Publications

We present a fast summation method for lattice sums of the type which arise when solving wave scattering problems with periodic boundary conditions. While there are a variety of effective algorithms in the literature for such calculations, the approach presented here is new and leads to a rigorous analysis of Wood's anomalies. These arise when illuminating a grating at specific combinations of the angle of incidence and the frequency of the wave, for which the lattice sums diverge. They were discovered by Wood in 1902 as singularities in the spectral response. The primary tools in our approach are the Euler-Maclaurin formula and a steepest descent argument. The resulting algorithm has super-algebraic convergence and requires only milliseconds of CPU time.

The inversion of the Laplace-Beltrami operator and the computation of the Hodge decomposition of a tangential vector field on smooth surfaces arise as computational tasks in many areas of science, from computer graphics to machine learning to com- putational physics. Here, we present a high-order accurate pseudo-spectral approach, applicable to closed surfaces of genus one in three dimensional space, with a view toward applications in plasma physics and fluid dynamics.

We consider the problems of compressed sensing and optimal denoising for signals $\mathbf{x_0}\in\mathbb{R}^N$ that are monotone, i.e., $\mathbf{x_0}(i+1) \geq \mathbf{x_0}(i)$, and sparsely varying, i.e., $\mathbf{x_0}(i+1) > \mathbf{x_0}(i)$ only for a small number $k$ of indices $i$. We approach the compressed sensing problem by minimizing the total variation norm restricted to the class of monotone signals subject to equality constraints obtained from a number of measurements $A\mathbf{x_0}$. For random Gaussian sensing matrices $A\in\mathbb{R}^{m\times N}$ we derive a closed form expression for the number of measurements $m$ required for successful reconstruction with high probability. We show that the probability undergoes a phase transition as $m$ varies, and depends not only on the number of change points, but also on their location. For denoising we regularize with the same norm and derive a formula for the optimal regularizer weight that depends only mildly on $\mathbf{x_0}$. We obtain our results using the statistical dimension tool.

Understanding gene regulatory networks is critical to understanding cellular differentiation and response to external stimuli. Methods for global network inference have been developed and applied to a variety of species. Most approaches consider the problem of network inference independently in each species, despite evidence that gene regulation can be conserved even in distantly related species. Further, network inference is often confined to single data-types (single platforms) and single cell types. We introduce a method for multi-source network inference that allows simultaneous estimation of gene regulatory networks in multiple species or biological processes through the introduction of priors based on known gene relationships such as orthology incorporated using fused regression. This approach improves network inference performance even when orthology mapping and conservation are incomplete. We refine this method by presenting an algorithm that extracts the true conserved subnetwork from a larger set of potentially conserved interactions and demonstrate the utility of our method in cross species network inference. Last, we demonstrate our method’s utility in learning from data collected on different experimental platforms.

We introduce a fast algorithm for computing volume potentials - that is, the convolution of a translation invariant, free-space Green's function with a compactly supported source distribution defined on a uniform grid. The algorithm relies on regularizing the Fourier transform of the Green's function by cutting off the interaction in physical space beyond the domain of interest. This permits the straightforward application of trapezoidal quadrature and the standard FFT, with superalgebraic convergence for smooth data. Moreover, the method can be interpreted as employing a Nystrom discretization of the corresponding integral operator, with matrix entries which can be obtained explicitly and rapidly. This is of use in the design of preconditioners or fast direct solvers for a variety of volume integral equations. The method proposed permits the computation of any derivative of the potential, at the cost of an additional FFT.

We present new integral representations in two dimensions for the elastance problem in electrostatics and the mobility problem in Stokes flow. These representations lead to resonance-free Fredholm integral equations of the second kind and well conditioned linear systems upon discretization. By coupling our integral equations with high order quadrature and fast multipole acceleration, large-scale problems can be solved with only modest computing resources. We also discuss some applications of these boundary value problems in applied physics.

We give a principled approach for the selection of a boundary integral, retarded potential representation for the solution of scattering problems for the wave equation in an exterior domain.

We present a fast direct solver for two dimensional scattering problems, where an incident wave impinges on a penetrable medium with compact support. We represent the scattered field using a volume potential whose kernel is the outgoing Green's function for the exterior domain. Inserting this representation into the governing partial differential equation, we obtain an integral equation of the Lippmann-Schwinger type. The principal contribution here is the development of an automatically adaptive, high-order accurate discretization based on a quad tree data structure which provides rapid access to arbitrary elements of the discretized system matrix. This permits the straightforward application of state-of-the-art algorithms for constructing compressed versions of the solution operator. These solvers typically require $O(N^{3/2})$ work, where $N$ denotes the number of degrees of freedom. We demonstrate the performance of the method for a variety of problems in both the low and high frequency regimes.

Selecting regularization parameters in penalized high-dimensional graphical models in a principled, data-driven, and computationally efficient manner continues to be one of the key challenges in high-dimensional statistics. We present substantial computational gains and conceptual generalizations of the Stability Approach to Regularization Selection (StARS), a state-of-the-art graphical model selection scheme. Using properties of the Poisson-Binomial distribution and convex non-asymptotic distributional modeling we propose lower and upper bounds on the StARS graph regularization path which results in greatly reduced computational cost without compromising regularization selection. We also generalize the StARS criterion from single edge to induced subgraph (graphlet) stability. We show that simultaneously requiring edge and graphlet stability leads to superior graph recovery performance independent of graph topology. These novel insights render Gaussian graphical model selection a routine task on standard multi-core computers.

The zebra finch brain features a set of clearly defined and hierarchically arranged motor nuclei that are selectively responsible for producing singing behavior. One of these regions, a critical forebrain structure called HVC, contains premotor neurons that are active at precise time points during song production. However, the neural representation of this behavior at a population level remains elusive. We used two-photon microscopy to monitor ensemble activity during singing, integrating across multiple trials by adopting a Bayesian inference approach to more precisely estimate burst timing. Additionally, we examined spiking and motor-related synaptic inputs using intracellular recordings during singing. With both experimental approaches, we find that premotor events do not occur preferentially at the onsets or offsets of song syllables or at specific subsyllabic motor landmarks. These results strongly support the notion that HVC projection neurons collectively exhibit a temporal sequence during singing that is uncoupled from ongoing movements.