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Real-world non-mydriatic retinal fundus photography is prone to artifacts, imperfections and low-quality when certain ocular or systemic co-morbidities exist. Artifacts may result in inaccuracy or ambiguity in clinical diagnoses. In this paper, we proposed a simple but effective end-to-end framework for enhancing poor-quality retinal fundus images. Leveraging the optimal transport theory, we proposed an unpaired image-to-image translation scheme for transporting low-quality images to their high-quality counterparts. We theoretically proved that a Generative Adversarial Networks (GAN) model with a generator and discriminator is sufficient for this task. Furthermore, to mitigate the inconsistency of information between the low-quality images and their enhancements, an information consistency mechanism was proposed to maximally maintain structural consistency (optical discs, blood vessels, lesions) between the source and enhanced domains. Extensive experiments were conducted on the EyeQ dataset to demonstrate the superiority of our proposed method perceptually and quantitatively.

In this paper, we present a flexible and probabilistic framework for tracking topological features in time-varying scalar fields using merge trees and partial optimal transport. Merge trees are topological descriptors that record the evolution of connected components in the sublevel sets of scalar fields. We present a new technique for modeling and comparing merge trees using tools from partial optimal transport. In particular, we model a merge tree as a measure network, that is, a network equipped with a probability distribution, and define a notion of distance on the space of merge trees inspired by partial optimal transport. Such a distance offers a new and flexible perspective for encoding intrinsic and extrinsic information in the comparative measures of merge trees. More importantly, it gives rise to a partial matching between topological features in time-varying data, thus enabling flexible topology tracking for scientific simulations. Furthermore, such partial matching may be interpreted as probabilistic coupling between features at adjacent time steps, which gives rise to probabilistic tracking graphs. We derive a stability result for our distance and provide numerous experiments indicating the efficacy of distance in extracting meaningful feature tracks.

Policymakers are required to evaluate the health benefits of reducing the National Ambient Air Quality Standards (NAAQS; i.e., the safety standards) for fine particulate matter PM 2.5 before implementing new policies. We formulate this objective as a shift-response function (SRF) and develop methods to analyze the problem using methods for causal inference, specifically under the stochastic interventions framework. SRFs model the average change in an outcome of interest resulting from a hypothetical shift in the observed exposure distribution. We propose a new broadly applicable doubly-robust method to learn SRFs using targeted regularization with neural networks. We evaluate our proposed method under various benchmarks specific for marginal estimates as a function of continuous exposure. Finally, we implement our estimator in the motivating application that considers the potential reduction in deaths from lowering the NAAQS from the current level of 12 $\mu g/m^3$ to levels that are recently proposed by the Environmental Protection Agency in the US (10, 9, and 8 $\mu g/m^3$).

Although a concept class may be learnt more efficiently using quantum samples as compared with classical samples in certain scenarios, Arunachalam and de Wolf (JMLR, 2018) proved that quantum learners are asymptotically no more efficient than classical ones in the quantum PAC and Agnostic learning models. They established lower bounds on sample complexity via quantum state identification and Fourier analysis. In this paper, we derive optimal lower bounds for quantum sample complexity in both the PAC and agnostic models via an information-theoretic approach. The proofs are arguably simpler, and the same ideas can potentially be used to derive optimal bounds for other problems in quantum learning theory. We then turn to a quantum analogue of the Coupon Collector problem, a classic problem from probability theory also of importance in the study of PAC learning. Arunachalam, Belovs, Childs, Kothari, Rosmanis, and de Wolf (TQC, 2020) characterized the quantum sample complexity of this problem up to constant factors. First, we show that the information-theoretic approach mentioned above provably does not yield the optimal lower bound. As a by-product, we get a natural ensemble of pure states in arbitrarily high dimensions which are not easily (simultaneously) distinguishable, while the ensemble has close to maximal Holevo information. Second, we discover that the information-theoretic approach yields an asymptotically optimal bound for an approximation variant of the problem. Finally, we derive a sharp lower bound for the Quantum Coupon Collector problem, with the exact leading order term, via the generalized Holevo-Curlander bounds on the distinguishability of an ensemble. All the aspects of the Quantum Coupon Collector problem we study rest on properties of the spectrum of the associated Gram matrix, which may be of independent interest.

A persistent challenge in conditional image synthesis has been to generate diverse output images from the same input image despite only one output image being observed per input image. GAN-based methods are prone to mode collapse, which leads to low diversity. To get around this, we leverage Implicit Maximum Likelihood Estimation (IMLE) which can overcome mode collapse fundamentally. IMLE uses the same generator as GANs but trains it with a different, non-adversarial objective which ensures each observed image has a generated sample nearby. Unfortunately, to generate high-fidelity images, prior IMLE-based methods require a large number of samples, which is expensive. In this paper, we propose a new method to get around this limitation, which we dub Conditional Hierarchical IMLE (CHIMLE), which can generate high-fidelity images without requiring many samples. We show CHIMLE significantly outperforms the prior best IMLE, GAN and diffusion-based methods in terms of image fidelity and mode coverage across four tasks, namely night-to-day, 16x single image super-resolution, image colourization and image decompression. Quantitatively, our method improves Fr\'echet Inception Distance (FID) by 36.9% on average compared to the prior best IMLE-based method, and by 27.5% on average compared to the best non-IMLE-based general-purpose methods.

Pair trading is one of the most effective statistical arbitrage strategies which seeks a neutral profit by hedging a pair of selected assets. Existing methods generally decompose the task into two separate steps: pair selection and trading. However, the decoupling of two closely related subtasks can block information propagation and lead to limited overall performance. For pair selection, ignoring the trading performance results in the wrong assets being selected with irrelevant price movements, while the agent trained for trading can overfit to the selected assets without any historical information of other assets. To address it, in this paper, we propose a paradigm for automatic pair trading as a unified task rather than a two-step pipeline. We design a hierarchical reinforcement learning framework to jointly learn and optimize two subtasks. A high-level policy would select two assets from all possible combinations and a low-level policy would then perform a series of trading actions. Experimental results on real-world stock data demonstrate the effectiveness of our method on pair trading compared with both existing pair selection and trading methods.

We introduce a method called MASCOT (Multi-Agent Shape Control with Optimal Transport) to compute optimal control solutions of agents with shape/formation/density constraints. For example, we might want to apply shape constraints on the agents -- perhaps we desire the agents to hold a particular shape along the path, or we want agents to spread out in order to minimize collisions. We might also want a proportion of agents to move to one destination, while the other agents move to another, and to do this in the optimal way, i.e. the source-destination assignments should be optimal. In order to achieve this, we utilize the Earth Mover's Distance from Optimal Transport to distribute the agents into their proper positions so that certain shapes can be satisfied. This cost is both introduced in the terminal cost and in the running cost of the optimal control problem.

We study the problem of robust distribution estimation under the Wasserstein metric, a popular discrepancy measure between probability distributions rooted in optimal transport (OT) theory. We introduce a new outlier-robust Wasserstein distance $\mathsf{W}_p^\varepsilon$ which allows for $\varepsilon$ outlier mass to be removed from its input distributions, and show that minimum distance estimation under $\mathsf{W}_p^\varepsilon$ achieves minimax optimal robust estimation risk. Our analysis is rooted in several new results for partial OT, including an approximate triangle inequality, which may be of independent interest. To address computational tractability, we derive a dual formulation for $\mathsf{W}_p^\varepsilon$ that adds a simple penalty term to the classic Kantorovich dual objective. As such, $\mathsf{W}_p^\varepsilon$ can be implemented via an elementary modification to standard, duality-based OT solvers. Our results are extended to sliced OT, where distributions are projected onto low-dimensional subspaces, and applications to homogeneity and independence testing are explored. We illustrate the virtues of our framework via applications to generative modeling with contaminated datasets.

Graph Neural Networks (GNNs), which generalize deep neural networks to graph-structured data, have drawn considerable attention and achieved state-of-the-art performance in numerous graph related tasks. However, existing GNN models mainly focus on designing graph convolution operations. The graph pooling (or downsampling) operations, that play an important role in learning hierarchical representations, are usually overlooked. In this paper, we propose a novel graph pooling operator, called Hierarchical Graph Pooling with Structure Learning (HGP-SL), which can be integrated into various graph neural network architectures. HGP-SL incorporates graph pooling and structure learning into a unified module to generate hierarchical representations of graphs. More specifically, the graph pooling operation adaptively selects a subset of nodes to form an induced subgraph for the subsequent layers. To preserve the integrity of graph's topological information, we further introduce a structure learning mechanism to learn a refined graph structure for the pooled graph at each layer. By combining HGP-SL operator with graph neural networks, we perform graph level representation learning with focus on graph classification task. Experimental results on six widely used benchmarks demonstrate the effectiveness of our proposed model.