Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions

In this paper we show that under some assumptions, for a $d$-dimensional fractional Brownian motion with Hurst parameter $H>1/2$, the density of solution of stochastic differential equation driven by it has a short-time expansion similar to that in the Brownian motion case

Upper bounds for the density of solutions of stochastic differential equations driven by fractional Brownian motions

In this paper we study upper bounds for the density of solution of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H > 1/3. We show that under some geometric conditions, in the regular case H > 1/2, the density of the solution satisfy the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case H > 1/3 and under the same geometric conditions, we show that the density of the solution is smooth and admits an upper sub-Gaussian bound

Gradient Bounds for Solutions of Stochastic Differential Equations Driven by Fractional Brownian Motions

We study some functional inequalities satisfied by the distribution of the solution of a stochastic differential equation driven by fractional Brownian motions. Such functional inequalities are obtained through new integration by parts formulas on the path space of a fractional Brownian motion.Comment: The paper is dedicated to Pr. David Nualart 60th's birthda

Generalizable deep learning based medical image segmentation

Deep learning is revolutionizing medical image analysis and interpretation. However, its real-world deployment is often hindered by the poor generalization to unseen domains (new imaging modalities and protocols). This lack of generalization ability is further exacerbated by the scarcity of labeled datasets for training: Data collection and annotation can be prohibitively expensive in terms of labor and costs because label quality heavily dependents on the expertise of radiologists. Additionally, unreliable predictions caused by poor model generalization pose safety risks to clinical downstream applications. To mitigate labeling requirements, we investigate and develop a series of techniques to strengthen the generalization ability and the data efficiency of deep medical image computing models. We further improve model accountability and identify unreliable predictions made on out-of-domain data, by designing probability calibration techniques. In the first and the second part of thesis, we discuss two types of problems for handling unexpected domains: unsupervised domain adaptation and single-source domain generalization. For domain adaptation we present a data-efficient technique that adapts a segmentation model trained on a labeled source domain (e.g., MRI) to an unlabeled target domain (e.g., CT), using a small number of unlabeled training images from the target domain. For domain generalization, we focus on both image reconstruction and segmentation. For image reconstruction, we design a simple and effective domain generalization technique for cross-domain MRI reconstruction, by reusing image representations learned from natural image datasets. For image segmentation, we perform causal analysis of the challenging cross-domain image segmentation problem. Guided by this causal analysis we propose an effective data-augmentation-based generalization technique for single-source domains. The proposed method outperforms existing approaches on a large variety of cross-domain image segmentation scenarios. In the third part of the thesis, we present a novel self-supervised method for learning generic image representations that can be used to analyze unexpected objects of interest. The proposed method is designed together with a novel few-shot image segmentation framework that can segment unseen objects of interest by taking only a few labeled examples as references. Superior flexibility over conventional fully-supervised models is demonstrated by our few-shot framework: it does not require any fine-tuning on novel objects of interest. We further build a publicly available comprehensive evaluation environment for few-shot medical image segmentation. In the fourth part of the thesis, we present a novel probability calibration model. To ensure safety in clinical settings, a deep model is expected to be able to alert human radiologists if it has low confidence, especially when confronted with out-of-domain data. To this end we present a plug-and-play model to calibrate prediction probabilities on out-of-domain data. It aligns the prediction probability in line with the actual accuracy on the test data. We evaluate our method on both artifact-corrupted images and images from an unforeseen MRI scanning protocol. Our method demonstrates improved calibration accuracy compared with the state-of-the-art method. Finally, we summarize the major contributions and limitations of our works. We also suggest future research directions that will benefit from the works in this thesis.Open Acces