Related papers: Fast Fourier Sparsity Testing
In high-dimensional settings, sparse structures are critical for efficiency in term of memory and computation complexity. For a linear system, to find the sparsest solution provided with an over-complete dictionary of features directly is…
We consider the problem of finding high dimensional approximate nearest neighbors. Suppose there are d independent rare features, each having its own independent statistics. A point x will have x_{i}=0 denote the absence of feature i, and…
Let $f:\{-1,1\}^n$ be a polynomial with at most $s$ non-zero real coefficients. We give an algorithm for exactly reconstructing f given random examples from the uniform distribution on $\{-1,1\}^n$ that runs in time polynomial in $n$ and…
We consider the problem of approximating $[0,1]^{d}$-periodic functions by convolution with a scaled Gaussian kernel. We start by establishing convergence rates to functions from periodic Sobolev spaces and we show that the saturation rate…
Many regression and classification procedures fit a parameterized function $f(x;w)$ of predictor variables $x$ to data $\{x_{i},y_{i}\}_1^N$ based on some loss criterion $L(y,f)$. Often, regularization is applied to improve accuracy by…
In the second part of the series papers, we set out to study the algorithmic efficiency of sparse sensing. Stemmed from co-prime sensing, we propose a generalized framework, termed Diophantine sensing, which utilizes generic Diophantine…
We provide a systematic deterministic numerical scheme to approximate the volume (i.e. the Lebesgue measure) of a basic semi-algebraic set whose description follows a sparsity pattern. As in previous works (without sparsity), the underlying…
The problem of finding \emph{distance} between \emph{pattern} of length $m$ and \emph{text} of length $n$ is a typical way of generalizing pattern matching to incorporate dissimilarity score. For both Hamming and $L_1$ distances only a…
The Fast Fourier Transform(FFT) is a classic signal processing algorithm that is utilized in a wide range of applications. For image processing, FFT computes on every pixel's value of an image, regardless of their properties in frequency…
We introduce two efficient algorithms for computing the partial Fourier transforms in one and two dimensions. Our study is motivated by the wave extrapolation procedure in reflection seismology. In both algorithms, the main idea is to…
We study the Fourier ratio of a signal $f:\mathbb Z_N\to\mathbb C$, \[ \mathrm{FR}(f)\ :=\ \sqrt{N}\,\frac{\|\widehat f\|_{L^1(\mu)}}{\|\widehat f\|_{L^2(\mu)}} \ =\ \frac{\|\widehat f\|_1}{\|\widehat f\|_2}, \] as a simple scalar parameter…
We consider the minimization of the number of non-zero coefficients (the $\ell_0$ "norm") of the representation of a data set in terms of a dictionary under a fidelity constraint. (Both the dictionary and the norm defining the constraint…
In this paper, we consider multiple signals sharing same instantaneous frequencies. This kind of data is very common in scientific and engineering problems. To take advantage of this special structure, we modify our data-driven…
In this work we are interested in the problems of supervised learning and variable selection when the input-output dependence is described by a nonlinear function depending on a few variables. Our goal is to consider a sparse nonparametric…
Incorporating sparsity priors in learning tasks can give rise to simple, and interpretable models for complex high dimensional data. Sparse models have found widespread use in structure discovery, recovering data from corruptions, and a…
This paper presents a comprehensive analysis of hyperparameter estimation within the empirical Bayes framework (EBF) for sparse learning. By studying the influence of hyperpriors on the solution of EBF, we establish a theoretical connection…
Computing the Sparse Fast Fourier Transform(sFFT) of a K-sparse signal of size N has emerged as a critical topic for a long time. There are mainly two stages in the sFFT: frequency bucketization and spectrum reconstruction. Frequency…
Approximation of high-dimensional functions is a problem in many scientific fields that is only feasible if advantageous structural properties, such as sparsity in a given basis, can be exploited. A relevant tool for analysing sparse…
Sparse Ising problems can be found in application areas such as logistics, condensed matter physics and training of deep Boltzmann networks, but can be very difficult to tackle with high efficiency and accuracy. This report presents new…
The problem of estimating the frequencies of an exponential sum has been studied extensively over the last years. It can be understood as a sparse estimation problem, as it strives to identify the sparse representation of a signal using…