arXiv subjects

Karthik S. Gurumoorthy

Publications and source records attributed to Karthik S. Gurumoorthy.

29 records · Page 2Linked to original sources

arXiv2018

Signal Recovery in Perturbed Fourier Compressed Sensing

In many applications in compressed sensing, the measurement matrix is a Fourier matrix, i.e., it measures the Fourier transform of the underlying signal at some specified `base' frequencies $\{u_i\}_{i=1}^M$, where $M$ is the number of measurements. However due to system calibration errors, the system may measure the Fourier transform at frequencies $\{u_i + δ_i\}_{i=1}^M$ that are different from the base frequencies and where $\{δ_i\}_{i=1}^M$ are unknown. Ignoring perturbations of this nature can lead to major errors in signal recovery. In this paper, we present a simple but effective alternating minimization algorithm to recover the perturbations in the frequencies \emph{in situ} with the signal, which we assume is sparse or compressible in some known basis. In many cases, the perturbations $\{δ_i\}_{i=1}^M$ can be expressed in terms of a small number of unique parameters $P \ll M$. We demonstrate that in such cases, the method leads to excellent quality results that are several times better than baseline algorithms (which are based on existing off-grid methods in the recent literature on direction of arrival (DOA) estimation, modified to suit the computational problem in this paper). Our results are also robust to noise in the measurement values. We also provide theoretical results for (1) the convergence of our algorithm, and (2) the uniqueness of its solution under some restrictions.

cs.IT↗

arXiv2017

Degenerate Kalman filter error covariances and their convergence onto the unstable subspace

The characteristics of the model dynamics are critical in the performance of (ensemble) Kalman filters. In particular, as emphasized in the seminal work of Anna Trevisan and co-authors, the error covariance matrix is asymptotically supported by the unstable-neutral subspace only, i.e., it is spanned by the backward Lyapunov vectors with non-negative exponents. This behavior is at the core of algorithms known as Assimilation in the Unstable Subspace, although a formal proof was still missing. This paper provides the analytical proof of the convergence of the Kalman filter covariance matrix onto the unstable-neutral subspace when the dynamics and the observation operator are linear and when the dynamical model is error-free, for any, possibly rank-deficient, initial error covariance matrix. The rate of convergence is provided as well. The derivation is based on an expression that explicitly relates the error covariances at an arbitrary time to the initial ones. It is also shown that if the unstable and neutral directions of the model are sufficiently observed and if the column space of the initial covariance matrix has a non-zero projection onto all of the forward Lyapunov vectors associated with the unstable and neutral directions of the dynamics, the covariance matrix of the Kalman filter collapses onto an asymptotic sequence which is independent of the initial covariances. Numerical results are also shown to illustrate and support the theoretical findings.

math.OC↗

arXiv2016

Error bounds for gradient density estimation computed from a finite sample set using the method of stationary phase

For a twice continuously differentiable function $S$, we define the density function of its gradient (derivative in one dimension) $s = S^{\prime}$ as a random variable transformation of a uniformly distributed random variable using $s$ as the transformation function. Given $N$ values of $S$ sampled at equally spaced locations, we demonstrate using the method of stationary phase that the approximation error between the integral of the scaled, discrete power spectrum of the wave function $ϕ^{D}_τ=\frac{1}{\sqrt{L}}\exp\left(\frac{iS}τ\right)$ and the integral of the true density function of $s$ over an arbitrarily small interval is bounded above by $O(1/N)$ as $N \rightarrow \infty$ ($τ\rightarrow 0$). In addition to its easy implementation and fast computability in $O(N \log N)$ that only requires computing the discrete Fourier transform, our framework for obtaining the derivative 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 histograms and Parzen windows.

stat.CO↗

arXiv2016

Rank deficiency of Kalman error covariance matrices in linear time-varying system with deterministic evolution

We prove that for linear, discrete, time-varying, deterministic system (perfect model) with noisy outputs, the Riccati transformation in the Kalman filter asymptotically bounds the rank of the forecast and the analysis error covariance matrices to be less than or equal to the number of non-negative Lyapunov exponents of the system. Further, the support of these error covariance matrices is shown to be confined to the space spanned by the unstable-neutral backward Lyapunov vectors, providing the theoretical justification for the methodology of the algorithms that perform assimilation only in the unstable-neutral subspace. The equivalent property of the autonomous system is investigated as a special case.

math.OC↗

arXiv2015

Sensitivity Analysis for additive STDP rule

Spike Timing Dependent Plasticity (STDP) is a Hebbian like synaptic learning rule. The basis of STDP has strong experimental evidences and it depends on precise input and output spike timings. In this paper we show that under biologically plausible spiking regime, slight variability in the spike timing leads to drastically different evolution of synaptic weights when its dynamics are governed by the additive STDP rule.

q-bio.NC↗

arXiv2015

A new variational principle for the Euclidean distance function: Linear approach to the non-linear eikonal problem

We present a fast convolution-based technique for computing an approximate, signed Euclidean distance function $S$ on a set of 2D and 3D grid locations. Instead of solving the non-linear, static Hamilton-Jacobi equation ($\|\nabla S\|=1$), our solution stems from first solving for a scalar field $ϕ$ in a linear differential equation and then deriving the solution for $S$ by taking the negative logarithm. In other words, when $S$ and $ϕ$ are related by $ϕ= \exp \left(-\frac{S}τ \right)$ and $ϕ$ satisfies a specific linear differential equation corresponding to the extremum of a variational problem, we obtain the approximate Euclidean distance function $S = -τ\log(ϕ)$ which converges to the true solution in the limit as $τ\rightarrow 0$. This is in sharp contrast to techniques like the fast marching and fast sweeping methods which directly solve the Hamilton-Jacobi equation by the Godunov upwind discretization scheme. Our linear formulation results in a closed-form solution to the approximate Euclidean distance function expressible as a discrete convolution, and hence efficiently computable using the fast Fourier transform (FFT). Our solution also circumvents the need for spatial discretization of the derivative operator. As $τ\rightarrow0$ we show the convergence of our results to the true solution and also bound the error for a given value of $τ$. The differentiability of our solution allows us to compute---using a set of convolutions---the first and second derivatives of the approximate distance function. In order to determine the sign of the distance function (defined to be positive inside a closed region and negative outside), we compute the winding number in 2D and the topological degree in 3D, whose computations can also be performed via fast convolutions. We demonstrate the efficacy of our method through a set of experimental results.

cs.CV↗

arXiv2015

A fast eikonal equation solver using the Schrodinger wave equation

We use a Schrödinger wave equation formalism to solve the eikonal equation. In our framework, a solution to the eikonal equation is obtained in the limit as Planck's constant $\hbar$ (treated as a free parameter) tends to zero of the solution to the corresponding linear Schrödinger equation. The Schrödinger equation corresponding to the eikonal turns out to be a \emph{generalized, screened Poisson equation}. Despite being linear, it does not have a closed-form solution for arbitrary forcing functions. We present two different techniques to solve the screened Poisson equation. In the first approach we use a standard perturbation analysis approach to derive a new algorithm which is guaranteed to converge provided the forcing function is bounded and positive. The perturbation technique requires a sequence of discrete convolutions which can be performed in $O(N\log N)$ using the Fast Fourier Transform (FFT) where $N$ is the number of grid points. In the second method we discretize the linear Laplacian operator by the finite difference method leading to a sparse linear system of equations which can be solved using the plethora of sparse solvers. The eikonal solution is recovered from the exponent of the resultant scalar field. Our approach eliminates the need to explicitly construct viscosity solutions as customary with direct solutions to the eikonal. Since the linear equation is computed for a small but non-zero $\hbar$, the obtained solution is an approximation. Though our solution framework is applicable to the general class of eikonal problems, we detail specifics for the popular vision applications of shape-from-shading, vessel segmentation, and path planning.

math.NA↗

arXiv2014

Gradient density estimation in arbitrary finite dimensions using the method of stationary phase

We prove that the density function of the gradient of a sufficiently smooth function $S : Ω\subset \mathbb{R}^d \rightarrow \mathbb{R}$, obtained via a random variable transformation of a uniformly distributed random variable, is increasingly closely approximated by the normalized power spectrum of $ϕ=\exp\left(\frac{iS}τ\right)$ as the free parameter $τ\rightarrow 0$. The result is shown using the stationary phase approximation and standard integration techniques and requires proper ordering of limits. We highlight a relationship with the well-known characteristic function approach to density estimation, and detail why our result is distinct from this approach.

stat.ML↗

arXiv2013

Distance Transform Gradient Density Estimation using the Stationary Phase Approximation

The complex wave representation (CWR) converts unsigned 2D distance transforms into their corresponding wave functions. Here, the distance transform S(X) appears as the phase of the wave function ϕ(X)---specifically, ϕ(X)=exp(iS(X)/τwhere τis a free parameter. In this work, we prove a novel result using the higher-order stationary phase approximation: we show convergence of the normalized power spectrum (squared magnitude of the Fourier transform) of the wave function to the density function of the distance transform gradients as the free parameter τ-->0. In colloquial terms, spatial frequencies are gradient histogram bins. Since the distance transform gradients have only orientation information (as their magnitudes are identically equal to one almost everywhere), as τ-->0, the 2D Fourier transform values mainly lie on the unit circle in the spatial frequency domain. The proof of the result involves standard integration techniques and requires proper ordering of limits. Our mathematical relation indicates that the CWR of distance transforms is an intriguing, new representation.

stat.ML↗

arXiv2013

An application of the stationary phase method for estimating probability densities of function derivatives

We prove a novel result wherein the density function of the gradients---corresponding to density function of the derivatives in one dimension---of a thrice differentiable function S (obtained via a random variable transformation of a uniformly distributed random variable) defined on a closed, bounded interval Ω\subset R is accurately approximated by the normalized power spectrum of ϕ=exp(iS/τ) as the free parameter τ-->0. The result is shown using the well known stationary phase approximation and standard integration techniques and requires proper ordering of limits. Experimental results provide anecdotal visual evidence corroborating the result.

stat.ML↗

arXiv2013

On the dynamic compressibility of sets

We define a new notion of compressibility of a set of numbers through the dynamics of a polynomial function. We provide approaches to solve the problem by reducing it to the multi-criteria traveling salesman problem through a series of transformations. We then establish computational complexity results by giving some NP-completeness proofs. We also discuss about a notion of $ε$ K-compressibility of a set, with regard to lossy compression and deduce the necessary condition for the given set to be $ε$ K-compressible. Finally, we conclude by providing a list of open problems solutions to which could extend the applicability the our technique.

cs.CC↗