Preprints/ Publications (by journal/ conference)

(* = equal contribution )
(** = alphabetical order )
(† = co-last author )

Journal Submissions

Generative models from the multivariate Fourier integral theorem . Under review.
Nhat Ho**, Stephen G. Walker**.

A general theory for softmax gating Gaussian mixture of experts . Under review.
Huy Nguyen, Nhat Ho.

Energy meets sliced Wasserstein distance: From linear to nonlinear projections . To be submitted.
Khai Nguyen, Nhat Ho.

Mixture model meets Transformer . To be submitted.
Tan Nguyen*, Tam Nguyen*, Nhat Ho†, Stanley Osher†.

Improving Transformer via general integral theorems . To be submitted.
Tan Nguyen*, Minh Pham*, Nhat Ho.

Revisiting sliced optimal transport via geometric methods. To be submitted.
Khai Nguyen, Nhat Ho.

Uniform convergence rates for maximum likelihood estimation under two-component Gaussian mixture models. Arxiv Preprint.
Tudor Manole, Nhat Ho.

Journal extension: Minimax optimal convergence rates for location-scale Gaussian mixture models. To be submitted.

On mean-field variational inference: Non-asymptotic approximation, Bayesian inference, and parameter estimation. To be submitted.
Wenlong Mou, Nhat Ho, Martin J. Wainwright, Peter L. Bartlett, Michael I. Jordan.

BONuS: Multiple multivariate testing with a data-adaptive test statistic . Arxiv Preprint.
Chiao-Yu Yang, Lihua Lei, Nhat Ho, Will Fithian.

Conference Submissions

Sigmoid gating is more sample efficient than softmax gating in mixture of experts . Under review.
Huy Nguyen, Nhat Ho†, Alessandro Rinaldo†.

Bayesian nonparametrics meets data-driven robust optimization . Under review.
Nicola Bariletto, Nhat Ho.

FuseMoE: Mixture-of-experts Transformers for fleximodal fusion . Under review.
Xing Han, Huy Nguyen, Carl William Harris, Nhat Ho†, Suchi Saria†.

Borrowing strength in distributionally robust optimization via hierarchical Dirichlet processes . Under review.
Nicola Bariletto, Khai Nguyen, Nhat Ho.

Statistical advantages of perturbing cosine router in sparse mixture of experts . Under review.
Huy Nguyen, Pedram Akbarian*, Trang Pham*, Trang Nguyen*, Shujian Zhang Nhat Ho.

Mixture of experts meets prompt-based continual learning . Under review.
Minh Le, An Nguyen*, Huy Nguyen*, Trang Nguyen*, Trang Pham*, Linh Van Ngo, Nhat Ho.

Hierarchical hybrid sliced Wasserstein: A scalable metric for heterogeneous joint distributions . Under review.
Khai Nguyen, Nhat Ho.

Marginal fairness sliced Wasserstein barycenter . Under review.
Khai Nguyen, Hai Nguyen, Nhat Ho.

Backdoor attack in prompt-based continual learning . Under review.
Trang Nguyen, Anh Tran, Nhat Ho.

CompeteSMoE - Effective training of sparse mixture of experts via competition . Under review.
Quang Pham, Truong Giang Do, Huy Nguyen, TrungTin Nguyen, Chenghao Liu, Mina Sartipi, Binh T. Nguyen, Savitha Ramasamy, Xiaoli Li, Steven Hoi†, Nhat Ho†.

Integrating efficient optimal transport and functional maps for unsupervised shape correspondence learning. Under review.
Tung Le, Khai Nguyen, Shanlin Sun, Nhat Ho, Xiaohui Xie.

Beyond EM algorithm on over-specified two-component location-scale Gaussian mixtures. Under review.
Tongzheng Ren*, Fuheng Cui*, Sujay Sanghavi, Nhat Ho.

Privacy preserving Bayesian federated learning in heterogeneous settings. Under review.
Disha Makhija, Nhat Ho†, Joydeep Ghosh†.

Federated self-supervised learning for heterogeneous clients. Under review.
Disha Makhija, Nhat Ho†, Joydeep Ghosh†.

On label shift in domain adaptation via Wasserstein distance . Under review.
Trung Le, Dat Do, Tuan Nguyen, Huy Nguyen, Hung Bui, Nhat Ho, Dinh Phung.

Improving generative flow networks with path regularization. Under review.
Anh Hoang, Duy Dinh, Tan Nguyen, Stanley Osher†, Nhat Ho†.

Efficient Transformer with compressive sensing . Under review.
Tan Minh Nguyen, Tho Tran Huu, Tam Minh Nguyen, Minh Pham, Stanley Osher†, Nhat Ho†.

Improving multi-task learning via seeking task-based flat regions. Under review.
Hoang Phan, Tung Tran, Ngoc Tran, Nhat Ho, Dinh Phung, Trung Le.

On scalable variant of Wasserstein barycenter . Under review.
Tam Le*, Viet Huynh*, Nhat Ho*, Dinh Phung, Makoto Yamada.

Journal Under Revision

Multivariate smoothing via the Fourier integral theorem and Fourier kernel . Under revision.
Nhat Ho**, Stephen G. Walker**.

An exponentially increasing step-size for parameter estimation in statistical models. Under revision.
Nhat Ho**, Tongzheng Ren**, Purnamrita Sarkar**, Sujay Sanghavi**, Rachel Ward**.

Probabilistic best subset selection via gradient-based optimization. Under revision.
Mingzhang Yin, Nhat Ho, Bowei Yan, Xiaoning Qian, Mingyuan Zhou.

On posterior inference for the number of clusters in Dirichlet process mixture models. Under revision.
Chiao-Yu Yang, Eric Xia, Nhat Ho, Michael I. Jordan.

Sampling for Bayesian mixture models: MCMC with polynomial-time mixing. Under revision.
Wenlong Mou, Nhat Ho, Martin J. Wainwright, Peter L. Bartlett, Michael I. Jordan.

Journal Publications

[J.18] Global optimality of the EM algorithm for mixtures of two linear regression . IEEE Transactions on Information Theory, 2024.
Jeongyeol Kwon, Wei Qian, Constantine Caramanis, Yudong Chen, Damek Davis, Nhat Ho.

[J.17] On the computational and statistical complexity of over-parameterized matrix sensing . Journal of Machine Learning Research (JMLR), 2024.
Jiacheng Zhuo, Jeongyeol Kwon, Nhat Ho, Constantine Caramanis.

[J.16] On integral theorems and their statistical properties . Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2024.
Nhat Ho**, Stephen G. Walker**.

[J.15] Statistical and computational complexities of BFGS quasi-Newton method for generalized linear models. Transactions on Machine Learning Research (TLMR), 2024.
Qiujiang Jin, Tongzheng Ren, Nhat Ho, Aryan Mokhtari.

[J.14] Instability, computational efficiency, and statistical accuracy . Accepted with minor revision at Journal of Machine Learning Research (JMLR), 2023.
Nhat Ho*, Raaz Dwivedi*, Koulik Khamaru*, Martin J. Wainwright, Michael I. Jordan, Bin Yu.

[J.13] A diffusion process perspective on the posterior contraction rates for parameters. SIAM Journal on Mathematics of Data Science (SIMODS), 2023.
Wenlong Mou, Nhat Ho, Martin J. Wainwright, Peter L. Bartlett, Michael I. Jordan.

[J.12] A Bayesian perspective of convolutional neural networks through a deconvolutional generative model. Journal of Machine Learning Research (JMLR), 2023.
Nhat Ho*, Tan Nguyen*, Ankit Patel, Anima Anandkumar, Michael I. Jordan, Richard Baraniuk.

[J.11] Bayesian consistency with the supremum metric . Statistica Sinica, 2022.
Nhat Ho**, Stephen G. Walker**.

[J.10] Convergence rates for Gaussian mixtures of experts. Journal of Machine Learning Research (JMLR), 2022.
Nhat Ho, Chiao-Yu Yang, Michael I. Jordan.

[J.9] On the efficiency of entropic regularized algorithms for optimal transport . Journal of Machine Learning Research (JMLR), 2022.
Tianyi Lin, Nhat Ho, Michael I. Jordan.

[J.8] On the complexity of approximating multi-marginal optimal transport. Journal of Machine Learning Research (JMLR), 2022.
Tianyi Lin*, Nhat Ho*, Marco Cuturi, Michael I. Jordan.

[J.7] On posterior contraction of parameters and interpretability in Bayesian mixture modeling . Bernoulli 27 (4), 2159-2188, 2021.
Aritra Guha, Nhat Ho, XuanLong Nguyen.

[J.6] Singularity, misspecification, and the convergence rate of EM. Annals of Statistics, 48(6), 3161-3182, 2020.
Raaz Dwivedi*, Nhat Ho*, Koulik Khamaru*, Martin J. Wainwright, Michael I. Jordan, Bin Yu.

[J.5] On efficient multilevel clustering via Wasserstein distances. Journal of Machine Learning Research (JMLR), 22, 1-43, 2021.
Viet Huynh*, Nhat Ho*, Nhan Dam, XuanLong Nguyen, Mikhail Yurochkin, Hung Bui, Dinh Phung.

[J.4] Robust estimation of mixing measures in finite mixture models. Bernoulli, 26(2), 828-857, 2020.
Nhat Ho, XuanLong Nguyen, Ya'acov Ritov.

[J.3] Singularity structures and impacts on parameter estimation behavior in finite mixtures of distributions. SIAM Journal on Mathematics of Data Science (SIMODS), 1(4), 730–758, 2019.
Nhat Ho and XuanLong Nguyen.

[J.2] Convergence rates of parameter estimation for some weakly identifiable finite mixtures. Annals of Statistics, 44(6), 2726-2755, 2016.
Nhat Ho and XuanLong Nguyen.

[J.1] On strong identifiability and convergence rates of parameter estimation in finite mixtures. Electronic Journal of Statistics, 10(1), 271-307, 2016.
Nhat Ho and XuanLong Nguyen.

Conference Publications (Machine Learning, Computer Vision, Signal Processing, Time Series Analysis)

[C.67] Is temperature sample efficient for softmax Gaussian mixture of experts? . Proceedings of the ICML, 2024.
Huy Nguyen, Pedram Akbarian, Nhat Ho.

[C.66] On least square estimation in softmax gating mixture of experts . Proceedings of the ICML, 2024.
Huy Nguyen, Nhat Ho†, Alessandro Rinaldo†.

[C.65] Sliced Wasserstein with random-path projecting directions . Proceedings of the ICML, 2024.
Khai Nguyen, Shujian Zhang, Tam Le, Nhat Ho.

[C.64] Neural collapse for cross-entropy class-imbalanced learning with unconstrained ReLU feature model . Proceedings of the ICML, 2024.
Hien Dang, Tho Tran, Tan Nguyen†, Nhat Ho†.

[C.63] Structure-aware E(3)-invariant molecular conformer Aggregation Networks . Proceedings of the ICML, 2024.
Duy Minh Ho Nguyen, Nina Lukashina, Tai Nguyen, An Thai Le, TrungTin Nguyen, Nhat Ho, Jan Peters, Daniel Sonntag, Viktor Zaverkin, Mathias Niepert.

[C.62] A general theory for softmax gating multinomial logistic mixture of experts . Proceedings of the ICML, 2024.
Huy Nguyen, Pedram Akbarian, Tin Nguyen, Nhat Ho.

[C.61] Improving computational complexity in statistical models with second-order information . Proceedings of the ICML, 2024.
Tongzheng Ren, Pedram Akbarian, Jiacheng Zhuo, Sujay Sanghavi, Nhat Ho.

[C.60] Quasi-Monte Carlo for 3D sliced Wasserstein . ICLR, 2024 (Spotlight).
Khai Nguyen, Nicola Bariletto, Nhat Ho.

[C.59] Statistical perspective of top-K sparse softmax gating mixture of experts . ICLR, 2024.
Huy Nguyen, Pedram Akbarian, Fanqi Yan, Nhat Ho.

[C.58] Beyond vanilla variational autoencoders: Detecting posterior collapse in conditional and hierarchical variational autoencoders . ICLR, 2024.
Hien Dang, Tho Tran, Tan Nguyen†, Nhat Ho†.

[C.57] Sliced Wasserstein estimation with control variates . ICLR, 2024.
Khai Nguyen, Nhat Ho.

[C.56] Diffeomorphic mesh deformation via efficient optimal transport for cortical surface reconstruction. ICLR, 2024.
Tung Le, Khai Nguyen, Shanlin Sun, Kun Han, Nhat Ho, Xiaohui Xie.

[C.55] Revisiting deep audio-text retrieval through the lens of transportation. ICLR, 2024.
Manh Luong, Khai Nguyen, Nhat Ho, Reza Haf, Dinh Phung, Lizhen Qu.

[C.54] Towards convergence rates for parameter estimation in Gaussian-gated mixture of experts . AISTATS, 2024.
Huy Nguyen, Tin Nguyen, Khai Nguyen, Nhat Ho.

[C.53] On parameter estimation in deviated Gaussian mixture of experts. AISTATS, 2024.
Huy Nguyen, Khai Nguyen, Nhat Ho.

[C.52] Integrating efficient optimal transport and functional maps For unsupervised shape correspondence learning. Conference on Computer Vision and Pattern Recognition (CVPR), 2024.
Thanh Tung Le, Khai Nguyen, shanlin sun, Nhat Ho, Xiaohui Xie.

[C.51] Fast approximation of the generalized sliced-Wasserstein distance. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2024.
Dung Le*, Huy Nguyen*, Khai Nguyen*, Trang Nguyen*, Nhat Ho.

[C.50] Demystifying softmax gating function in Gaussian mixture of experts . Advances in NeurIPS, 2023 (Spotlight).
Huy Nguyen, Tin Nguyen, Nhat Ho.

[C.49] Energy-based sliced Wasserstein distance . Advances in NeurIPS, 2023.
Khai Nguyen, Nhat Ho.

[C.48] Minimax optimal rate for parameter estimation in multivariate deviated models. Advances in NeurIPS, 2023.
Dat Do*, Huy Nguyen*, Khai Nguyen, Nhat Ho.

[C.47] Markovian sliced Wasserstein distances: Beyond independent projections . Advances in NeurIPS, 2023.
Khai Nguyen, Tongzheng Ren, Nhat Ho.

[C.46] Designing robust transformers using robust kernel density estimation. Advances in NeurIPS, 2023.
Xing Han, Tongzheng Ren, Tan Minh Nguyen, Khai Nguyen, Joydeep Ghosh, Nhat Ho.

[C.45] LVM-Med: Learning large-Scale self-Supervised vision models for medical imaging via second-order graph matching. Advances in NeurIPS, 2023.
Duy Nguyen, Hoang Nguyen, Nghiem Diep, Tan Pham, Tri Cao, Binh T. Nguyen, Paul Swoboda, Nhat Ho, Shadi Albarqouni, Pengtao Xie, Mathias Niepert, Daniel Sonntag.

[C.44] On excess mass behavior in Gaussian mixture models with Orlicz Wasserstein distances . Proceedings of the ICML, 2023.
Aritra Guha, Nhat Ho, Long Nguyen.

[C.43] Neural collapse in deep linear network: from balanced to imbalanced data . Proceedings of the ICML, 2023.
Hien Dang*, Tan Nguyen*, Tho Tran, Stanley Osher, Hung Tran, Nhat Ho†.

[C.42] Self-attention amortized distributional projection optimization for sliced Wasserstein point-clouds reconstruction . Proceedings of the ICML, 2023.
Khai Nguyen*, Dang Nguyen*, Nhat Ho.

[C.41] Revisiting over-smoothing and over-squashing using Ollivier's Ricci curvature . Proceedings of the ICML, 2023.
Khang Nguyen, Tan Minh Nguyen, Nong Minh Hieu, Vinh Duc Nguyen, Nhat Ho†, Stanley Osher†.

[C.40] Hierarchical sliced Wasserstein distance. ICLR, 2023.
Khai Nguyen, Tongzheng Ren, Huy Nguyen, Litu Rout, Tan Nguyen, Nhat Ho.

[C.39] A primal-dual framework for transformers and neural networks. ICLR, 2023 (Spotlight).
Tan Minh Nguyen, Tam Minh Nguyen, Nhat Ho, Andrea L. Bertozzi, Richard Baraniuk, Stanley Osher.

[C.38] Global-local regularization via distributional robustness. AISTATS, 2023.
Hoang Phan, Trung Le, Trung Phung, Tuan Anh Bui, Nhat Ho, Dinh Phung.

[C.37] Model fusion of heterogeneous neural networks via cross-layer alignment . IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2023 (Top 3%).
Dang Nguyen, Trang Nguyen, Khai Nguyen, Dinh Phung, Hung Bui, Nhat Ho.

[C.36] A probabilistic framework for pruning transformers via a finite admixture of keys . IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2023 (Top 3%).
Tam Nguyen*, Tan Nguyen*, Long Bui, Hai Do, Dung Le, Hung Tran-The, Khuong Nguyen, Nhat Ho, Stanley Osher, Richard Baraniuk.

[C.35] Joint self-supervised image-volume representation learning with intra-inter contrastive clustering . AAAI, 2023.
Duy Nguyen, Nguyen Hoang, Truong Mai, Cao Tri, Binh Nguyen, Nhat Ho, Paul Swoboda, Shadi Albarqouni, Pengtao Xie, Daniel Sonntag.

[C.34] Efficient forecasting of large scale hierarchical time series via multilevel clustering . International conference on Time Series and Forecasting (ITISE), 2023.
Xing Han, Tongzheng Ren, Jing Hu, Joydeep Ghosh, Nhat Ho.

[C.33] Revisiting projected Wasserstein metric on images: from vectorization to convolution. Advances in NeurIPS, 2022.
Khai Nguyen, Nhat Ho.

[C.32] FourierFormer: Transformer meets generalized Fourier integral attentions . Advances in NeurIPS, 2022.
Tan Nguyen*, Minh Pham*, Tam Nguyen, Khai Nguyen, Stanley Osher, Nhat Ho.

[C.31] Amortized projection optimization for sliced Wasserstein generative models. Advances in NeurIPS, 2022.
Khai Nguyen, Nhat Ho.

[C.30] Improving Transformer with an admixture of attention heads . Advances in NeurIPS, 2022.
Tam Nguyen*, Tan Nguyen*, Hai Do, Khai Nguyen, Vishwanath Saragadam, Minh Pham, Khuong Nguyen,Stanley Osher†, Nhat Ho†.

[C.29] Beyond black box densities: Parameter learning for the deviated components. Advances in NeurIPS, 2022.
Dat Do*, Nhat Ho*, Long Nguyen.

[C.28] Stochastic multiple target sampling gradient descent . Advances in NeurIPS, 2022.
Hoang Phan, Ngoc Tran, Trung Le, Toan Tran, Nhat Ho, Dinh Phung.

[C.27] Refined convergence rates for maximum likelihood estimation under finite mixture models. Proceedings of the ICML, 2022 (Long Presentation).
Tudor Manole, Nhat Ho.

[C.26] Improving Transformers with probabilistic attention keys . Proceedings of the ICML, 2022.
Tam Nguyen*, Tan Nguyen*, Dung Le, Khuong Nguyen, Anh Tran, Richard Baraniuk, Stanley Osher†, Nhat Ho†.

[C.25] Entropic Gromov-Wasserstein between Gaussian distributions. Proceedings of the ICML, 2022.
Khang Le*, Dung Le*, Huy Nguyen*, Dat Do, Tung Pham, Nhat Ho.

[C.24] Improving minibatch optimal transport via partial transportation. Proceedings of the ICML, 2022.
Khai Nguyen, Dang Nguyen, Anh Le, Tung Pham, Nhat Ho.

[C.23] On transportation of mini-batches: A hierarchical approach . Proceedings of the ICML, 2022.
Khai Nguyen, Dang Nguyen, Quoc Nguyen, Tung Pham, Dinh Phung, Hung Bui, Trung Le, Nhat Ho.

[C.22] Architecture agnostic federated learning for neural networks. Proceedings of the ICML, 2022.
Disha Makhija, Xing Han, Nhat Ho, Joydeep Ghosh.

[C.21] Towards statistical and computational complexities of Polyak step size gradient descent . AISTATS, 2022.
Tongzheng Ren*, Fuheng Cui*, Alexia Atsidakou*, Sujay Sanghavi, Nhat Ho.

[C.20] On multimarginal partial optimal transport: Equivalent forms and computational complexity. AISTATS, 2022.
Khang Le*, Huy Nguyen*, Tung Pham, Nhat Ho.

[C.19] On structured filtering-clustering: Global error bound and optimal first-order algorithms. AISTATS, 2022.
Nhat Ho*, Tianyi Lin*, Michael I. Jordan.

[C.18] Weak separation in mixture models and implications for principal stratification. AISTATS, 2022.
Nhat Ho**, Avi Feller**, Evan Greif**, Luke W. Miratrix**, Natesh S. Pillai**.

[C.17] On robust optimal transport: Computational complexity and barycenter computation . Advances in NeurIPS, 2021.
Khang Le*, Huy Nguyen*, Quang Nguyen, Tung Pham, Hung Bui†, Nhat Ho†.

[C.16] Structured Dropout variational inference for Bayesian neural networks . Advances in NeurIPS, 2021.
Son Nguyen*, Duong Nguyen*, Khai Nguyen, Khoat Than, Hung Bui†, Nhat Ho†.

[C.15] LAMDA: Label matching deep domain adaptation. Proceedings of the ICML, 2021.
Trung Le, Tuan Nguyen, Nhat Ho, Hung Bui, Dinh Phung.

[C.14] Point-set distances for learning representations of 3D point clouds. International Conference on Computer Vision (ICCV), 2021.
Trung Nguyen, Hieu Pham, Tam Le, Tung Pham, Nhat Ho, Son Hua.

[C.13] On the minimax optimality of the EM algorithm for learning two-component mixed linear regression. AISTATS, 2021.
Jeong Y. Kwon, Nhat Ho, Constantine Caramanis.

[C.12] Flow-based alignment approaches for probability measures in different spaces. AISTATS, 2021.
Tam Le*, Nhat Ho*, Makoto Yamada.

[C.11] Distributional sliced-Wasserstein and applications to deep generative modeling. ICLR, 2021 (Spotlight).
Khai Nguyen, Nhat Ho, Tung Pham, Hung Bui.

[C.10] Improving relational regularized autoencoders with spherical sliced fused Gromov Wasserstein. ICLR, 2021.
Khai Nguyen, Son Nguyen, Nhat Ho, Tung Pham, Hung Bui.

[C.9] Projection robust Wasserstein distance and Riemannian optimization . Advances in NeurIPS, 2020 (Spotlight).
Tianyi Lin*, Chenyou Fan*, Nhat Ho, Marco Cuturi, Michael I. Jordan.

[C.8] Fixed-support Wasserstein barycenters: computational hardness and fast algorithm . Advances in NeurIPS, 2020.
Tianyi Lin, Nhat Ho, Xi Chen, Marco Cuturi, Michael I. Jordan.

[C.7] On unbalanced optimal transport: an analysis of Sinkhorn algorithm. Proceedings of the ICML, 2020.
Khiem Pham*, Khang Le*, Nhat Ho, Tung Pham, Hung Bui.

[C.6] Sharp analysis of Expectation-Maximization for weakly identifiable models. AISTATS, 2020.
Raaz Dwivedi*, Nhat Ho*, Koulik Khamaru*, Martin J. Wainwright, Michael I. Jordan, Bin Yu.

[C.5] Fast algorithms for computational optimal transport and Wasserstein barycenter. AISTATS, 2020.
Wenshuo Guo, Nhat Ho, Michael I. Jordan.

[C.4] On efficient optimal transport: an analysis of greedy and accelerated mirror descent algorithms. Proceedings of the ICML, 2019.
Tianyi Lin*, Nhat Ho*, Michael I. Jordan.

[C.3] Probabilistic multilevel clustering via composite transportation distance. AISTATS, 2019.
Nhat Ho*, Viet Huynh*, Dinh Phung, Michael I. Jordan.

[C.2] Theoretical guarantees for EM under misspecified Gaussian mixture models . Advances in NeurIPS, 2018.
Raaz Dwivedi*, Nhat Ho*, Koulik Khamaru*, Martin J. Wainwright, Michael I. Jordan.

[C.1] Multilevel clustering via Wasserstein means. Proceedings of the ICML, 2017.
Nhat Ho, XuanLong Nguyen, Mikhail Yurochkin, Hung Hai Bui, Viet Huynh, and Dinh Phung.

Phd Thesis/ Manuscripts

Parameter Estimation and Multilevel Clustering with Mixture and Hierarchical Models . Phd Thesis, University of Michigan, Ann Arbor. Advisors: Professor Long Nguyen and Professor Ya'acov Ritov.