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Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields including computer vision, medical imaging, and machine learning. The dominant geometric paradigm for such applications has consisted of a few Riemannian geometries associated with spectral computations that are costly at high scale and in high dimensions. We present a route to a scalable geometric framework for the analysis and processing of SPD-valued data based on the efficient computation of extreme generalized eigenvalues through the Hilbert and Thompson geometries of the semidefinite cone. We explore a particular geodesic space structure based on Thompson geometry in detail and establish several properties associated with this structure. Furthermore, we define a novel iterative mean of SPD matrices based on this geometry and prove its existence and uniqueness for a given finite collection of points. Finally, we state and prove a number of desirable properties that are satisfied by this mean.

Contrastive loss has been increasingly used in learning representations from multiple modalities. In the limit, the nature of the contrastive loss encourages modalities to exactly match each other in the latent space. Yet it remains an open question how the modality alignment affects the downstream task performance. In this paper, based on an information-theoretic argument, we first prove that exact modality alignment is sub-optimal in general for downstream prediction tasks. Hence we advocate that the key of better performance lies in meaningful latent modality structures instead of perfect modality alignment. To this end, we propose three general approaches to construct latent modality structures. Specifically, we design 1) a deep feature separation loss for intra-modality regularization; 2) a Brownian-bridge loss for inter-modality regularization; and 3) a geometric consistency loss for both intra- and inter-modality regularization. Extensive experiments are conducted on two popular multi-modal representation learning frameworks: the CLIP-based two-tower model and the ALBEF-based fusion model. We test our model on a variety of tasks including zero/few-shot image classification, image-text retrieval, visual question answering, visual reasoning, and visual entailment. Our method achieves consistent improvements over existing methods, demonstrating the effectiveness and generalizability of our proposed approach on latent modality structure regularization.

Classic algorithms and machine learning systems like neural networks are both abundant in everyday life. While classic computer science algorithms are suitable for precise execution of exactly defined tasks such as finding the shortest path in a large graph, neural networks allow learning from data to predict the most likely answer in more complex tasks such as image classification, which cannot be reduced to an exact algorithm. To get the best of both worlds, this thesis explores combining both concepts leading to more robust, better performing, more interpretable, more computationally efficient, and more data efficient architectures. The thesis formalizes the idea of algorithmic supervision, which allows a neural network to learn from or in conjunction with an algorithm. When integrating an algorithm into a neural architecture, it is important that the algorithm is differentiable such that the architecture can be trained end-to-end and gradients can be propagated back through the algorithm in a meaningful way. To make algorithms differentiable, this thesis proposes a general method for continuously relaxing algorithms by perturbing variables and approximating the expectation value in closed form, i.e., without sampling. In addition, this thesis proposes differentiable algorithms, such as differentiable sorting networks, differentiable renderers, and differentiable logic gate networks. Finally, this thesis presents alternative training strategies for learning with algorithms.

This PhD thesis contains several contributions to the field of statistical causal modeling. Statistical causal models are statistical models embedded with causal assumptions that allow for the inference and reasoning about the behavior of stochastic systems affected by external manipulation (interventions). This thesis contributes to the research areas concerning the estimation of causal effects, causal structure learning, and distributionally robust (out-of-distribution generalizing) prediction methods. We present novel and consistent linear and non-linear causal effects estimators in instrumental variable settings that employ data-dependent mean squared prediction error regularization. Our proposed estimators show, in certain settings, mean squared error improvements compared to both canonical and state-of-the-art estimators. We show that recent research on distributionally robust prediction methods has connections to well-studied estimators from econometrics. This connection leads us to prove that general K-class estimators possess distributional robustness properties. We, furthermore, propose a general framework for distributional robustness with respect to intervention-induced distributions. In this framework, we derive sufficient conditions for the identifiability of distributionally robust prediction methods and present impossibility results that show the necessity of several of these conditions. We present a new structure learning method applicable in additive noise models with directed trees as causal graphs. We prove consistency in a vanishing identifiability setup and provide a method for testing substructure hypotheses with asymptotic family-wise error control that remains valid post-selection. Finally, we present heuristic ideas for learning summary graphs of nonlinear time-series models.

Knowledge enhanced pre-trained language models (K-PLMs) are shown to be effective for many public tasks in the literature but few of them have been successfully applied in practice. To address this problem, we propose K-AID, a systematic approach that includes a low-cost knowledge acquisition process for acquiring domain knowledge, an effective knowledge infusion module for improving model performance, and a knowledge distillation component for reducing the model size and deploying K-PLMs on resource-restricted devices (e.g., CPU) for real-world application. Importantly, instead of capturing entity knowledge like the majority of existing K-PLMs, our approach captures relational knowledge, which contributes to better-improving sentence-level text classification and text matching tasks that play a key role in question answering (QA). We conducted a set of experiments on five text classification tasks and three text matching tasks from three domains, namely E-commerce, Government, and Film&TV, and performed online A/B tests in E-commerce. Experimental results show that our approach is able to achieve substantial improvement on sentence-level question answering tasks and bring beneficial business value in industrial settings.

Analyzing observational data from multiple sources can be useful for increasing statistical power to detect a treatment effect; however, practical constraints such as privacy considerations may restrict individual-level information sharing across data sets. This paper develops federated methods that only utilize summary-level information from heterogeneous data sets. Our federated methods provide doubly-robust point estimates of treatment effects as well as variance estimates. We derive the asymptotic distributions of our federated estimators, which are shown to be asymptotically equivalent to the corresponding estimators from the combined, individual-level data. We show that to achieve these properties, federated methods should be adjusted based on conditions such as whether models are correctly specified and stable across heterogeneous data sets.

Object detectors usually achieve promising results with the supervision of complete instance annotations. However, their performance is far from satisfactory with sparse instance annotations. Most existing methods for sparsely annotated object detection either re-weight the loss of hard negative samples or convert the unlabeled instances into ignored regions to reduce the interference of false negatives. We argue that these strategies are insufficient since they can at most alleviate the negative effect caused by missing annotations. In this paper, we propose a simple but effective mechanism, called Co-mining, for sparsely annotated object detection. In our Co-mining, two branches of a Siamese network predict the pseudo-label sets for each other. To enhance multi-view learning and better mine unlabeled instances, the original image and corresponding augmented image are used as the inputs of two branches of the Siamese network, respectively. Co-mining can serve as a general training mechanism applied to most of modern object detectors. Experiments are performed on MS COCO dataset with three different sparsely annotated settings using two typical frameworks: anchor-based detector RetinaNet and anchor-free detector FCOS. Experimental results show that our Co-mining with RetinaNet achieves 1.4%~2.1% improvements compared with different baselines and surpasses existing methods under the same sparsely annotated setting.

Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.

In this paper, we adopt 3D Convolutional Neural Networks to segment volumetric medical images. Although deep neural networks have been proven to be very effective on many 2D vision tasks, it is still challenging to apply them to 3D tasks due to the limited amount of annotated 3D data and limited computational resources. We propose a novel 3D-based coarse-to-fine framework to effectively and efficiently tackle these challenges. The proposed 3D-based framework outperforms the 2D counterpart to a large margin since it can leverage the rich spatial infor- mation along all three axes. We conduct experiments on two datasets which include healthy and pathological pancreases respectively, and achieve the current state-of-the-art in terms of Dice-S{\o}rensen Coefficient (DSC). On the NIH pancreas segmentation dataset, we outperform the previous best by an average of over 2%, and the worst case is improved by 7% to reach almost 70%, which indicates the reliability of our framework in clinical applications.

We consider the task of weakly supervised one-shot detection. In this task, we attempt to perform a detection task over a set of unseen classes, when training only using weak binary labels that indicate the existence of a class instance in a given example. The model is conditioned on a single exemplar of an unseen class and a target example that may or may not contain an instance of the same class as the exemplar. A similarity map is computed by using a Siamese neural network to map the exemplar and regions of the target example to a latent representation space and then computing cosine similarity scores between representations. An attention mechanism weights different regions in the target example, and enables learning of the one-shot detection task using the weaker labels alone. The model can be applied to detection tasks from different domains, including computer vision object detection. We evaluate our attention Siamese networks on a one-shot detection task from the audio domain, where it detects audio keywords in spoken utterances. Our model considerably outperforms a baseline approach and yields a 42.6% average precision for detection across 10 unseen classes. Moreover, architectural developments from computer vision object detection models such as a region proposal network can be incorporated into the model architecture, and results show that performance is expected to improve by doing so.