Model selection is indispensable to high-dimensional sparse modeling in selecting the best set of covariates among a sequence of candidate models. Most existing work assumes implicitly that the model is correctly specified or of fixed dimensions. Yet model misspecification and high dimensionality are common in real applications. In this paper, we investigate two classical Kullback-Leibler divergence and Bayesian principles of model selection in the setting of high-dimensional misspecified models. Asymptotic expansions of these principles reveal that the effect of model misspecification is crucial and should be taken into account, leading to the generalized AIC and generalized BIC in high dimensions. With a natural choice of prior probabilities, we suggest the generalized BIC with prior probability which involves a logarithmic factor of the dimensionality in penalizing model complexity. We further establish the consistency of the covariance contrast matrix estimator in a general setting. Our results and new method are supported by numerical studies.

We consider a non-stationary variant of a sequential stochastic optimization problem, in which the underlying cost functions may change along the horizon. We propose a measure, termed variation budget, that controls the extent of said change, and study how restrictions on this budget impact achievable performance. We identify sharp conditions under which it is possible to achieve long-run-average optimality and more refined performance measures such as rate optimality that fully characterize the complexity of such problems. In doing so, we also establish a strong connection between two rather disparate strands of literature: adversarial online convex optimization; and the more traditional stochastic approximation paradigm (couched in a non-stationary setting). This connection is the key to deriving well performing policies in the latter, by leveraging structure of optimal policies in the former. Finally, tight bounds on the minimax regret allow us to quantify the "price of non-stationarity," which mathematically captures the added complexity embedded in a temporally changing environment versus a stationary one.

Several new estimation methods have been recently proposed for the linear regression model with observation error in the design. Different assumptions on the data generating process have motivated different estimators and analysis. In particular, the literature considered (1) observation errors in the design uniformly bounded by some $\bar \delta$, and (2) zero mean independent observation errors. Under the first assumption, the rates of convergence of the proposed estimators depend explicitly on $\bar \delta$, while the second assumption has been applied when an estimator for the second moment of the observational error is available. This work proposes and studies two new estimators which, compared to other procedures for regression models with errors in the design, exploit an additional $l_{\infty}$-norm regularization. The first estimator is applicable when both (1) and (2) hold but does not require an estimator for the second moment of the observational error. The second estimator is applicable under (2) and requires an estimator for the second moment of the observation error. Importantly, we impose no assumption on the accuracy of this pilot estimator, in contrast to the previously known procedures. As the recent proposals, we allow the number of covariates to be much larger than the sample size. We establish the rates of convergence of the estimators and compare them with the bounds obtained for related estimators in the literature. These comparisons show interesting insights on the interplay of the assumptions and the achievable rates of convergence.

In this on-going work, I explore certain theoretical and empirical implications of data transformations under the PCA. In particular, I state and prove three theorems about PCA, which I paraphrase as follows: 1). PCA without discarding eigenvector rows is injective, but looses this injectivity when eigenvector rows are discarded 2). PCA without discarding eigen- vector rows preserves pair-wise distances, but tends to cause pair-wise distances to shrink when eigenvector rows are discarded. 3). For any pair of points, the shrinkage in pair-wise distance is bounded above by an L1 norm reconstruction error associated with the points. Clearly, 3). suggests that there might exist some correlation between shrinkages in pair-wise distances and mean square reconstruction error which is defined as the sum of those eigenvalues associated with the discarded eigenvectors. I therefore decided to perform numerical experiments to obtain the corre- lation between the sum of those eigenvalues and shrinkages in pair-wise distances. In addition, I have also performed some experiments to check respectively the effect of the sum of those eigenvalues and the effect of the shrinkages on classification accuracies under the PCA map. So far, I have obtained the following results on some publicly available data from the UCI Machine Learning Repository: 1). There seems to be a strong cor- relation between the sum of those eigenvalues associated with discarded eigenvectors and shrinkages in pair-wise distances. 2). Neither the sum of those eigenvalues nor pair-wise distances have any strong correlations with classification accuracies. 1

Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.

This paper considers statistical estimation problems where the probability distribution of the observed random variable is invariant with respect to actions of a finite topological group. It is shown that any such distribution must satisfy a restricted finite mixture representation. When specialized to the case of distributions over the sphere that are invariant to the actions of a finite spherical symmetry group $\mathcal G$, a group-invariant extension of the Von Mises Fisher (VMF) distribution is obtained. The $\mathcal G$-invariant VMF is parameterized by location and scale parameters that specify the distribution's mean orientation and its concentration about the mean, respectively. Using the restricted finite mixture representation these parameters can be estimated using an Expectation Maximization (EM) maximum likelihood (ML) estimation algorithm. This is illustrated for the problem of mean crystal orientation estimation under the spherically symmetric group associated with the crystal form, e.g., cubic or octahedral or hexahedral. Simulations and experiments establish the advantages of the extended VMF EM-ML estimator for data acquired by Electron Backscatter Diffraction (EBSD) microscopy of a polycrystalline Nickel alloy sample.

Principal Component Analysis (PCA) has wide applications in machine learning, text mining and computer vision. Classical PCA based on a Gaussian noise model is fragile to noise of large magnitude. Laplace noise assumption based PCA methods cannot deal with dense noise effectively. In this paper, we propose Cauchy Principal Component Analysis (Cauchy PCA), a very simple yet effective PCA method which is robust to various types of noise. We utilize Cauchy distribution to model noise and derive Cauchy PCA under the maximum likelihood estimation (MLE) framework with low rank constraint. Our method can robustly estimate the low rank matrix regardless of whether noise is large or small, dense or sparse. We analyze the robustness of Cauchy PCA from a robust statistics view and present an efficient singular value projection optimization method. Experimental results on both simulated data and real applications demonstrate the robustness of Cauchy PCA to various noise patterns.

The generalisation properties of a kernel method are entirely controlled by a kernel function, which represents an inner product of arbitrarily many basis functions. Kernel methods typically face a tradeoff between speed and flexibility. Methods which learn a kernel lead to slow and expensive to compute function classes, whereas many fast function classes are not adaptive. This problem is compounded by the fact that expressive kernel learning methods are most needed on large modern datasets, which provide unprecedented opportunities to automatically learn rich statistical representations. For example, the recent spectral kernels proposed by Wilson and Adams [2013] are flexible, but require an arbitrarily large number of basis functions, combined with many free hyperparameters, which can lead to major computational restrictions.

Face verification remains a challenging problem in very complex conditions with large variations such as pose, illumination, expression, and occlusions. This problem is exacerbated when we rely unrealistically on a single training data source, which is often insufficient to cover the intrinsically complex face variations. This paper proposes a principled multi-task learning approach based on Discriminative Gaussian Process Latent Variable Model, named GaussianFace, to enrich the diversity of training data. In comparison to existing methods, our model exploits additional data from multiple source-domains to improve the generalization performance of face verification in an unknown target-domain. Importantly, our model can adapt automatically to complex data distributions, and therefore can well capture complex face variations inherent in multiple sources. Extensive experiments demonstrate the effectiveness of the proposed model in learning from diverse data sources and generalize to unseen domain. Specifically, the accuracy of our algorithm achieves an impressive accuracy rate of 98.52% on the well-known and challenging Labeled Faces in the Wild (LFW) benchmark [23]. For the first time, the human-level performance in face verification (97.53%) [28] on LFW is surpassed.

Machine learning offers novel ways and means to design personalized learning systems wherein each student's educational experience is customized in real time depending on their background, learning goals, and performance to date. SPARse Factor Analysis (SPARFA) is a novel framework for machine learning-based learning analytics, which estimates a learner's knowledge of the concepts underlying a domain, and content analytics, which estimates the relationships among a collection of questions and those concepts. SPARFA jointly learns the associations among the questions and the concepts, learner concept knowledge profiles, and the underlying question difficulties, solely based on the correct/incorrect graded responses of a population of learners to a collection of questions. In this paper, we extend the SPARFA framework significantly to enable: (i) the analysis of graded responses on an ordinal scale (partial credit) rather than a binary scale (correct/incorrect); (ii) the exploitation of tags/labels for questions that partially describe the question-concept associations. The resulting Ordinal SPARFA-Tag framework greatly enhances the interpretability of the estimated concepts. We demonstrate using real educational data that Ordinal SPARFA-Tag outperforms both SPARFA and existing collaborative filtering techniques in predicting missing learner responses.