Dimensionality reduction techniques aim at representing high-dimensional data in low-dimensional spaces to extract hidden and useful information or facilitate visual understanding and interpretation of the data. However, few of them take into consideration the potential cluster information contained implicitly in the high-dimensional data. In this paper, we propose LaptSNE, a new graph-layout nonlinear dimensionality reduction method based on t-SNE, one of the best techniques for visualizing high-dimensional data as 2D scatter plots. Specifically, LaptSNE leverages the eigenvalue information of the graph Laplacian to shrink the potential clusters in the low-dimensional embedding when learning to preserve the local and global structure from high-dimensional space to low-dimensional space. It is nontrivial to solve the proposed model because the eigenvalues of normalized symmetric Laplacian are functions of the decision variable. We provide a majorization-minimization algorithm with convergence guarantee to solve the optimization problem of LaptSNE and show how to calculate the gradient analytically, which may be of broad interest when considering optimization with Laplacian-composited objective. We evaluate our method by a formal comparison with state-of-the-art methods, both visually and via established quantitative measurements. The results demonstrate the superiority of our method over baselines such as t-SNE and UMAP. We also extend our method to spectral clustering and establish an accurate and parameter-free clustering algorithm, which provides us high reliability and convenience in real applications.
相關內容
Mixtures of experts (MoE) models are a popular framework for modeling heterogeneity in data, for both regression and classification problems in statistics and machine learning, due to their flexibility and the abundance of available statistical estimation and model choice tools. Such flexibility comes from allowing the mixture weights (or gating functions) in the MoE model to depend on the explanatory variables, along with the experts (or component densities). This permits the modeling of data arising from more complex data generating processes when compared to the classical finite mixtures and finite mixtures of regression models, whose mixing parameters are independent of the covariates. The use of MoE models in a high-dimensional setting, when the number of explanatory variables can be much larger than the sample size, is challenging from a computational point of view, and in particular from a theoretical point of view, where the literature is still lacking results for dealing with the curse of dimensionality, for both the statistical estimation and feature selection problems. We consider the finite MoE model with soft-max gating functions and Gaussian experts for high-dimensional regression on heterogeneous data, and its $l_1$-regularized estimation via the Lasso. We focus on the Lasso estimation properties rather than its feature selection properties. We provide a lower bound on the regularization parameter of the Lasso function that ensures an $l_1$-oracle inequality satisfied by the Lasso estimator according to the Kullback--Leibler loss.
In this paper, we propose a new covering technique localized for the trajectories of SGD. This localization provides an algorithm-specific complexity measured by the covering number, which can have dimension-independent cardinality in contrast to standard uniform covering arguments that result in exponential dimension dependency. Based on this localized construction, we show that if the objective function is a finite perturbation of a piecewise strongly convex and smooth function with $P$ pieces, i.e. non-convex and non-smooth in general, the generalization error can be upper bounded by $O(\sqrt{(\log n\log(nP))/n})$, where $n$ is the number of data samples. In particular, this rate is independent of dimension and does not require early stopping and decaying step size. Finally, we employ these results in various contexts and derive generalization bounds for multi-index linear models, multi-class support vector machines, and $K$-means clustering for both hard and soft label setups, improving the known state-of-the-art rates.
In recent years, deep learning has been a topic of interest in almost all disciplines due to its impressive empirical success in analyzing complex data sets, such as imaging, genetics, climate, and medical data. While most of the developments are treated as black-box machines, there is an increasing interest in interpretable, reliable, and robust deep learning models applicable to a broad class of applications. Feature-selected deep learning is proven to be promising in this regard. However, the recent developments do not address the situations of ultra-high dimensional and highly correlated feature selection in addition to the high noise level. In this article, we propose a novel screening and cleaning strategy with the aid of deep learning for the cluster-level discovery of highly correlated predictors with a controlled error rate. A thorough empirical evaluation over a wide range of simulated scenarios demonstrates the effectiveness of the proposed method by achieving high power while having a minimal number of false discoveries. Furthermore, we implemented the algorithm in the riboflavin (vitamin $B_2$) production dataset in the context of understanding the possible genetic association with riboflavin production. The gain of the proposed methodology is illustrated by achieving lower prediction error compared to other state-of-the-art methods.
Experienced users often have useful knowledge and intuition in solving real-world optimization problems. User knowledge can be formulated as inter-variable relationships to assist an optimization algorithm in finding good solutions faster. Such inter-variable interactions can also be automatically learned from high-performing solutions discovered at intermediate iterations in an optimization run - a process called innovization. These relations, if vetted by the users, can be enforced among newly generated solutions to steer the optimization algorithm towards practically promising regions in the search space. Challenges arise for large-scale problems where the number of such variable relationships may be high. This paper proposes an interactive knowledge-based evolutionary multi-objective optimization (IK-EMO) framework that extracts hidden variable-wise relationships as knowledge from evolving high-performing solutions, shares them with users to receive feedback, and applies them back to the optimization process to improve its effectiveness. The knowledge extraction process uses a systematic and elegant graph analysis method which scales well with number of variables. The working of the proposed IK-EMO is demonstrated on three large-scale real-world engineering design problems. The simplicity and elegance of the proposed knowledge extraction process and achievement of high-performing solutions quickly indicate the power of the proposed framework. The results presented should motivate further such interaction-based optimization studies for their routine use in practice.
In data science, vector autoregression (VAR) models are popular in modeling multivariate time series in the environmental sciences and other applications. However, these models are computationally complex with the number of parameters scaling quadratically with the number of time series. In this work, we propose a so-called neighborhood vector autoregression (NVAR) model to efficiently analyze large-dimensional multivariate time series. We assume that the time series have underlying neighborhood relationships, e.g., spatial or network, among them based on the inherent setting of the problem. When this neighborhood information is available or can be summarized using a distance matrix, we demonstrate that our proposed NVAR method provides a computationally efficient and theoretically sound estimation of model parameters. The performance of the proposed method is compared with other existing approaches in both simulation studies and a real application of stream nitrogen study.
As a special infinite-order vector autoregressive (VAR) model, the vector autoregressive moving average (VARMA) model can capture much richer temporal patterns than the widely used finite-order VAR model. However, its practicality has long been hindered by its non-identifiability, computational intractability, and relative difficulty of interpretation. This paper introduces a novel infinite-order VAR model which, with only a little sacrifice of generality, inherits the essential temporal patterns of the VARMA model but avoids all of the above drawbacks. As another attractive feature, the temporal and cross-sectional dependence structures of this model can be interpreted separately, since they are characterized by different sets of parameters. For high-dimensional time series, this separation motivates us to impose sparsity on the parameters determining the cross-sectional dependence. As a result, greater statistical efficiency and interpretability can be achieved, while no loss of temporal information is incurred by the imposed sparsity. We introduce an $\ell_1$-regularized estimator for the proposed model and derive the corresponding nonasymptotic error bounds. An efficient block coordinate descent algorithm and a consistent model order selection method are developed. The merit of the proposed approach is supported by simulation studies and a real-world macroeconomic data analysis.
Modern biomedical studies often collect multi-view data, that is, multiple types of data measured on the same set of objects. A popular model in high-dimensional multi-view data analysis is to decompose each view's data matrix into a low-rank common-source matrix generated by latent factors common across all data views, a low-rank distinctive-source matrix corresponding to each view, and an additive noise matrix. We propose a novel decomposition method for this model, called decomposition-based generalized canonical correlation analysis (D-GCCA). The D-GCCA rigorously defines the decomposition on the L2 space of random variables in contrast to the Euclidean dot product space used by most existing methods, thereby being able to provide the estimation consistency for the low-rank matrix recovery. Moreover, to well calibrate common latent factors, we impose a desirable orthogonality constraint on distinctive latent factors. Existing methods, however, inadequately consider such orthogonality and may thus suffer from substantial loss of undetected common-source variation. Our D-GCCA takes one step further than generalized canonical correlation analysis by separating common and distinctive components among canonical variables, while enjoying an appealing interpretation from the perspective of principal component analysis. Furthermore, we propose to use the variable-level proportion of signal variance explained by common or distinctive latent factors for selecting the variables most influenced. Consistent estimators of our D-GCCA method are established with good finite-sample numerical performance, and have closed-form expressions leading to efficient computation especially for large-scale data. The superiority of D-GCCA over state-of-the-art methods is also corroborated in simulations and real-world data examples.
Stochastic kriging has been widely employed for simulation metamodeling to predict the response surface of complex simulation models. However, its use is limited to cases where the design space is low-dimensional because, in general, the sample complexity (i.e., the number of design points required for stochastic kriging to produce an accurate prediction) grows exponentially in the dimensionality of the design space. The large sample size results in both a prohibitive sample cost for running the simulation model and a severe computational challenge due to the need to invert large covariance matrices. Based on tensor Markov kernels and sparse grid experimental designs, we develop a novel methodology that dramatically alleviates the curse of dimensionality. We show that the sample complexity of the proposed methodology grows only slightly in the dimensionality, even under model misspecification. We also develop fast algorithms that compute stochastic kriging in its exact form without any approximation schemes. We demonstrate via extensive numerical experiments that our methodology can handle problems with a design space of more than 10,000 dimensions, improving both prediction accuracy and computational efficiency by orders of magnitude relative to typical alternative methods in practice.
We introduce two new tools to assess the validity of statistical distributions. These tools are based on components derived from a new statistical quantity, the $comparison$ $curve$. The first tool is a graphical representation of these components on a $bar$ $plot$ (B plot), which can provide a detailed appraisal of the validity of the statistical model, in particular when supplemented by acceptance regions related to the model. The knowledge gained from this representation can sometimes suggest an existing $goodness$-$of$-$fit$ test to supplement this visual assessment with a control of the type I error. Otherwise, an adaptive test may be preferable and the second tool is the combination of these components to produce a powerful $\chi^2$-type goodness-of-fit test. Because the number of these components can be large, we introduce a new selection rule to decide, in a data driven fashion, on their proper number to take into consideration. In a simulation, our goodness-of-fit tests are seen to be powerwise competitive with the best solutions that have been recommended in the context of a fully specified model as well as when some parameters must be estimated. Practical examples show how to use these tools to derive principled information about where the model departs from the data.
Despite the success of large-scale empirical risk minimization (ERM) at achieving high accuracy across a variety of machine learning tasks, fair ERM is hindered by the incompatibility of fairness constraints with stochastic optimization. We consider the problem of fair classification with discrete sensitive attributes and potentially large models and data sets, requiring stochastic solvers. Existing in-processing fairness algorithms are either impractical in the large-scale setting because they require large batches of data at each iteration or they are not guaranteed to converge. In this paper, we develop the first stochastic in-processing fairness algorithm with guaranteed convergence. For demographic parity, equalized odds, and equal opportunity notions of fairness, we provide slight variations of our algorithm--called FERMI--and prove that each of these variations converges in stochastic optimization with any batch size. Empirically, we show that FERMI is amenable to stochastic solvers with multiple (non-binary) sensitive attributes and non-binary targets, performing well even with minibatch size as small as one. Extensive experiments show that FERMI achieves the most favorable tradeoffs between fairness violation and test accuracy across all tested setups compared with state-of-the-art baselines for demographic parity, equalized odds, equal opportunity. These benefits are especially significant with small batch sizes and for non-binary classification with large number of sensitive attributes, making FERMI a practical fairness algorithm for large-scale problems.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191