Publications

Several recent works have introduced highly compact representations of single-particle Green's functions in the imaginary time and Matsubara frequency domains, as well as efficient interpolation grids used to recover the representations. In particular, the intermediate representation with sparse sampling and the discrete Lehmann representation (DLR) make use of low rank compression techniques to obtain optimal approximations with controllable accuracy. We consider the use of the DLR in dynamical mean-field theory (DMFT) calculations, and in particular show that the standard full Matsubara frequency grid can be replaced by the compact grid of DLR Matsubara frequency nodes. We test the performance of the method for a DMFT calculation of Sr$_2$RuO$_4$ at temperature $50$K using a continuous-time quantum Monte Carlo impurity solver, and demonstrate that Matsubara frequency quantities can be represented on a grid of only 36 nodes with no reduction in accuracy, or increase in the number of self-consistent iterations, despite the presence of significant Monte Carlo noise.

Cryo-electron microscopy (cryo-EM) has recently become a leading method for obtaining high-resolution structures of biological macromolecules. However, cryo-EM is limited to biomolecular samples with low conformational heterogeneity, where most conformations can be well-sampled at various projection angles. While cryo-EM provides single-molecule data for heterogeneous molecules, most existing reconstruction tools cannot retrieve the ensemble distribution of possible molecular conformations from these data. To overcome these limitations, we build on a previous Bayesian approach and develop an ensemble refinement framework that estimates the ensemble density from a set of cryo-EM particle images by reweighting a prior conformational ensemble, e.g., from molecular dynamics simulations or structure prediction tools. Our work provides a general approach to recovering the equilibrium probability density of the biomolecule directly in conformational space from single-molecule data. To validate the framework, we study the extraction of state populations and free energies for a simple toy model and from synthetic cryo-EM particle images of a simulated protein that explores multiple folded and unfolded conformations.

The 3D pose estimation problem – aligning pairs of noisy 3D point clouds – is a problem with a wide variety of real- world applications. Here we focus on the use of quaternion- based neural network approaches to this problem and ap- parent anomalies that have arisen in previous efforts to re- solve them. In addressing these anomalies, we draw heav- ily from the extensive literature on closed-form methods to solve this problem. We suggest that the major concerns that have been put forward could be resolved using a sim- ple multi-valued training target derived from rigorous theo- retical properties of the rotation-to-quaternion map of Bar- Itzhack. This multi-valued training target is then demon- strated to have good performance for both simulated and ModelNet targets. We provide a comprehensive theoretical context, using the quaternion adjugate, to confirm and es- tablish the necessity of replacing single-valued quaternion functions by quaternions treated in the extended domain of multiple-charted manifolds.

We introduce a Fourier-based fast algorithm for Gaussian process regression in low dimensions. It approximates a translationally-invariant covariance kernel by complex exponentials on an equispaced Cartesian frequency grid of $M$ nodes. This results in a weight-space $M\times M$ system matrix with Toeplitz structure, which can thus be applied to a vector in ${\mathcal O}(M \log{M})$ operations via the fast Fourier transform (FFT), independent of the number of data points $N$. The linear system can be set up in ${\mathcal O}(N + M \log{M})$ operations using nonuniform FFTs. This enables efficient massive-scale regression via an iterative solver, even for kernels with fat-tailed spectral densities (large $M$). We provide bounds on both kernel approximation and posterior mean errors. Numerical experiments for squared-exponential and Matérn kernels in one, two and three dimensions often show 1-2 orders of magnitude acceleration over state-of-the-art rank-structured solvers at comparable accuracy. Our method allows 2D Matérn-$\mbox{$\frac{3}{2}$}$ regression from $N=10^9$ data points to be performed in 2 minutes on a standard desktop, with posterior mean accuracy $10^{-3}$. This opens up spatial statistics applications 100 times larger than previously possible.

We present a new version of the fast Gauss transform (FGT) for discrete and continuous sources. Classical Hermite expansions are avoided entirely, making use only of the plane-wave representation of the Gaussian kernel and a new hierarchical merging scheme. For continuous source distributions sampled on adaptive tensor-product grids, we exploit the separable structure of the Gaussian kernel to accelerate the computation. For discrete sources, the scheme relies on the nonuniform fast Fourier transform (NUFFT) to construct near field plane wave representations. The scheme has been implemented for either free-space or periodic boundary conditions. In many regimes, the speed is comparable to or better than that of the conventional FFT in work per gridpoint, despite being fully adaptive.

Recently, the stochastic Polyak step size (SPS) has emerged as a competitive adaptive step size scheme for stochastic gradient descent. Here we develop ProxSPS, a proximal variant of SPS that can handle regularization terms. Developing a proximal variant of SPS is particularly important, since SPS requires a lower bound of the objective function to work well. When the objective function is the sum of a loss and a regularizer, available estimates of a lower bound of the sum can be loose. In contrast, ProxSPS only requires a lower bound for the loss which is often readily available. As a consequence, we show that ProxSPS is easier to tune and more stable in the presence of regularization. Furthermore for image classification tasks, ProxSPS performs as well as AdamW with little to no tuning, and results in a network with smaller weight parameters. We also provide an extensive convergence analysis for ProxSPS that includes the non-smooth, smooth, weakly convex and strongly convex setting.

Randomized sketches of a tensor product of pvectors follow a tradeoff between statistical efficiency and computational acceleration. Commonly used approaches avoid computing the high-dimensional tensor product explicitly, resulting in a suboptimal dependence of O(3p) in the embedding dimension. We propose a simple Complex-to-Real (CtR) modification of well-known sketches that replaces real random projections by complex ones, incurring a lower O(2p)factor in the embedding dimension. The output of our sketches is real-valued, which renders
their downstream use straightforward. In particular, we apply our sketches to p-fold self-tensored inputs corresponding to the feature maps of the
polynomial kernel. We show that our method achieves state-of-the-art performance in terms of accuracy and speed compared to other randomized approximations from the literature.

This paper reviews the growing field of Bayesian prediction. Bayes point and interval prediction are defined and exemplified and situated in statistical prediction more generally. Then, four general approaches to Bayes prediction are defined and we turn to predictor selection. This can be done predictively or non-predictively and predictors can be based on single models or multiple models. We call these latter cases unitary predictors and model average predictors, respectively. Then we turn to the most recent aspect of prediction to emerge, namely prediction in the context of large observational data sets and discuss three further classes of techniques. We conclude with a summary and statement of several current open problems.

We introduce a layer potential representation for the solution of the transmission problem defined by two dielectric channels, or open waveguides, meeting along the straight-line interface, {x1=0}. The main observation is that the outgoing fundamental solution for the operator Δ+k21+q(x2), acting on functions defined in ℝ2, is easily constructed using the Fourier transform in the x1-variable and the elementary theory of ordinary differential equations. These fundamental solutions can then be used to represent the solution to the transmission problem in half planes. The transmission boundary conditions lead to integral equations along the intersection of the half planes, which, in our normalization, is the x2-axis. We show that, in appropriate Banach spaces, these integral equations are Fredholm equations of second kind, which are therefore generically solvable. We then show that the solutions satisfy an analogue of the Sommerfeld radiation condition that follows from work of Isozaki, Melrose, Vasy, et al. This formulation suggests practicable numerical methods to approximately solve this class of problems.