Yihao Fang

Yihao Fang, Ilsang Ohn, Vijay Gupta et al.

We propose extrinsic and intrinsic deep neural network architectures as general frameworks for deep learning on manifolds. Specifically, extrinsic deep neural networks (eDNNs) preserve geometric features on manifolds by utilizing an equivariant embedding from the manifold to its image in the Euclidean space. Moreover, intrinsic deep neural networks (iDNNs) incorporate the underlying intrinsic geometry of manifolds via exponential and log maps with respect to a Riemannian structure. Consequently, we prove that the empirical risk of the empirical risk minimizers (ERM) of eDNNs and iDNNs converge in optimal rates. Overall, The eDNNs framework is simple and easy to compute, while the iDNNs framework is accurate and fast converging. To demonstrate the utilities of our framework, various simulation studies, and real data analyses are presented with eDNNs and iDNNs.

Yihao Fang, Mu Niu, Pokman Cheung et al.

We propose an extrinsic Bayesian optimization (eBO) framework for general optimization problems on manifolds. Bayesian optimization algorithms build a surrogate of the objective function by employing Gaussian processes and quantify the uncertainty in that surrogate by deriving an acquisition function. This acquisition function represents the probability of improvement based on the kernel of the Gaussian process, which guides the search in the optimization process. The critical challenge for designing Bayesian optimization algorithms on manifolds lies in the difficulty of constructing valid covariance kernels for Gaussian processes on general manifolds. Our approach is to employ extrinsic Gaussian processes by first embedding the manifold onto some higher dimensional Euclidean space via equivariant embeddings and then constructing a valid covariance kernel on the image manifold after the embedding. This leads to efficient and scalable algorithms for optimization over complex manifolds. Simulation study and real data analysis are carried out to demonstrate the utilities of our eBO framework by applying the eBO to various optimization problems over manifolds such as the sphere, the Grassmannian, and the manifold of positive definite matrices.

Sudhandar Balakrishnan, Yihao Fang, Xioadan Zhu

The invention of transformer-based models such as BERT, GPT, and RoBERTa has enabled researchers and financial companies to finetune these powerful models and use them in different downstream tasks to achieve state-of-the-art performance. Recently, a lightweight alternative (approximately 0.1% - 3% of the original model parameters) to fine-tuning, known as prefix tuning has been introduced. This method freezes the model parameters and only updates the prefix to achieve performance comparable to full fine-tuning. Prefix tuning enables researchers and financial practitioners to achieve similar results with much fewer parameters. In this paper, we explore the robustness of prefix tuning when facing noisy data. Our experiments demonstrate that fine-tuning is more robust to noise than prefix tuning -- the latter method faces a significant decrease in performance on most corrupted data sets with increasing noise levels. Furthermore, prefix tuning has high variances in the F1 scores compared to fine-tuning in many corruption methods. We strongly advocate that caution should be carefully taken when applying the state-of-the-art prefix tuning method to noisy data.