Publications

Tensor networks are a compressed format for multi-dimensional data. One-dimensional tensor networks -- often referred to as tensor trains (TT) or matrix product states (MPS) -- are increasingly being used as a numerical ansatz for continuum functions by "quantizing" the inputs into discrete binary digits. Here we demonstrate the power of more general tree tensor networks for this purpose. We provide direct constructions of a number of elementary functions as generic tree tensor networks and interpolative constructions for more complicated functions via a generalization of the tensor cross interpolation algorithm. For a range of multi-dimensional functions we show how more structured tree tensor networks offer a significantly more efficient ansatz than the commonly used tensor train. We demonstrate an application of our methods to solving multi-dimensional, non-linear Fredholm equations, providing a rigorous bound on the rank of the solution which, in turn, guarantees exponentially scaling accuracy with the size of the tree tensor network for certain problems.

Image classification and denoising suffer from complementary issues of lack of robustness or partially ignoring conditioning information. We argue that they can be alleviated by unifying both tasks through a model of the joint probability of (noisy) images and class labels. Classification is performed with a forward pass followed by conditioning. Using the Tweedie-Miyasawa formula, we evaluate the denoising function with the score, which can be computed by marginalization and back-propagation. The training objective is then a combination of cross-entropy loss and denoising score matching loss integrated over noise levels. Numerical experiments on CIFAR-10 and ImageNet show competitive classification and denoising performance compared to reference deep convolutional classifiers/denoisers, and significantly improves efficiency compared to previous joint approaches. Our model shows an increased robustness to adversarial perturbations compared to a standard discriminative classifier, and allows for a novel interpretation of adversarial gradients as a difference of denoisers.

2D template matching (2DTM) can be used to detect molecules and their assemblies in cellular cryo-EM images with high positional and orientational accuracy. While 2DTM successfully detects spherical targets such as large ribosomal subunits, challenges remain in detecting smaller and more aspherical targets in various environments. In this work, a novel 2DTM metric, referred to as the 2DTM p-value, is developed to extend the 2DTM framework to more complex applications. The 2DTM p-value combines information from two previously used 2DTM metrics, namely the 2DTM signal-to-noise ratio (SNR) and z-score, which are derived from the cross-correlation coefficient between the target and the template. The 2DTM p-value demonstrates robust detection accuracies under various imaging and sample conditions and outperforms the 2DTM SNR and z-score alone. Specifically, the 2DTM p-value improves the detection of aspherical targets such as a modified artificial tubulin patch particle (500 kDa) and a much smaller clathrin monomer (193 kDa) in simulated data. It also accurately recovers mature 60S ribosomes in yeast lamellae samples, even under conditions of increased Gaussian noise. The new metric will enable the detection of a wider variety of targets in both purified and cellular samples through 2DTM.

2D template matching (2DTM) can be used to detect molecules and their assemblies in cellular cryo-EM images with high positional and orientational accuracy. While 2DTM successfully detects spherical targets such as large ribosomal subunits, challenges remain in detecting smaller and more aspherical targets in various environments. In this work, a novel 2DTM metric, referred to as the 2DTM p-value, is developed to extend the 2DTM framework to more complex applications. The 2DTM p-value combines information from two previously used 2DTM metrics, namely the 2DTM signal-to-noise ratio (SNR) and z-score, which are derived from the cross-correlation coefficient between the target and the template. The 2DTM p-value demonstrates robust detection accuracies under various imaging and sample conditions and outperforms the 2DTM SNR and z-score alone. Specifically, the 2DTM p-value improves the detection of aspherical targets such as a modified artificial tubulin patch particle (500 kDa) and a much smaller clathrin monomer (193 kDa) in simulated data. It also accurately recovers mature 60S ribosomes in yeast lamellae samples, even under conditions of increased Gaussian noise. The new metric will enable the detection of a wider variety of targets in both purified and cellular samples through 2DTM.

The terminal cells of the Drosophila larval tracheal system are perhaps the simplest delivery networks, providing an analogue for mammalian vascular growth and function in a system with many fewer components. These cells are a prime example of single-cell morphogenesis, branching significantly over time to adapt to the needs of the growing tissue they supply. While the genetic mechanisms governing local branching decisions have been studied extensively, an understanding of the emergence of a global network architecture is still lacking. Mapping out the full network architecture of populations of terminal cells at different developmental times of Drosophila larvae, we find that cell growth follows scaling laws relating the total edge length, supply area, and branch density. Using time-lapse imaging of individual terminal cells, we identify that the cells grow in three ways: by extending branches, by the side budding of new branches, and by internally growing existing branches. A generative model based on these modes of growth recapitulates statistical properties of the terminal cell network data. These results suggest that the scaling laws arise from the coupled contributions of branching and internal growth. This study establishes the terminal cell as a uniquely tractable model system for further studies of transportation and distribution networks.

In vivo molecular imaging tools hold immense potential to drive transformative breakthroughs by enabling researchers to visualize cellular and molecular interactions in real-time and/or at high resolution. These advancements will facilitate a deeper understanding of fundamental biological processes and their dysregulation in disease states. Here, we develop and characterize a self-assembling protein nanomicelle called collagen type I binding – thermoresponsive assembled protein (Col1-TRAP) that binds tightly to type I collagen in vitro with nanomolar affinity. For ex vivo visualization, Col1-TRAP is labeled with a near-infrared fluorescent dye (NIR-Col1-TRAP). Both Col1-TRAP and NIR-Col1-TRAP display approximately a 3.8-fold greater binding to type I collagen compared to TRAP when measured by surface plasmon resonance (SPR). We present a proof-of-concept study using NIR-Col1-TRAP to detect fibrotic type I collagen deposition ex vivo in the livers of mice with non-alcoholic steatohepatitis (NASH). We show that NIR-Col1-TRAP demonstrates significantly decreased plasma recirculation time as well as increased liver accumulation in the NASH mice compared to mice without disease over 4 hours. As a result, NIR-Col1-TRAP shows potential as an imaging probe for NASH with in vivo targeting performance after injection in mice.

Multiomic studies at a single cell and spatial resolution are powerful approaches to define molecular and cellular landscape of the human kidney and understand etiology of failed or successful repair in acute or chronic injury. We expand KPMP AtlasV1 with clinicopathological correlations and maps of immune-fibroblast-vascular niches with insights into AKI-CKD transition.

A central question in deep learning is to understand the functions learned by deep networks. What is their approximation class? Do the learned weights and representations depend on initialization? Previous empirical work has evidenced that kernels defined by network acti- vations are similar across initializations. For shallow networks, this has been theoretically studied with random feature models, but an extension to deep networks has remained elu- sive. Here, we provide a deep extension of such random feature models, which we call the rainbow model. We prove that rainbow networks define deterministic (hierarchical) kernels in the infinite-width limit. The resulting functions thus belong to a data-dependent RKHS which does not depend on the weight randomness. We also verify numerically our mod- eling assumptions on deep CNNs trained on image classification tasks, and show that the trained networks approximately satisfy the rainbow hypothesis. In particular, rainbow net- works sampled from the corresponding random feature model achieve similar performance as the trained networks. Our results highlight the central role played by the covariances of network weights at each layer, which are observed to be low-rank as a result of feature learning.

We present nifty-ls, a software package for fast and accurate evaluation of the Lomb–Scargle periodogram. nifty-ls leverages the fact that Lomb–Scargle can be computed using a non-uniform fast Fourier transform (NUFFT), which we evaluate with the Flatiron Institute NUFFT package (finufft). This approach achieves a many-fold speedup over the Press & Rybicki method as implemented in Astropy and is simultaneously many orders of magnitude more accurate. nifty-ls also supports fast evaluation on GPUs via CUDA and integrates with the Astropy Lomb–Scargle interface. nifty-ls is publicly available at https://github.com/flatironinstitute/nifty-ls/.

The unusual structure and symmetry of low-energy states in twisted transition metal dichalcogenides leads to large in-plane spin-exchange interactions between spin-valley locked holes. We demonstrate that this exchange interaction can stabilize a gapped spin-liquid phase with a quantized spin-Chern number of three when the twist angle is sufficiently small and the system lies in a Mott insulating phase. The gapped spin liquid may be understood as arising from spinon pairing in the DIII Altland-Zirnbauer symmetry class. Applying an out of plane electric field or increasing the twist angle is shown to drive a transition respectively to an anomalous Hall insulator or an in-plane antiferromagnet. Recent experiments indicate that a spin-Chern number three phase occurs in twisted MoTe2 at small twist angles with a transition to a quantum anomalous Hall phase as the twist angle is increased above a critical value of about 2.5∘ in absence of applied electric field.