Predicting properties of crystals from their structures is a fundamental yet challenging task in materials science. Unlike molecules, crystal structures exhibit infinite periodic arrangements of atoms, requiring methods capable of capturing both local and global information effectively. However, most current works fall short of capturing long-range interactions within periodic structures. To address this limitation, we leverage reciprocal space to efficiently encode long-range interactions with learnable filters within Fourier transforms. We introduce Reciprocal Geometry Network (ReGNet), a novel architecture that integrates geometric GNNs and reciprocal blocks to model short-range and long-range interactions, respectively. Additionally, we introduce ReGNet-MT, a multi-task extension that employs mixture of experts (MoE) for multi-property prediction. Experimental results on the JARVIS and Materials Project benchmarks demonstrate that ReGNet achieves significant performance improvements. Moreover, ReGNet-MT attains state-of-the-art results on two bandgap properties due to positive transfer, while maintaining high computational efficiency. These findings highlight the potential of our model as a scalable and accurate solution for crystal property prediction. The code will be released upon paper acceptance.

This technology has important applications in many challenges of the industry such as content moderation, tracing AI-generated content, and monitoring the usage of AI models. The contributions of this thesis include the development of new watermarking techniques for images, audio, and text. We first introduce methods for active moderation of images on social platforms. We then develop specific techniques for AI-generated content. We specifically demonstrate methods to adapt latent generative models to embed watermarks in all generated content, identify watermarked sections in speech, and improve watermarking in large language models with tests that ensure low false positive rates. Furthermore, we explore the use of digital watermarking to detect model misuse, including the detection of watermarks in language models fine-tuned on watermarked text, and introduce training-free watermarks for the weights of large transformers. Through these contributions, the thesis provides effective solutions for the challenges posed by the increasing use of generative AI models and the need for model monitoring and content moderation. It finally examines the challenges and limitations of watermarking techniques and discuss potential future directions for research in this area.

Functional MRI (fMRI) and single-cell transcriptomics are pivotal in Alzheimer's disease (AD) research, each providing unique insights into neural function and molecular mechanisms. However, integrating these complementary modalities remains largely unexplored. Here, we introduce scBIT, a novel method for enhancing AD prediction by combining fMRI with single-nucleus RNA (snRNA). scBIT leverages snRNA as an auxiliary modality, significantly improving fMRI-based prediction models and providing comprehensive interpretability. It employs a sampling strategy to segment snRNA data into cell-type-specific gene networks and utilizes a self-explainable graph neural network to extract critical subgraphs. Additionally, we use demographic and genetic similarities to pair snRNA and fMRI data across individuals, enabling robust cross-modal learning. Extensive experiments validate scBIT's effectiveness in revealing intricate brain region-gene associations and enhancing diagnostic prediction accuracy. By advancing brain imaging transcriptomics to the single-cell level, scBIT sheds new light on biomarker discovery in AD research. Experimental results show that incorporating snRNA data into the scBIT model significantly boosts accuracy, improving binary classification by 3.39% and five-class classification by 26.59%. The codes were implemented in Python and have been released on GitHub (https://github.com/77YQ77/scBIT) and Zenodo (https://zenodo.org/records/11599030) with detailed instructions.

Cosine similarity is a popular distance measure that measures the similarity between two vectors in the inner product space. It is widely used in many data classification algorithms like K-Nearest Neighbors, Clustering etc. This study demonstrates limitations of application of cosine similarity. Particularly, this study demonstrates that traditional cosine similarity metric is valid only in the Euclidean space, whereas the original data resides in a random variable space. When there is variance and correlation in the data, then cosine distance is not a completely accurate measure of similarity. While new similarity and distance metrics have been developed to make up for the limitations of cosine similarity, these metrics are used as substitutes to cosine distance, and do not make modifications to cosine distance to overcome its limitations. Subsequently, we propose a modified cosine similarity metric, where cosine distance is adjusted by variance-covariance of the data. Application of variance-adjusted cosine distance gives better similarity performance compared to traditional cosine distance. KNN modelling on the Wisconsin Breast Cancer Dataset is performed using both traditional and modified cosine similarity measures and compared. The modified formula shows 100% test accuracy on the data.

In mixture models, nonspherical (anisotropic) noise within each cluster is widely present in real-world data. We study both the minimax rate and optimal statistical procedure for clustering under high-dimensional nonspherical mixture models. In high-dimensional settings, we first establish the information-theoretic limits for clustering under Gaussian mixtures. The minimax lower bound unveils an intriguing informational dimension-reduction phenomenon: there exists a substantial gap between the minimax rate and the oracle clustering risk, with the former determined solely by the projected centers and projected covariance matrices in a low-dimensional space. Motivated by the lower bound, we propose a novel computationally efficient clustering method: Covariance Projected Spectral Clustering (COPO). Its key step is to project the high-dimensional data onto the low-dimensional space spanned by the cluster centers and then use the projected covariance matrices in this space to enhance clustering. We establish tight algorithmic upper bounds for COPO, both for Gaussian noise with flexible covariance and general noise with local dependence. Our theory indicates the minimax-optimality of COPO in the Gaussian case and highlights its adaptivity to a broad spectrum of dependent noise. Extensive simulation studies under various noise structures and real data analysis demonstrate our method's superior performance.

Parameter shift rules (PSRs) are key techniques for efficient gradient estimation in variational quantum eigensolvers (VQEs). In this paper, we propose its Bayesian variant, where Gaussian processes with appropriate kernels are used to estimate the gradient of the VQE objective. Our Bayesian PSR offers flexible gradient estimation from observations at arbitrary locations with uncertainty information and reduces to the generalized PSR in special cases. In stochastic gradient descent (SGD), the flexibility of Bayesian PSR allows the reuse of observations in previous steps, which accelerates the optimization process. Furthermore, the accessibility to the posterior uncertainty, along with our proposed notion of gradient confident region (GradCoRe), enables us to minimize the observation costs in each SGD step. Our numerical experiments show that the VQE optimization with Bayesian PSR and GradCoRe significantly accelerates SGD and outperforms the state-of-the-art methods, including sequential minimal optimization.

We address the challenge of minimizing true risk in multi-node distributed learning. These systems are frequently exposed to both inter-node and intra-node label shifts, which present a critical obstacle to effectively optimizing model performance while ensuring that data remains confined to each node. To tackle this, we propose the Versatile Robust Label Shift (VRLS) method, which enhances the maximum likelihood estimation of the test-to-train label density ratio. VRLS incorporates Shannon entropy-based regularization and adjusts the density ratio during training to better handle label shifts at the test time. In multi-node learning environments, VRLS further extends its capabilities by learning and adapting density ratios across nodes, effectively mitigating label shifts and improving overall model performance. Experiments conducted on MNIST, Fashion MNIST, and CIFAR-10 demonstrate the effectiveness of VRLS, outperforming baselines by up to 20% in imbalanced settings. These results highlight the significant improvements VRLS offers in addressing label shifts. Our theoretical analysis further supports this by establishing high-probability bounds on estimation errors.

ASR systems are deployed across diverse environments, each with specific hardware constraints. We use supernet training to jointly train multiple encoders of varying sizes, enabling dynamic model size adjustment to fit hardware constraints without redundant training. Moreover, we introduce a novel method called OrthoSoftmax, which applies multiple orthogonal softmax functions to efficiently identify optimal subnets within the supernet, avoiding resource-intensive search. This approach also enables more flexible and precise subnet selection by allowing selection based on various criteria and levels of granularity. Our results with CTC on Librispeech and TED-LIUM-v2 show that FLOPs-aware component-wise selection achieves the best overall performance. With the same number of training updates from one single job, WERs for all model sizes are comparable to or slightly better than those of individually trained models. Furthermore, we analyze patterns in the selected components and reveal interesting insights.

TabPFN has emerged as a promising in-context learning model for tabular data, capable of directly predicting the labels of test samples given labeled training examples. It has demonstrated competitive performance, particularly on small-scale classification tasks. However, despite its effectiveness, TabPFN still requires further refinement in several areas, including handling high-dimensional features, aligning with downstream datasets, and scaling to larger datasets. In this paper, we revisit existing variants of TabPFN and observe that most approaches focus either on reducing bias or variance, often neglecting the need to address the other side, while also increasing inference overhead. To fill this gap, we propose Beta (Bagging and Encoder-based Fine-tuning for TabPFN Adaptation), a novel and effective method designed to minimize both bias and variance. To reduce bias, we introduce a lightweight encoder to better align downstream tasks with the pre-trained TabPFN. By increasing the number of encoders in a lightweight manner, Beta mitigate variance, thereby further improving the model's performance. Additionally, bootstrapped sampling is employed to further reduce the impact of data perturbations on the model, all while maintaining computational efficiency during inference. Our approach enhances TabPFN's ability to handle high-dimensional data and scale to larger datasets. Experimental results on over 200 benchmark classification datasets demonstrate that Beta either outperforms or matches state-of-the-art methods.

In many practical applications, regression models are employed to uncover relationships between predictors and a response variable, yet the common assumption of constant error variance is frequently violated. This issue is further compounded in high-dimensional settings where the number of predictors exceeds the sample size, necessitating regularization for effective estimation and variable selection. To address this problem, we propose the Heteroscedastic Double Bayesian Elastic Net (HDBEN), a novel framework that jointly models the mean and log-variance using hierarchical Bayesian priors incorporating both $\ell_1$ and $\ell_2$ penalties. Our approach simultaneously induces sparsity and grouping in the regression coefficients and variance parameters, capturing complex variance structures in the data. Theoretical results demonstrate that proposed HDBEN achieves posterior concentration, variable selection consistency, and asymptotic normality under mild conditions which justifying its behavior. Simulation studies further illustrate that HDBEN outperforms existing methods, particularly in scenarios characterized by heteroscedasticity and high dimensionality.