arXiv subjects

Ziwei Zhu

Publications and source records attributed to Ziwei Zhu.

At least 73 records · Page 4Linked to original sources

arXiv2022

Rank-Constrained Least-Squares: Prediction and Inference

In this work, we focus on the high-dimensional trace regression model with a low-rank coefficient matrix. We establish a nearly optimal in-sample prediction risk bound for the rank-constrained least-squares estimator under no assumptions on the design matrix. Lying at the heart of the proof is a covering number bound for the family of projection operators corresponding to the subspaces spanned by the design. By leveraging this complexity result, we perform a power analysis for a permutation test on the existence of a low-rank signal under the high-dimensional trace regression model. We show that the permutation test based on the rank-constrained least-squares estimator achieves non-trivial power with no assumptions on the minimum (restricted) eigenvalue of the covariance matrix of the design. Finally, we use alternating minimization to approximately solve the rank-constrained least-squares problem to evaluate its empirical in-sample prediction risk and power of the resulting permutation test in our numerical study.

math.ST↗

arXiv2022

VarRCWA: An Adaptive High-Order Rigorous Coupled Wave Analysis Method

Semi-analytical methods, such as rigorous coupled wave analysis, have been pivotal for numerical analysis of photonic structures. In comparison to other methods, they offer much faster computation, especially for structures with constant cross-sectional shapes (such as metasurface units). However, when the cross-sectional shape varies even mildly (such as a taper), existing semi-analytical methods suffer from high computational cost. We show that the existing methods can be viewed as a zeroth-order approximation with respect to the structure's cross-sectional variation. We instead derive a high-order perturbative expansion with respect to the cross-sectional variation. Based on this expansion, we propose a new semi-analytical method that is fast to compute even in presence of large cross-sectional shape variation. Furthermore, we design an algorithm that automatically discretizes the structure in a way that achieves a user specified accuracy level while at the same time reducing the computational cost.

math.NA↗

arXiv2022

Supervised Homogeneity Fusion: a Combinatorial Approach

Fusing regression coefficients into homogenous groups can unveil those coefficients that share a common value within each group. Such groupwise homogeneity reduces the intrinsic dimension of the parameter space and unleashes sharper statistical accuracy. We propose and investigate a new combinatorial grouping approach called $L_0$-Fusion that is amenable to mixed integer optimization (MIO). On the statistical aspect, we identify a fundamental quantity called grouping sensitivity that underpins the difficulty of recovering the true groups. We show that $L_0$-Fusion achieves grouping consistency under the weakest possible requirement of the grouping sensitivity: if this requirement is violated, then the minimax risk of group misspecification will fail to converge to zero. Moreover, we show that in the high-dimensional regime, one can apply $L_0$-Fusion coupled with a sure screening set of features without any essential loss of statistical efficiency, while reducing the computational cost substantially. On the algorithmic aspect, we provide a MIO formulation for $L_0$-Fusion along with a warm start strategy. Simulation and real data analysis demonstrate that $L_0$-Fusion exhibits superiority over its competitors in terms of grouping accuracy.

stat.ML↗

arXiv2021

Session-based Recommendation with Hypergraph Attention Networks

Session-based recommender systems aim to improve recommendations in short-term sessions that can be found across many platforms. A critical challenge is to accurately model user intent with only limited evidence in these short sessions. For example, is a flower bouquet being viewed meant as part of a wedding purchase or for home decoration? Such different perspectives greatly impact what should be recommended next. Hence, this paper proposes a novel session-based recommendation system empowered by hypergraph attention networks. Three unique properties of the proposed approach are: (i) it constructs a hypergraph for each session to model the item correlations defined by various contextual windows in the session simultaneously, to uncover item meanings; (ii) it is equipped with hypergraph attention layers to generate item embeddings by flexibly aggregating the contextual information from correlated items in the session; and (iii) it aggregates the dynamic item representations for each session to infer the general purpose and current need, which is decoded to infer the next interesting item in the session. Through experiments on three benchmark datasets, we find the proposed model is effective in generating informative dynamic item embeddings and providing more accurate recommendations compared to the state-of-the-art.

cs.IR↗

arXiv2021

High-dimensional inference via hybrid orthogonalization

The past decade has witnessed a surge of endeavors in statistical inference for high-dimensional sparse regression, particularly via de-biasing or relaxed orthogonalization. Nevertheless, these techniques typically require a more stringent sparsity condition than needed for estimation consistency, which seriously limits their practical applicability. To alleviate such constraint, we propose to exploit the identifiable features to residualize the design matrix before performing debiasing-based inference over the parameters of interest. This leads to a hybrid orthogonalization (HOT) technique that performs strict orthogonalization against the identifiable features but relaxed orthogonalization against the others. Under an approximately sparse model with a mixture of identifiable and unidentifiable signals, we establish the asymptotic normality of the HOT test statistic while accommodating as many identifiable signals as consistent estimation allows. The efficacy of the proposed test is also demonstrated through simulation and analysis of a stock market dataset.

math.ST↗

arXiv2021

End-to-end Learning for Fair Ranking Systems

The learning-to-rank problem aims at ranking items to maximize exposure of those most relevant to a user query. A desirable property of such ranking systems is to guarantee some notion of fairness among specified item groups. While fairness has recently been considered in the context of learning-to-rank systems, current methods cannot provide guarantees on the fairness of the proposed ranking policies. This paper addresses this gap and introduces Smart Predict and Optimize for Fair Ranking (SPOFR), an integrated optimization and learning framework for fairness-constrained learning to rank. The end-to-end SPOFR framework includes a constrained optimization sub-model and produces ranking policies that are guaranteed to satisfy fairness constraints while allowing for fine control of the fairness-utility tradeoff. SPOFR is shown to significantly improve current state-of-the-art fair learning-to-rank systems with respect to established performance metrics.

cs.LG↗

arXiv2021

Volatility prediction comparison via robust volatility proxies: An empirical deviation perspective

Volatility forecasting is crucial to risk management and portfolio construction. One particular challenge of assessing volatility forecasts is how to construct a robust proxy for the unknown true volatility. In this work, we show that the empirical loss comparison between two volatility predictors hinges on the deviation of the volatility proxy from the true volatility. We then establish non-asymptotic deviation bounds for three robust volatility proxies, two of which are based on clipped data, and the third of which is based on exponentially weighted Huber loss minimization. In particular, in order for the Huber approach to adapt to non-stationary financial returns, we propose to solve a tuning-free weighted Huber loss minimization problem to jointly estimate the volatility and the optimal robustification parameter at each time point. We then inflate this robustification parameter and use it to update the volatility proxy to achieve optimal balance between the bias and variance of the global empirical loss. We also extend this Huber method to construct volatility predictors. Finally, we exploit the proposed robust volatility proxy to compare different volatility predictors on the Bitcoin market data. It turns out that when the sample size is limited, applying the robust volatility proxy gives more consistent and stable evaluation of volatility forecasts.

math.ST↗

arXiv2021

Best subset selection is robust against design dependence

Best subset selection (BSS) is widely known as the holy grail for high-dimensional variable selection. Nevertheless, the notorious NP-hardness of BSS substantially restricts its practical application and also discourages its theoretical development to some extent, particularly in the current era of big data. In this paper, we investigate the variable selection properties of BSS when its target sparsity is greater than or equal to the true sparsity. Our main message is that BSS is robust against design dependence in terms of achieving model consistency and sure screening, and more importantly, that such robustness can be propagated to the near best subsets that are computationally tangible. Specifically, we introduce an identifiability margin condition that is free of restricted eigenvalues and show that it is sufficient and nearly necessary for BSS to exactly recover the true model. A relaxed version of this condition is also sufficient for BSS to achieve the sure screening property. Moreover, taking optimization error into account, we find that all the established statistical properties for the exact best subset carry over to any near best subset whose residual sum of squares is close enough to that of the best one. In particular, a two-stage fully corrective iterative hard thresholding (IHT) algorithm can provably find a sparse sure screening subset within logarithmic steps; another round of exact BSS within this set can recover the true model. The simulation studies and real data examples show that IHT yields lower false discovery rates and higher true positive rates than the competing approaches including LASSO, SCAD and Sure Independence Screening (SIS), especially under highly correlated design.

stat.ME↗

arXiv2021

Fairness-aware Personalized Ranking Recommendation via Adversarial Learning

Recommendation algorithms typically build models based on historical user-item interactions (e.g., clicks, likes, or ratings) to provide a personalized ranked list of items. These interactions are often distributed unevenly over different groups of items due to varying user preferences. However, we show that recommendation algorithms can inherit or even amplify this imbalanced distribution, leading to unfair recommendations to item groups. Concretely, we formalize the concepts of ranking-based statistical parity and equal opportunity as two measures of fairness in personalized ranking recommendation for item groups. Then, we empirically show that one of the most widely adopted algorithms -- Bayesian Personalized Ranking -- produces unfair recommendations, which motivates our effort to propose the novel fairness-aware personalized ranking model. The debiased model is able to improve the two proposed fairness metrics while preserving recommendation performance. Experiments on three public datasets show strong fairness improvement of the proposed model versus state-of-the-art alternatives. This is paper is an extended and reorganized version of our SIGIR 2020~\cite{zhu2020measuring} paper. In this paper, we re-frame the studied problem as `item recommendation fairness' in personalized ranking recommendation systems, and provide more details about the training process of the proposed model and details of experiment setup.

cs.IR↗

arXiv2020

Learning Markov models via low-rank optimization

Modeling unknown systems from data is a precursor of system optimization and sequential decision making. In this paper, we focus on learning a Markov model from a single trajectory of states. Suppose that the transition model has a small rank despite of having a large state space, meaning that the system admits a low-dimensional latent structure. We show that one can estimate the full transition model accurately using a trajectory of length that is proportional to the total number of states. We propose two maximum likelihood estimation methods: a convex approach with nuclear-norm regularization and a nonconvex approach with rank constraint. We explicitly derive the statistical rates of both estimators in terms of the Kullback-Leiber divergence and the $\ell_2$ error and also establish a minimax lower bound to assess the tightness of these rates. For computing the nonconvex estimator, we develop a novel DC (difference of convex function) programming algorithm that starts with the convex M-estimator and then successively refines the solution till convergence. Empirical experiments demonstrate consistent superiority of the nonconvex estimator over the convex one.

stat.ME↗

arXiv2020

Infusing Disease Knowledge into BERT for Health Question Answering, Medical Inference and Disease Name Recognition

Knowledge of a disease includes information of various aspects of the disease, such as signs and symptoms, diagnosis and treatment. This disease knowledge is critical for many health-related and biomedical tasks, including consumer health question answering, medical language inference and disease name recognition. While pre-trained language models like BERT have shown success in capturing syntactic, semantic, and world knowledge from text, we find they can be further complemented by specific information like knowledge of symptoms, diagnoses, treatments, and other disease aspects. Hence, we integrate BERT with disease knowledge for improving these important tasks. Specifically, we propose a new disease knowledge infusion training procedure and evaluate it on a suite of BERT models including BERT, BioBERT, SciBERT, ClinicalBERT, BlueBERT, and ALBERT. Experiments over the three tasks show that these models can be enhanced in nearly all cases, demonstrating the viability of disease knowledge infusion. For example, accuracy of BioBERT on consumer health question answering is improved from 68.29% to 72.09%, while new SOTA results are observed in two datasets. We make our data and code freely available.

cs.CL↗

arXiv2020

Differentiable Scattering Matrix for Optimization of Photonic Structures

The scattering matrix, which quantifies the optical reflection and transmission of a photonic structure, is pivotal for understanding the performance of the structure. In many photonic design tasks, it is also desired to know how the structure's optical performance changes with respect to design parameters, that is, the scattering matrix's derivatives (or gradient). Here we address this need. We present a new algorithm for computing scattering matrix derivatives accurately and robustly. In particular, we focus on the computation in semi-analytical methods (such as rigorous coupled-wave analysis). To compute the scattering matrix of a structure, these methods must solve an eigen-decomposition problem. However, when it comes to computing scattering matrix derivatives, differentiating the eigen-decomposition poses significant numerical difficulties. We show that the differentiation of the eigen-decomposition problem can be completely sidestepped, and thereby propose a robust algorithm. To demonstrate its efficacy, we use our algorithm to optimize metasurface structures and reach various optical design goals.

physics.optics↗

arXiv2020

Taming heavy-tailed features by shrinkage

In this work, we focus on a variant of the generalized linear model (GLM) called corrupted GLM (CGLM) with heavy-tailed features and responses. To robustify the statistical inference on this model, we propose to apply $\ell_4$-norm shrinkage to the feature vectors in the low-dimensional regime and apply elementwise shrinkage to them in the high-dimensional regime. Under bounded fourth moment assumptions, we show that the maximum likelihood estimator (MLE) based on the shrunk data enjoys nearly the minimax optimal rate with an exponential deviation bound. Our simulations demonstrate that the proposed feature shrinkage significantly enhances the statistical performance in linear regression and logistic regression on heavy-tailed data. Finally, we apply our shrinkage principle to guard against mislabeling and image noise in the human-written digit recognition problem. We add an $\ell_4$-norm shrinkage layer to the original neural net and reduce the testing misclassification rate by more than $30\%$ relatively in the presence of mislabeling and image noise.

stat.ME↗

arXiv2019

High-dimensional principal component analysis with heterogeneous missingness

We study the problem of high-dimensional Principal Component Analysis (PCA) with missing observations. In simple, homogeneous missingness settings with a noise level of constant order, we show that an existing inverse-probability weighted (IPW) estimator of the leading principal components can (nearly) attain the minimax optimal rate of convergence. However, deeper investigation reveals both that, particularly in more realistic settings where the missingness mechanism is heterogeneous, the empirical performance of the IPW estimator can be unsatisfactory, and moreover that, in the noiseless case, it fails to provide exact recovery of the principal components. Our main contribution, then, is to introduce a new method for high-dimensional PCA, called `primePCA', that is designed to cope with situations where observations may be missing in a heterogeneous manner. Starting from the IPW estimator, primePCA iteratively projects the observed entries of the data matrix onto the column space of our current estimate to impute the missing entries, and then updates our estimate by computing the leading right singular space of the imputed data matrix. It turns out that the interaction between the heterogeneity of missingness and the low-dimensional structure is crucial in determining the feasibility of the problem. We therefore introduce an incoherence condition on the principal components and prove that in the noiseless case, the error of primePCA converges to zero at a geometric rate when the signal strength is not too small. An important feature of our theoretical guarantees is that they depend on average, as opposed to worst-case, properties of the missingness mechanism. Our numerical studies on both simulated and real data reveal that primePCA exhibits very encouraging performance across a wide range of scenarios.

stat.ME↗

arXiv2019

Pseudo-Implicit Feedback for Alleviating Data Sparsity in Top-K Recommendation

We propose PsiRec, a novel user preference propagation recommender that incorporates pseudo-implicit feedback for enriching the original sparse implicit feedback dataset. Three of the unique characteristics of PsiRec are: (i) it views user-item interactions as a bipartite graph and models pseudo-implicit feedback from this perspective; (ii) its random walks-based approach extracts graph structure information from this bipartite graph, toward estimating pseudo-implicit feedback; and (iii) it adopts a Skip-gram inspired measure of confidence in pseudo-implicit feedback that captures the pointwise mutual information between users and items. This pseudo-implicit feedback is ultimately incorporated into a new latent factor model to estimate user preference in cases of extreme sparsity. PsiRec results in improvements of 21.5% and 22.7% in terms of Precision@10 and Recall@10 over state-of-the-art Collaborative Denoising Auto-Encoders. Our implementation is available at https://github.com/heyunh2015/PsiRecICDM2018.

cs.IR↗

arXiv2018

Fairness-Aware Recommendation of Information Curators

This paper highlights our ongoing efforts to create effective information curator recommendation models that can be personalized for individual users, while maintaining important fairness properties. Concretely, we introduce the problem of information curator recommendation, provide a high-level overview of a fairness-aware recommender, and introduce some preliminary experimental evidence over a real-world Twitter dataset. We conclude with some thoughts on future directions.

cs.IR↗

arXiv2018

Robust high dimensional factor models with applications to statistical machine learning

Factor models are a class of powerful statistical models that have been widely used to deal with dependent measurements that arise frequently from various applications from genomics and neuroscience to economics and finance. As data are collected at an ever-growing scale, statistical machine learning faces some new challenges: high dimensionality, strong dependence among observed variables, heavy-tailed variables and heterogeneity. High-dimensional robust factor analysis serves as a powerful toolkit to conquer these challenges. This paper gives a selective overview on recent advance on high-dimensional factor models and their applications to statistics including Factor-Adjusted Robust Model selection (FarmSelect) and Factor-Adjusted Robust Multiple testing (FarmTest). We show that classical methods, especially principal component analysis (PCA), can be tailored to many new problems and provide powerful tools for statistical estimation and inference. We highlight PCA and its connections to matrix perturbation theory, robust statistics, random projection, false discovery rate, etc., and illustrate through several applications how insights from these fields yield solutions to modern challenges. We also present far-reaching connections between factor models and popular statistical learning problems, including network analysis and low-rank matrix recovery.

stat.ME↗

arXiv2018

Distributed Estimation of Principal Eigenspaces

Principal component analysis (PCA) is fundamental to statistical machine learning. It extracts latent principal factors that contribute to the most variation of the data. When data are stored across multiple machines, however, communication cost can prohibit the computation of PCA in a central location and distributed algorithms for PCA are thus needed. This paper proposes and studies a distributed PCA algorithm: each node machine computes the top $K$ eigenvectors and transmits them to the central server; the central server then aggregates the information from all the node machines and conducts a PCA based on the aggregated information. We investigate the bias and variance for the resulting distributed estimator of the top $K$ eigenvectors. In particular, we show that for distributions with symmetric innovation, the empirical top eigenspaces are unbiased and hence the distributed PCA is "unbiased". We derive the rate of convergence for distributed PCA estimators, which depends explicitly on the effective rank of covariance, eigen-gap, and the number of machines. We show that when the number of machines is not unreasonably large, the distributed PCA performs as well as the whole sample PCA, even without full access of whole data. The theoretical results are verified by an extensive simulation study. We also extend our analysis to the heterogeneous case where the population covariance matrices are different across local machines but share similar top eigen-structures.

stat.CO↗