Showing papers on "Hadamard transform published in 2019"

Hadamard Response: Estimating Distributions Privately, Efficiently, and with Little Communication

Abstract: We study the problem of estimating $k$-ary distributions under $\eps$-local differential privacy. $n$ samples are distributed across users who send privatized versions of their sample to a central server. All previously known sample optimal algorithms require linear (in $k$) communication from each user in the high privacy regime $(\eps=O(1))$, and run in time that grows as $n\cdot k$, which can be prohibitive for large domain size $k$. We propose Hadamard Response (HR), a local privatization scheme that requires no shared randomness and is symmetric with respect to the users. Our scheme has order optimal sample complexity for all $\eps$, a communication of at most $\log k+2$ bits per user, and nearly linear running time of $\tilde{O}(n + k)$. Our encoding and decoding are based on Hadamard matrices and are simple to implement. The statistical performance relies on the coding theoretic aspects of Hadamard matrices, ie, the large Hamming distance between the rows. An efficient implementation of the algorithm using the Fast Walsh-Hadamard transform gives the computational gains. We compare our approach with Randomized Response (RR), RAPPOR, and subset-selection mechanisms (SS), both theoretically, and experimentally. For $k=10000$, our algorithm runs about 100x faster than SS, and RAPPOR.

Super Sub-Nyquist Single-Pixel Imaging by Means of Cake-Cutting Hadamard Basis Sort.

Abstract: Single-pixel imaging via compressed sensing can reconstruct high-quality images from a few linear random measurements of an object known a priori to be sparse or compressive, by using a point/bucket detector without spatial resolution. Nevertheless, random measurements still have blindness, limiting the sampling ratios and leading to a harsh trade-off between the acquisition time and the spatial resolution. Here, we present a new compressive imaging approach by using a strategy we call cake-cutting, which can optimally reorder the deterministic Hadamard basis. The proposed method is capable of recovering images of large pixel-size with dramatically reduced sampling ratios, realizing super sub-Nyquist sampling and significantly decreasing the acquisition time. Furthermore, such kind of sorting strategy can be easily combined with the structured characteristic of the Hadamard matrix to accelerate the computational process and to simultaneously reduce the memory consumption of the matrix storage. With the help of differential modulation/measurement technology, we demonstrate this method with a single-photon single-pixel camera under the ulta-weak light condition and retrieve clear images through partially obscuring scenes. Thus, this method complements the present single-pixel imaging approaches and can be applied to many fields.

ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity

Abstract: We present a new graphical calculus that is sound and complete for a universal family of quantum circuits, which can be seen as the natural string-diagrammatic extension of the approximately (real-valued) universal family of Hadamard+CCZ circuits. The diagrammatic language is generated by two kinds of nodes: the so-called 'spider' associated with the computational basis, as well as a new arity-N generalisation of the Hadamard gate, which satisfies a variation of the spider fusion law. Unlike previous graphical calculi, this admits compact encodings of non-linear classical functions. For example, the AND gate can be depicted as a diagram of just 2 generators, compared to ~25 in the ZX-calculus. Consequently, N-controlled gates, hypergraph states, Hadamard+Toffoli circuits, and diagonal circuits at arbitrary levels of the Clifford hierarchy also enjoy encodings with low constant overhead. This suggests that this calculus will be significantly more convenient for reasoning about the interplay between classical non-linear behaviour (e.g. in an oracle) and purely quantum operations. After presenting the calculus, we will prove it is sound and complete for universal quantum computation by demonstrating the reduction of any diagram to an easily describable normal form.

Hermite-Hadamard Type Inequalities for the Class of Convex Functions on Time Scale

Abstract: We investigate a time scale version of two auxiliary functions for the class of convex functions. We derive several novel dynamic inequalities for these classes of convex functions. Applications of these consequences are taken into consideration in special means. Furthermore, illustrative examples are introduced to help our outcomes. Meanwhile, we communicate a few particular cases which may be deduced from our main outcomes.

On q-Hermite-Hadamard Inequalities for Differentiable Convex Functions

Abstract: In this paper, we establish some new results on the left-hand side of the q-Hermite–Hadamard inequality for differentiable convex functions with a critical point. Our work extends the results of Alp et. al (q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions, J. King Saud Univ. Sci., 2018, 30, 193-203), by considering the critical point-type inequalities.

Some new Hermite–Hadamard type inequalities for s-convex functions and their applications

Abstract: In this paper, we establish some new integral inequalities of Hermite–Hadamard type for s-convex functions by using the Holder–Iscan integral inequality. We also compare our new results with the known results and show that the results which we obtained are better than the known results. Finally, we give some applications to trapezoidal formula and to special means.

On the Hermite–Hadamard type inequality for ψ-Riemann–Liouville fractional integrals via convex functions

Abstract: In this paper, we establish a new Hermite–Hadamard inequality involving left-sided and right-sided ψ-Riemann–Liouville fractional integrals via convex functions. We also show two basic ψ-Riemann–Liouville fractional integral identities including the first order derivative of a given convex function, and these will be used to derive estimates for some fractional Hermite–Hadamard inequalities. Finally, we give some applications to special means of real numbers.

Hyers–Ulam stability of a coupled system of fractional differential equations of Hilfer–Hadamard type

Abstract: Abstract In this paper, existence and uniqueness of solution for a coupled impulsive Hilfer–Hadamard type fractional differential system are obtained by using Kransnoselskii’s fixed point theorem. Different types of Hyers–Ulam stability are also discussed.We provide an example demonstrating consistency to the theoretical findings.

Regularization of proximal point algorithms in Hadamard manifolds

Abstract: In this paper, we consider the regularization method for exact as well as for inexact proximal point algorithms for finding the singularities of maximal monotone set-valued vector fields. We prove that the sequences generated by these algorithms converge to an element of the set of singularities of a maximal monotone set-valued vector field. A numerical example is provided to illustrate the inexact proximal point algorithm with regularization. Applications of our results to minimization problems and saddle point problems are given in the setting of Hadamard manifolds.

Certain inequalities via generalized proportional Hadamard fractional integral operators

Abstract: In the article, we introduce the generalized proportional Hadamard fractional integrals and establish several inequalities for convex functions in the framework of the defined class of fractional integrals. The given results are generalizations of some known results.

Conformable Integral Inequalities of the Hermite-Hadamard Type in terms of GG- and GA-Convexities

Abstract: In the article, we present several conformable fractional integrals’ versions of the Hermite-Hadamard type inequalities for GG- and GA-convex functions and provide their applications in special bivariate means.

Stability analysis of nonlinear Hadamard fractional differential system

Abstract: The stability of the zero solution of a class of nonlinear Hadamard type fractional differential system is investigated by utilizing a new fractional comparison principle. The novelty of this paper is based on some new fractional differential inequalities along the given nonlinear Hadamard fractional differential system. A comparison principle employing the new fractional differential inequality for scalar Hadamard fractional differential system is presented. Based on the new comparison principle, some sufficient conditions for the (generalized) stability and the (generalized) Mittag-Leffler stability are given.

Positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions

Abstract: In this paper we use the fixed point index to study the existence of positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions. Here we use appropriate nonnegative matrices to depict the coupling behavior for our nonlinearities.

Some Quantum Estimates of Hermite-Hadamard Inequalities for Quasi-Convex Functions

Abstract: In this paper, we develop some quantum estimates of Hermite-Hadamard type inequalities for quasi-convex functions. In some special cases, these quantum estimates reduce to the known results.

Hermite–Hadamard type inequalities for exponentially p-convex functions and exponentially s-convex functions in the second sense with applications

Abstract: In this paper, we introduce the notion of exponentially p-convex function and exponentially s-convex function in the second sense. We establish several Hermite–Hadamard type inequalities for exponentially p-convex functions and exponentially s-convex functions in second sense. The present investigation is an extension of several well known results.

A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space

Abstract: The main purpose of this paper is to introduce a viscosity-type proximal point algorithm, comprising of a nonexpansive mapping and a finite sum of resolvent operators associated with monotone bifunctions. A strong convergence of the proposed algorithm to a common solution of a finite family of equilibrium problems and fixed point problem for a nonexpansive mapping is established in a Hadamard space. We further applied our results to solve some optimization problems in Hadamard spaces.

Hermite-Hadamard Type Inequalities for Interval (h1, h2)-Convex Functions

Abstract: We introduce the concept of interval ( h 1 , h 2 ) -convex functions. Under the new concept, we establish some new interval Hermite-Hadamard type inequalities, which generalize those in the literature. Also, we give some interesting examples.

Building Efficient Deep Neural Networks With Unitary Group Convolutions

Abstract: We propose unitary group convolutions (UGConvs), a building block for CNNs which compose a group convolution with unitary transforms in feature space to learn a richer set of representations than group convolution alone. UGConvs generalize two disparate ideas in CNN architecture, channel shuffling (i.e. ShuffleNet) and block-circulant networks (i.e. CirCNN), and provide unifying insights that lead to a deeper understanding of each technique. We experimentally demonstrate that dense unitary transforms can outperform channel shuffling in DNN accuracy. On the other hand, different dense transforms exhibit comparable accuracy performance. Based on these observations we propose HadaNet, a UGConv network using Hadamard transforms. HadaNets achieve similar accuracy to circulant networks with lower computation complexity, and better accuracy than ShuffleNets with the same number of parameters and floating-point multiplies.

Some new hermite-hadamard type inequalities and their applications

Abstract: In this paper first, we prove some new generalizations of Hermite-Hadamard type inequalities for the convex function f and for (s, m)-convex function f in the second sense in conformable f...

Analysis of coupled systems of implicit impulsive fractional differential equations involving Hadamard derivatives

Abstract: We present some results on the existence, uniqueness and Hyers–Ulam stability to the solution of an implicit coupled system of impulsive fractional differential equations having Hadamard type fractional derivative. Using a fixed point theorem of Kransnoselskii’s type, the existence and uniqueness results are obtained. Along these lines, different kinds of Hyers–Ulam stability are discussed. An example is given to illustrate the main theorems.

Positive Solutions for a System of Fractional Integral Boundary Value Problems Involving Hadamard-Type Fractional Derivatives

Abstract: In this paper, we use fixed-point index to study the existence of positive solutions for a system of Hadamard fractional integral boundary value problems involving nonnegative nonlinearities. By virtue of integral-type Jensen inequalities, some appropriate concave and convex functions are used to depict the coupling behaviors for our nonlinearities .

Asymptotics for Sketching in Least Squares Regression

Abstract: We consider a least squares regression problem where the data has been generated from a linear model, and we are interested to learn the unknown regression parameters. We consider "sketch-and-solve" methods that randomly project the data first, and do regression after. Previous works have analyzed the statistical and computational performance of such methods. However, the existing analysis is not fine-grained enough to show the fundamental differences between various methods, such as the Subsampled Randomized Hadamard Transform (SRHT) and Gaussian projections. In this paper, we make progress on this problem, working in an asymptotic framework where the number of datapoints and dimension of features goes to infinity. We find the limits of the accuracy loss (for estimation and test error) incurred by popular sketching methods. We show separation between different methods, so that SRHT is better than Gaussian projections. Our theoretical results are verified on both real and synthetic data. The analysis of SRHT relies on novel methods from random matrix theory that may be of independent interest.

Uniform recovery in infinite-dimensional compressed sensing and applications to structured binary sampling

Abstract: Infinite-dimensional compressed sensing deals with the recovery of analog signals (functions) from linear measurements, often in the form of integral transforms such as the Fourier transform. This framework is well-suited to many real-world inverse problems, which are typically modelled in infinite-dimensional spaces, and where the application of finite-dimensional approaches can lead to noticeable artefacts. Another typical feature of such problems is that the signals are not only sparse in some dictionary, but possess a so-called local sparsity in levels structure. Consequently, the sampling scheme should be designed so as to exploit this additional structure. In this paper, we introduce a series of uniform recovery guarantees for infinite-dimensional compressed sensing based on sparsity in levels and so-called multilevel random subsampling. By using a weighted $\ell^1$-regularizer we derive measurement conditions that are sharp up to log factors, in the sense they agree with those of certain oracle estimators. These guarantees also apply in finite dimensions, and improve existing results for unweighted $\ell^1$-regularization. To illustrate our results, we consider the problem of binary sampling with the Walsh transform using orthogonal wavelets. Binary sampling is an important mechanism for certain imaging modalities. Through carefully estimating the local coherence between the Walsh and wavelet bases, we derive the first known recovery guarantees for this problem.

Analysis of Nonlinear Coupled Systems of Impulsive Fractional Differential Equations with Hadamard Derivatives

Abstract: This work is committed to establishing the assumptions essential for at least one and unique solution of a switched coupled system of impulsive fractional differential equations having derivative of Hadamard type. Using Krasnoselskii’s fixed point theorem, the existence, as well as uniqueness results, is obtained. Along with this, different kinds of Hyers–Ulam stability are discussed. For supporting the theory, example is provided.

Changing the circuit-depth complexity of measurement-based quantum computation with hypergraph states

Abstract: While the circuit model of quantum computation defines its logical depth or "computational time" in terms of temporal gate sequences, the measurement-based model could allow totally different temporal ordering and parallelization of logical gates. By developing techniques to analyze Pauli measurements on multi-qubit hypergraph states generated by the Controlled-Controlled-Z (CCZ) gates, we introduce a deterministic scheme of universal measurement-based computation. In contrast to the cluster-state scheme, where the Clifford gates are parallelizable, our scheme enjoys massive parallelization of CCZ and SWAP gates, so that the computational depth grows with the number of global applications of Hadamard gates, or, in other words, with the number of changing computational bases. A logarithmic-depth implementation of an N-times Controlled-Z gate illustrates a novel trade-off between space and time complexity.