Publications

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 two-dimensional Matérn-\(\frac{3}{2}\) regression from \(N=10^9\) data points to be performed in two minutes on a standard desktop, with posterior mean accuracy \(10^{-3}\). This opens up spatial statistics applications 100 times larger than previously possible.

The discovery of superconductivity in twisted bilayer and twisted trilayer graphene has generated tremendous interest. The key feature of these systems is an interplay between interlayer coupling and a moiré superlattice that gives rise to low-energy flat bands with strong correlations. Flat bands can also be induced by moiré patterns in lattice-mismatched and or twisted heterostructures of other two-dimensional materials such as transition metal dichalcogenides (TMDs). Although a wide range of correlated phenomenon have indeed been observed in the moiré TMDs, robust demonstration of superconductivity has remained absent. Here we report superconductivity in 5 degree twisted bilayer WSe

We present a generalization of the phaseless auxiliary-field quantum Monte Carlo (AFQMC) method to cavity quantum-electrodynamical (QED) matter systems. The method can be formulated in both the Coulomb and the dipole gauge. We verify its accuracy by benchmarking calculations on a set of small molecules against full configuration interaction and state-of-the-art QED coupled cluster (QED-CCSD) calculations. Our results show that (i) gauge invariance can be achieved within correlation-consistent Gaussian basis sets, (ii) the accuracy of QED-CCSD can be enhanced significantly by adding the standard perturbative triples correction without light-matter coupling, and (iii) there is a straightforward way to evaluate the differential expression for the photon occupation number that works in any gauge. The high accuracy and favorable computational scaling of our AFQMC approach will enable a broad range of applications. Besides polaritonic chemistry, the method opens a way to simulate extended QED matter systems.

We solve a model of electrons with Hubbard-U Coulomb repulsion and a random Yukawa coupling to a two-dimensional bosonic bath, using an extended dynamical mean field theory scheme. Our model exhibits a quantum critical point, at which the repulsive component of the electron interactions strongly enhances the effects of the quantum critical bosonic fluctuations on the electrons, leading to a breakdown of Fermi liquid physics and the formation of a strange metal with `Planckian' (O(k

Tensor network contractions are widely used in statistical physics, quantum computing, and computer science. We introduce a method to efficiently approximate tensor network contractions using low-rank approximations, where each intermediate tensor generated during the contractions is approximated as a low-rank binary tree tensor network. The proposed algorithm has the flexibility to incorporate a large portion of the environment when performing low-rank approximations, which can lead to high accuracy for a given rank. Here, the environment refers to the remaining set of tensors in the network, and low-rank approximations with larger environments can generally provide higher accuracy. For contracting tensor networks defined on lattices, the proposed algorithm can be viewed as a generalization of the standard boundary-based algorithms. In addition, the algorithm includes a cost-efficient density matrix algorithm for approximating a tensor network with a general graph structure into a tree structure, whose computational cost is asymptotically upper-bounded by that of the standard algorithm that uses canonicalization. Experimental results indicate that the proposed technique outperforms previously proposed approximate tensor network contraction algorithms for multiple problems in terms of both accuracy and efficiency.

Cells must limit RNA–RNA interactions to avoid irreversible RNA entanglement. Cells may prevent deleterious RNA-RNA interactions by genome organization to avoid complementarity however, RNA viruses generate long, perfectly complementary antisense RNA during replication. How do viral RNAs avoid irreversible entanglement? One possibility is RNA sequestration into biomolecular condensates. To test this, we reconstituted critical SARS-CoV-2 RNA–RNA interactions in Nucleocapsid condensates. We observed that RNAs with low propensity RNA–RNA interactions resulted in more round, liquid-like condensates while those with high sequence complementarity resulted in more heterogeneous networked morphology independent of RNA structure stability. Residue-resolution molecular simulations and direct sequencing-based detection of RNA–RNA interactions support that these properties arise from degree of trans RNA contacts. We propose that extensive RNA–RNA interactions in cell and viral replication are controlled via a combination of genome organization, timing, RNA sequence content, RNA production ratios, and emergent biomolecular condensate material properties.

This work addresses the question of uniqueness and regularity of the minimizers of a convex but not strictly convex integral functional with linear growth in a two-dimensional setting. The integrand -- whose precise form derives directly from the theory of perfect plasticity -- behaves quadratically close to the origin and grows linearly once a specific threshold is reached. Thus, in contrast with the only existing literature on uniqueness for functionals with linear growth, that is that which pertains to the generalized least gradient, the integrand is not a norm. We make use of hyperbolic conservation laws hidden in the structure of the problem to tackle uniqueness. Our argument strongly relies on the regularity of a vector field -- the Cauchy stress in the terminology of perfect plasticity -- which allows us to define characteristic lines, and then to employ the method of characteristics. Using the detailed structure of the characteristic landscape evidenced in our preliminary study BF, we show that this vector field is actually continuous, save for possibly two points. The different behaviors of the energy density at zero and at infinity imply an inequality constraint on the Cauchy stress. Under a barrier type convexity assumption on the set where the inequality constraint is saturated, we show that uniqueness holds for pure Dirichlet boundary data devoid of any regularity properties, a stronger result than that of uniqueness for a given trace on the whole boundary since our minimizers can fail to attain the boundary data. We also show a partial regularity result for the minimizer.

This work addresses the question of uniqueness and regularity of the minimizers of a convex but not strictly convex integral functional with linear growth in a two-dimensional setting. The integrand -- whose precise form derives directly from the theory of perfect plasticity -- behaves quadratically close to the origin and grows linearly once a specific threshold is reached. Thus, in contrast with the only existing literature on uniqueness for functionals with linear growth, that is that which pertains to the generalized least gradient, the integrand is not a norm. We make use of hyperbolic conservation laws hidden in the structure of the problem to tackle uniqueness. Our argument strongly relies on the regularity of a vector field -- the Cauchy stress in the terminology of perfect plasticity -- which allows us to define characteristic lines, and then to employ the method of characteristics. Using the detailed structure of the characteristic landscape evidenced in our preliminary study BF, we show that this vector field is actually continuous, save for possibly two points. The different behaviors of the energy density at zero and at infinity imply an inequality constraint on the Cauchy stress. Under a barrier type convexity assumption on the set where the inequality constraint is saturated, we show that uniqueness holds for pure Dirichlet boundary data devoid of any regularity properties, a stronger result than that of uniqueness for a given trace on the whole boundary since our minimizers can fail to attain the boundary data. We also show a partial regularity result for the minimizer.

The Gibbs sampler, also known as the coordinate hit-and-run algorithm, is a Markov chain that is widely used to draw samples from probability distributions in arbitrary dimensions. At each iteration of the algorithm, a randomly selected coordinate is resampled from the distribution that results from conditioning on all the other coordinates. We study the behavior of the Gibbs sampler on the class of log-smooth and strongly log-concave target distributions supported on ℝ

We study the problem of gradient descent learning of a single-index target function f∗(x) = σ∗(⟨x,θ⟩) under isotropic Gaussian data in Rd, where the unknown link function σ∗ : R → R has information exponent p (defined as the lowest degree in the Hermite expansion). Prior works showed that gradientbased training of neural networks can learn this target with n ≳ dΘ(p) samples, and such complexity is predicted to be necessary by the correlational statistical query lower bound. Surprisingly, we prove that a two-layer neural network optimized by an SGD-based algorithm (on the squared loss) learns f∗ with a complexity that is not governed by the information exponent. Specifically, for arbitrary polynomial single-index models, we establish a sample and runtime complexity of n ≃ T = Θ(d·polylogd), where Θ(·) hides a constant only depending on the degree of σ∗; this dimension dependence matches the information theoretic limit up to polylogarithmic factors. More generally, we show that n ≳ d(p∗−1)∨1 samples are sufficient to achieve low generalization error, where p∗ ≤ p is the generative exponent of the link function. Core to our analysis is the reuse of minibatch in the gradient computation, which gives rise to higher-order information beyond correlational queries.