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A compactness of the Revuz map is established in the sense that the locally uniform convergence of a sequence of positive continuous additive functionals is derived in terms of their smooth measures. To this end, we first introduce a metric…
We show that rough isometries between metric spaces X, Y can be lifted to the spaces of real valued 1-Lipschitz functions over X and Y with supremum metric and apply this to their scaling limits. For the inverse, we show how rough…
Metric embedding is a powerful tool used extensively in mathematics and computer science. We devise a new method of using metric embeddings recursively, which turns out to be particularly effective in $\ell_p$ spaces, $p>2$, yielding…
The recently introduced full-history recursive multilevel Picard (MLP) approximation methods have turned out to be quite successful in the numerical approximation of solutions of high-dimensional nonlinear PDEs. In particular, there are…
The advancement of artificial intelligence demands flexible multimodal data processing with high throughput and energy efficiency. Photonic integrated circuits (PIC) has demonstrated promising potentials in terms of low latency and low…
Recent research suggested that the embeddings produced by CLIP-like contrastive language-image training are suboptimal for image-only tasks. The main theory is that the inter-modal (language-image) alignment loss ignores intra-modal…
CLIP is a seminal multimodal model that maps images and text into a shared representation space through contrastive learning on billions of image-caption pairs. Inspired by the rapid progress of large language models (LLMs), we investigate…
Judging the similarity of visualizations is crucial to various applications, such as visualization-based search and visualization recommendation systems. Recent studies show deep-feature-based similarity metrics correlate well with…
Vision-language models (VLMs) such as CLIP are trained via contrastive learning between text and image pairs, resulting in aligned image and text embeddings that are useful for many downstream tasks. A notable drawback of CLIP, however, is…
In this short note, we show that, in any given metric space, every Lipschitz open-map image of every subset of a given metric space whose boundary is Hausdorff-null is Hausdorff-measurable with respect to the same dimension. The main…
An Axial Shifted MLP architecture (AS-MLP) is proposed in this paper. Different from MLP-Mixer, where the global spatial feature is encoded for information flow through matrix transposition and one token-mixing MLP, we pay more attention to…
The MMT Adaptive Optics exoPlanet Characterization System (MAPS) is a comprehensive update to the first generation MMT adaptive optics system (MMTAO), designed to produce a facility class suite of instruments whose purpose is to image…
We propose depth from coupled optical differentiation, a low-computation passive-lighting 3D sensing mechanism. It is based on our discovery that per-pixel object distance can be rigorously determined by a coupled pair of optical…
We address the problem of merging graph and feature-space information while learning a metric from structured data. Existing algorithms tackle the problem in an asymmetric way, by either extracting vectorized summaries of the graph…
Symmetric positive definite (SPD) matrices are useful for capturing second-order statistics of visual data. To compare two SPD matrices, several measures are available, such as the affine-invariant Riemannian metric, Jeffreys divergence,…
Data-sensitive metrics adapt distances locally based the density of data points with the goal of aligning distances and some notion of similarity. In this paper, we give the first exact algorithm for computing a data-sensitive metric called…
Answering a question about triangle inequality suggested by R. Li, A. Barrlund introduced a distance function which is a metric on a subdomain of ${\mathbb R}^n\,.$ We study this Barrlund metric and give sharp bounds for it in terms of…
We study the metric structure of walks on graphs, understood as Lipschitz sequences. To this end, a weighted metric is introduced to handle sequences, enabling the definition of distances between walks based on stepwise vertex distances and…
In this paper, we study a natural extension of Multi-Layer Perceptrons (MLP) to functional inputs. We show that fundamental results for classical MLP can be extended to functional MLP. We obtain universal approximation results that show the…
Recovering sharp images from dual-pixel (DP) pairs with disparity-dependent blur is a challenging task.~Existing blur map-based deblurring methods have demonstrated promising results. In this paper, we propose, to the best of our knowledge,…