Publications

While effective, the backpropagation (BP) algorithm exhibits limitations in terms of biological plausibility, computational cost, and suitability for online learning. As a result, there has been a growing interest in developing alternative biologically plausible learning approaches that rely on local learning rules. This study focuses on the primarily unsupervised similarity matching (SM) framework, which aligns with observed mechanisms in biological systems and offers online, localized, and biologically plausible algorithms. i) To scale SM to large datasets, we propose an implementation of Convolutional Nonnegative SM using PyTorch. ii) We introduce a localized supervised SM objective reminiscent of canonical correlation analysis, facilitating stacking SM layers. iii) We leverage the PyTorch implementation for pre-training architectures such as LeNet and compare the evaluation of features against BP-trained models. This work combines biologically plausible algorithms with computational efficiency opening multiple avenues for further explorations.

Spectral clustering and diffusion maps are celebrated dimensionality reduction algorithms built on eigen-elements related to the diffusive structure of the data. The core of these procedures is the approximation of a Laplacian through a graph kernel approach, however this local average construction is known to be cursed by the high-dimension 𝑑. In this article, we build a different estimator of the Laplacian, via a reproducing kernel Hilbert spaces method, which adapts naturally to the regularity of the problem. We provide non-asymptotic statistical rates proving that the kernel estimator we build can circumvent the curse of dimensionality. Finally we discuss techniques (Nyström subsampling, Fourier features) that enable to reduce the computational cost of the estimator while not degrading its overall performance.

We present an algorithm to solve the dispersive depth-averaged Serre-Green-Naghdi (SGN) equations using patch-based adaptive mesh refinement. These equations require adding additional higher derivative terms to the nonlinear shallow water equations. This has been implemented as a new component of the open source GeoClaw software that is widely used for modeling tsunamis, storm surge, and related hazards, improving its accuracy on shorter wavelength phenomena. The equations require the solution of an elliptic system at each time step. The adaptive algorithm allows different time steps on different refinement levels, and solves the implicit equations level by level. Computational examples are presented to illustrate the stability and accuracy on a radially symmetric test case and two realistic tsunami modeling problems, including a hypothetical asteroid impact creating a short wavelength tsunami for which dispersive terms are necessary.

The hierarchical prior used in Latent Gaussian models (LGMs) induces a posterior geometry prone to frustrate inference algorithms. Marginalizing out the latent Gaussian variable using an integrated Laplace approximation removes the offending geometry, allowing us to do efficient inference on the hyperparameters. To use gradient-based inference we need to compute the approximate marginal likelihood and its gradient. The adjoint-differentiated Laplace approximation differentiates the marginal likelihood and scales well with the dimension of the hyperparameters. While this method can be applied to LGMs with any prior covariance, it only works for likelihoods with a diagonal Hessian. Furthermore, the algorithm requires methods which compute the first three derivatives of the likelihood with current implementations relying on analytical derivatives. I propose a generalization which is applicable to a broader class of likelihoods and does not require analytical derivatives of the likelihood. Numerical experiments suggest the added flexibility comes at no computational cost: on a standard LGM, the new method is in fact slightly faster than the existing adjoint-differentiated Laplace approximation. I also apply the general method to an LGM with an unconventional likelihood. This example highlights the algorithm's potential, as well as persistent challenges.

Autism spectrum disorder (ASD) is a neurodevelopmental disorder characterized by heterogeneous cognitive, behavioral and communication impairments. Disruption of the gut–brain axis (GBA) has been implicated in ASD although with limited reproducibility across studies. In this study, we developed a Bayesian differential ranking algorithm to identify ASD-associated molecular and taxa profiles across 10 cross-sectional microbiome datasets and 15 other datasets, including dietary patterns, metabolomics, cytokine profiles and human brain gene expression profiles. We found a functional architecture along the GBA that correlates with heterogeneity of ASD phenotypes, and it is characterized by ASD-associated amino acid, carbohydrate and lipid profiles predominantly encoded by microbial species in the genera Prevotella, Bifidobacterium, Desulfovibrio and Bacteroides and correlates with brain gene expression changes, restrictive dietary patterns and pro-inflammatory cytokine profiles. The functional architecture revealed in age-matched and sex-matched cohorts is not present in sibling-matched cohorts. We also show a strong association between temporal changes in microbiome composition and ASD phenotypes. In summary, we propose a framework to leverage multi-omic datasets from well-defined cohorts and investigate how the GBA influences ASD.

Differentiable simulation enables gradients to be back-propagated through physics simulations. In this way, one can learn the dynamics and properties of a physics system by gradient-based optimization or embed the whole differentiable simulation as a layer in a deep learning model for downstream tasks, such as planning and control. However, differentiable simulation at its current stage is not perfect and might provide wrong gradients that deteriorate its performance in learning tasks. In this paper, we study differentiable rigid-body simulation with contacts. We find that existing differentiable simulation methods provide inaccurate gradients when the contact normal direction is not fixed - a general situation when the contacts are between two moving objects. We propose to improve gradient computation by continuous collision detection and leverage the time-of-impact (TOI) to calculate the post-collision velocities. We demonstrate our proposed method, referred to as TOI-Velocity, on two optimal control problems. We show that with TOI-Velocity, we are able to learn an optimal control sequence that matches the analytical solution, while without TOI-Velocity, existing differentiable simulation methods fail to do so.

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.