational ideas in quantum computing, such as Richard Feynman's early 1980s vision of quantum simulation using unitary operations. It became essential in quantum algorithms starting with the Deutsch-Jozsa algorithm in 1992, where applications of H^{\otimes n} at the beginning and end create and interfere superpositions to determine function properties exponentially faster than classical methods.

Signal Processing and Coding

In signal processing, the Hadamard transform, particularly its discrete Walsh-Hadamard variant, is employed for spectral analysis of switching or binary signals, where the sequency spectrum provides an analogous representation to the frequency domain in Fourier analysis. This enables efficient detection of patterns in telemetry data and other digital signals by decomposing them into orthogonal components ordered by sequency, facilitating filtering and power spectrum estimation without multiplications. For instance, in processing speech and medical signals, the transform's use of only additions and subtractions allows for real-time analysis with low computational overhead.^[25] The Hadamard transform plays a key role in image and audio compression, especially for binary or low-contrast data, as its basis functions concentrate signal energy in low-sequency components, enabling effective block-based coding similar to JPEG but optimized for discrete-valued inputs. In half-tone medical imaging, such as MRI scans of the brain, Hadamard decomposition allows efficient storage and transmission by exploiting the transform's orthogonality to separate significant features from noise.^[26] For seismic data processing, the transform's uniform energy distribution across its spectrum supports compression techniques, where Walsh-Hadamard coding has been applied to reduce data volume while preserving waveform integrity, outperforming other methods like the Paley transform in certain sparse signal scenarios.^[27] In coding theory, Hadamard matrices generate simplex codes, also known as Hadamard codes, which are linear error-correcting codes with optimal properties for detection and correction in noisy channels. A Hadamard code of order $2^m consists of $2^m codewords of length $2^{m+1}, achieving a minimum Hamming distance of $2^m, which provides strong error resilience. These codes were instrumental in early space communications, such as the Mariner and Voyager missions, where they encoded telemetry data transmitted from Mars and outer planets back to Earth, ensuring reliable reception over vast distances.^[28]^[29] Walsh-Hadamard sequences derived from the transform underpin spread-spectrum communication systems, serving as orthogonal codes for multiple access schemes that predate modern CDMA implementations. In direct-sequence CDMA, these sequences spread user signals across a wider bandwidth while maintaining orthogonality to minimize inter-user interference, enabling synchronous downlink transmission in cellular networks. Their binary nature and perfect orthogonality make them ideal for environments with low Doppler shifts, as demonstrated in IS-95 CDMA standards where 64 Walsh codes per base station support channel separation.^[25]^[30]

Molecular Phylogenetics

The Hadamard conjugation, developed by Hendy and Penny in the late 1980s, serves as a key analytical tool in molecular phylogenetics for detecting incompatible characters within phylogenetic datasets. This method applies the Hadamard transform to a compatibility matrix derived from sequence data, transforming it into a conjugate spectrum that highlights deviations indicative of reticulation or conflicting evolutionary signals. By leveraging the orthogonal properties of Hadamard matrices, the transform separates underlying evolutionary patterns, enabling identification of sites that do not conform to a single tree topology. In practice, Hadamard conjugation is particularly effective for analyzing four-taxon datasets, where the Hadamard matrix processes site pattern counts to resolve quartet inconsistencies and assess compatibility across possible tree topologies. For larger datasets, a full Hadamard transform on the counts of site patterns can flag evidence of recombination by revealing non-tree-like signals in the conjugate spectrum, such as negative variances that suggest mosaic evolution. This approach addresses limitations in traditional parsimony methods by quantifying "noise" from processes like horizontal gene transfer, providing a spectral signature for evolutionary conflicts.^[31] Within distance-based phylogenetic methods, Hadamard conjugation facilitates the computation of spectral distances for additive trees through iterative Hadamard powering, which involves repeated matrix squaring to estimate branch lengths efficiently. The distance matrix D is transformed into the Hadamard domain to achieve linearity with respect to branch lengths, as expressed by:

\hat{D} = H D H

where H is the normalized Hadamard matrix, and \hat{D} represents the conjugate distances that add up along tree edges under group-based substitution models. This transformation linearizes the expected distances, simplifying tree reconstruction.^[32] The method is implemented in software packages like PHYLIP, which uses Hadamard transformations for bipartition spectra and distance corrections to handle such analyses.^[33] In evolutionary biology, Hadamard conjugation has proven valuable for studying complex scenarios, such as viral phylogenies where recombination is prevalent; for instance, it has been applied to HIV datasets to detect mosaic genomes by identifying incompatible site patterns across genomic regions. Recent extensions post-2000 have incorporated the transform into supertree methods, combining multiple partial phylogenies while accounting for reticulate signals to build comprehensive species trees.^[34]^[35]

Other Fields

In optics and imaging, the Hadamard transform enables efficient multiplexing techniques for spectroscopy, where it reconstructs spectral information from encoded measurements using orthogonal masks, as pioneered in Hadamard transform spectrometry during the 1970s.^[36] This approach improves signal-to-noise ratios in dispersive instruments by allowing simultaneous interrogation of multiple spectral channels, with applications in infrared and ultraviolet spectrometry for chemical analysis.^[37] In holography, the transform supports lensless imaging reconstruction, such as through delta-Hadamard operations on selected regions of Fourier holograms to recover object details without physical lenses.^[38] For phase retrieval, Hadamard-based coded apertures minimize the condition number of measurement matrices, reducing the number of required exposures from over 20 to as few as four while enabling accurate recovery of complex-valued fields in computational imaging systems.^[39] In cryptography, the Hadamard transform, often via its Walsh variant, serves as a foundation for designing lightweight ciphers by facilitating fast diffusion layers through binary orthogonal transformations, which enhance keystream randomness in resource-constrained environments like IoT devices.^[40] It computes Walsh spectra to evaluate nonlinear bias in filtering functions, aiding the construction of bent functions with optimal cryptographic properties such as high nonlinearity and low autocorrelation.^[40] Specifically, in stream ciphers, Walsh sequences derived from Hadamard matrices generate pseudorandom sequences for key streams, where the transform assesses correlation immunity against linear attacks by revealing probabilistic decryption risks.^[41] In machine learning, the Hadamard transform extracts features from binary data by projecting onto the Walsh basis, which decomposes high-dimensional {+1, -1}-valued inputs into uncorrelated components for tasks like classification of sparse signals.^[42] Post-2010 developments integrated it into random projection sketches for dimensionality reduction, where subsampled randomized Hadamard transforms (SRHT) preserve pairwise distances with high probability while reducing computational overhead compared to dense Gaussian projections, enabling scalable processing of large datasets in regression and kernel methods.^[43] For instance, SRHT-based embeddings accelerate approximate nearest neighbor searches in binary feature spaces, achieving near-Johnson-Lindenstrauss guarantees with O(d log d) time complexity for d-dimensional data.^[44] In physics simulations, the Hadamard transform analyzes spin correlations in Ising models by representing the partition function as a Walsh-Hadamard transform over interaction subgraphs, allowing exact computation of thermodynamic properties in spin-glass variants without exhaustive enumeration.^[45] For Monte Carlo methods on hypercubes, it diagonalizes the Fourier-Walsh basis for sampling spin configurations, efficiently estimating correlation functions in high-dimensional lattices by transforming between spin and momentum representations.^[46] This is particularly useful for Ising-like models on arbitrary graphs, where the energy spectrum emerges as the Hadamard transform of a graph-associated vector, aiding simulations of phase transitions and ground-state searches.^[46] Emerging applications in the 2020s leverage the Hadamard transform in AI hardware, such as GPU tensor cores for accelerated computations in deep learning frameworks, enabling efficient tensor operations.^[47] In quantum-inspired classical algorithms, it accelerates matrix sketches and low-rank approximations via fast Walsh-Hadamard convolutions, enabling efficient solutions to linear systems and recommendation tasks on classical hardware with polynomial speedups over naive methods.

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[PDF] A Survey of Dimensionality Reduction Techniques Based on ... - arXiv
FEATURE EXTRACTION APPROACHES. Feature extraction transforms data from high-dimensional space lower-dimensional space, which is the most commonly used ...
[45]
[PDF] Random Projections For Large-Scale Regression - arXiv
Jan 19, 2017 · Random projections are extensively used as a dimension reduction tool in machine learning and statistics. ... Hadamard Transform (SRHT) (Tropp, ...
[46]
The Fast Johnson–Lindenstrauss Transform and Approximate ...
The FJLT is faster than standard random projections and just as easy to implement. It is based upon the preconditioning of a sparse projection matrix with a ...
[47]
On the quantum computational complexity of the Ising spin glass ...
Nov 16, 2004 · The spin-glass partition function is a Walsh–Hadamard transform (or discrete. Fourier transform with addition modulo 2), over the subgraph ...
[48]
Ising-like models on arbitrary graphs: The Hadamard way
Aug 6, 2025 · We propose a generic framework to analyse classical Ising-like models defined on arbitrary graphs. The energy spectrum is shown to be the ...
[49]
HadaCore: Tensor Core Accelerated Hadamard Transform Kernel
Dec 2, 2024 · In this blog, we present HadaCore, a Hadamard Transform CUDA kernel that achieves state-of-the-art performance on NVIDIA A100 and H100 GPUs.