Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces

We establish a regularity theorem for second-order elliptic PDEs on $\mathbb{R}^{d}$ in spectral Barron spaces. Under mild ellipticity and smallness assumptions, the solution gains two additional orders of Barron regularity. As a corollary, we identify a class of PDEs whose solutions can be approximated by two-layer neural networks with cosine activation functions, where the width of the neural network is independent of the spatial dimension.

mViSE: A Visual Search Engine for Analyzing Multiplex IHC Brain Tissue Images

Whole-slide multiplex imaging of brain tissue generates massive information-dense images that are challenging to analyze and require custom software. We present an alternative query-driven programming-free strategy using a multiplex visual search engine (mViSE) that learns the multifaceted brain tissue chemoarchitecture, cytoarchitecture, and myeloarchitecture. Our divide-and-conquer strategy organizes the data into panels of related molecular markers and uses self-supervised learning to train a multiplex encoder for each panel with explicit visual confirmation of successful learning. Multiple panels can be combined to process visual queries for retrieving similar communities of individual cells or multicellular niches using information-theoretic methods. The retrievals can be used for diverse purposes including tissue exploration, delineating brain regions and cortical cell layers, profiling and comparing brain regions without computer programming. We validated mViSE's ability to retrieve single cells, proximal cell pairs, tissue patches, delineate cortical layers, brain regions and sub-regions. mViSE is provided as an open-source QuPath plug-in.

Weak-to-Strong Generalization Enables Fully Automated De Novo Training of Multi-head Mask-RCNN Model for Segmenting Densely Overlapping Cell Nuclei in Multiplex Whole-slice Brain Images

We present a weak to strong generalization methodology for fully automated training of a multi-head extension of the Mask-RCNN method with efficient channel attention for reliable segmentation of overlapping cell nuclei in multiplex cyclic immunofluorescent (IF) whole-slide images (WSI), and present evidence for pseudo-label correction and coverage expansion, the key phenomena underlying weak to strong generalization. This method can learn to segment de novo a new class of images from a new instrument and/or a new imaging protocol without the need for human annotations. We also present metrics for automated self-diagnosis of segmentation quality in production environments, where human visual proofreading of massive WSI images is unaffordable. Our method was benchmarked against five current widely used methods and showed a significant improvement. The code, sample WSI images, and high-resolution segmentation results are provided in open form for community adoption and adaptation.

Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions

We study the approximation complexity of high-dimensional second-order elliptic PDEs with homogeneous boundary conditions on the unit hypercube, within the framework of Barron spaces. Under the assumption that the coefficients belong to suitably defined Barron spaces, we prove that the solution can be efficiently approximated by two-layer neural networks, circumventing the curse of dimensionality. Our results demonstrate the expressive power of shallow networks in capturing high-dimensional PDE solutions under appropriate structural assumptions.