arXiv:1404.1147v1 [stat.CO] 4 Apr 2014 ERRORS BOUNDS FOR THE GRADIENT DENSITY ESTIMATION COMPUTED FROM A FINITE SAMPLE SET USING THE METHOD OF STATIONARY PHASE KARTHIK S. GURUMOORTHY §∗ AND ANAND RANGARAJAN ¶† Abstract. We demonstrate that our gradient density estimator—corresponding to estimating the density function of the derivatives in one dimension—obtained from a finite sample set of size N using the method of stationary phase converges at the rate of O(1/N) as N →∞. For a thrice differentiable function S, the density function of its derivative s = S ′ is obtained via a random variable transformation of a uniformly distributed random variable defined on a closed, bounded interval Ω = [0,L] ⊂ R using s as the transformation function. Given N i.i.d. samples of S we prove that the integral of the scaled, discrete power spectrum of φ = exp  iS τ  increasingly approximates the integral of the density function of s over an arbitrarily small interval Nα at the rate of O(1/N). In addition to its fast computability in O(N log N), our framework for obtaining the density does not involve any parameter selection like the number of histogram bins, width of the histogram bins, width of the kernel parameter, number of mixture components etc. as required by other widely applied methods like histogramming and Parzen windows. Key words. Keywords: Stationary phase approximation; Density estimation; Fourier transform; Convergence rate; Error bounds AMS subject classifications. 42B10; 41A60 1. Introduction. Density estimation methods attempt to estimate an unob- servable probability density function using observed data [12, 13, 15]. The observed data are treated as random samples from a large population which is assumed to be distributed according to the underlying density function. The aim of our current work is to compute the density function of the gradient—corresponding to derivative in one dimension—of a thrice differentiable function S (density function of S ′ ) from a finite set of N samples of S using the method of stationary phase [3, 8, 10, 17, 18] and bound the error between the estimated and the unknown true density as a function of N . If s = S ′ represent the derivative of the function S, the density function of s is defined via a random variable transformation of the uniformly distributed random variable X using s as the transformation function. In other words, if we define a random variable Y = s(X ) where the random variable X has a uniform distribution on the interval Ω = [0,L], the density function of Y represents the density function of s. In the field of computer vision many applications arise where the density of the gradient of the image, also popularly known as the histogram of oriented gradients (HOG), directly estimated from samples of the image are employed for human and object detection [4, 19]. Here the image intensity plays the role of the function S and the distribution of intensity gradients or edge directions are used as the feature descriptors to characterize the object appearance and shape within an image. In the recent article [7], an adaption of the HOG descriptor called the Gradient Field HOF (GF-HOG) is used for sketch based image retrieval. ∗ This work is fully supported by EADS Prize Postdoctoral Fellowship. Email: karthik.gurumoorthy@icts.res.in † Email: anand@cise.ufl.edu § International Center for Theoretical Sciences, Tata Institute of Fundamental Research, Banga- lore, Karnataka, India ¶ Department of Computer and Information Science and Engineering, University of Florida, Gainesville, Florida, USA 1