More Like this

1. Covariance Matrix Estimation under Total Positivity for Portfolio Selection*

   https://doi.org/10.1093/jjfinec/nbaa018  

   Agrawal, Raj ; Roy, Uma ; Uhler, Caroline ( September 2020 , Journal of Financial Econometrics)

   null (Ed.)

   Abstract Selecting the optimal Markowitz portfolio depends on estimating the covariance matrix of the returns of N assets from T periods of historical data. Problematically, N is typically of the same order as T, which makes the sample covariance matrix estimator perform poorly, both empirically and theoretically. While various other general-purpose covariance matrix estimators have been introduced in the financial economics and statistics literature for dealing with the high dimensionality of this problem, we here propose an estimator that exploits the fact that assets are typically positively dependent. This is achieved by imposing that the joint distribution of returns be multivariate totally positive of order 2 (MTP2). This constraint on the covariance matrix not only enforces positive dependence among the assets but also regularizes the covariance matrix, leading to desirable statistical properties such as sparsity. Based on stock market data spanning 30 years, we show that estimating the covariance matrix under MTP2 outperforms previous state-of-the-art methods including shrinkage estimators and factor models. 

   more » « less
2. Fundamental limits for rank-one matrix estimation with groupwise heteroskedasticity

   Behne, J. ( January 2022 , International Conference on Artificial Intelligence and Statistics)

   Low-rank matrix recovery problems involving high-dimensional and heterogeneous data appear in applications throughout statistics and machine learning. The contribution of this paper is to establish the fundamental limits of recovery for a broad class of these problems. In particular, we study the problem of estimating a rank-one matrix from Gaussian observations where different blocks of the matrix are observed under different noise levels. In the setting where the number of blocks is fixed while the number of variables tends to infinity, we prove asymptotically exact formulas for the minimum mean-squared error in estimating both the matrix and underlying factors. These results are based on a novel reduction from the low-rank matrix tensor product model (with homogeneous noise) to a rank-one model with heteroskedastic noise. As an application of our main result, we show that show recently proposed methods based on applying principal component analysis (PCA) to weighted combinations of the data are optimal in some settings but sub-optimal in others. We also provide numerical results comparing our asymptotic formulas with the performance of methods based weighted PCA, gradient descent, and approximate message passing. 

   more » « less
3. Joint Generative Moment-Matching Network for Learning Structural Latent Code

   https://doi.org/10.24963/ijcai.2018/293  

   Gao, Hongchang ; Huang, Heng ( July 2018 , 27th International Joint Conference on Artificial Intelligence (IJCAI 2018))

   Generative Moment-Matching Network (GMMN) is a deep generative model, which employs maximum mean discrepancy as the objective to learn model parameters. However, this model can only generate samples, failing to infer the latent code from samples for downstream tasks. In this paper, we propose a novel Joint Generative Moment-Matching Network (JGMMN), which learns the structural latent code for unsupervised inference. Specifically, JGMMN has a generation network for the generation task and an inference network for the inference task. We first reformulate this model as the two joint distributions matching problem. To solve this problem, we propose to use the Joint Maximum Mean Discrepancy (JMMD) as the objective to learn these two networks simultaneously. Furthermore, to enforce the consistency between the sample distribution and the inferred latent code distribution, we propose a novel multi-modal regularization to enforce this consistency. At last, extensive experiments on both synthetic and real-world datasets have verified the effectiveness and correctness of our proposed JGMMN.

    

   more » « less
4. Phase retrieval of low-rank matrices by anchored regression

   https://doi.org/10.1093/imaiai/iaaa018  

   Lee, Kiryung ; Bahmani, Sohail ; Eldar, Yonina C ; Romberg, Justin ( September 2020 , Information and Inference: A Journal of the IMA)

   Abstract We study the low-rank phase retrieval problem, where our goal is to recover a $d_1\times d_2$ low-rank matrix from a series of phaseless linear measurements. This is a fourth-order inverse problem, as we are trying to recover factors of a matrix that have been observed, indirectly, through some quadratic measurements. We propose a solution to this problem using the recently introduced technique of anchored regression. This approach uses two different types of convex relaxations: we replace the quadratic equality constraints for the phaseless measurements by a search over a polytope and enforce the rank constraint through nuclear norm regularization. The result is a convex program in the space of $d_1 \times d_2$ matrices. We analyze two specific scenarios. In the first, the target matrix is rank-$1$, and the observations are structured to correspond to a phaseless blind deconvolution. In the second, the target matrix has general rank, and we observe the magnitudes of the inner products against a series of independent Gaussian random matrices. In each of these problems, we show that anchored regression returns an accurate estimate from a near-optimal number of measurements given that we have access to an anchor matrix of sufficient quality. We also show how to create such an anchor in the phaseless blind deconvolution problem from an optimal number of measurements and present a partial result in this direction for the general rank problem. 

   more » « less
5. Distributionally Robust Structure Learning for Discrete Pairwise Markov Networks

   Li, Yeshu ; Shi, Zhan ; Zhang, Xinhua ; Ziebart, Brian ( March 2022 , International Conference on Artificial Intelligence and Statistics)

   We consider the problem of learning the underlying structure of a general discrete pairwise Markov network. Existing approaches that rely on empirical risk minimization may perform poorly in settings with noisy or scarce data. To overcome these limitations, we propose a computationally efficient and robust learning method for this problem with near-optimal sample complexities. Our approach builds upon distributionally robust optimization (DRO) and maximum conditional log-likelihood. The proposed DRO estimator minimizes the worst-case risk over an ambiguity set of adversarial distributions within bounded transport cost or f-divergence of the empirical data distribution. We show that the primal minimax learning problem can be efficiently solved by leveraging sufficient statistics and greedy maximization in the ostensibly intractable dual formulation. Based on DRO’s approximation to Lipschitz and variance regularization, we derive near-optimal sample complexities matching existing results. Extensive empirical evidence with different corruption models corroborates the effectiveness of the proposed methods. 

   more » « less