We describe a nonparametric topic model for labeled data. The model uses a mixture of random measures (MRM) as a base distribution of the Dirichlet process (DP) of the HDP framework, so we call it the DP-MRM. To model labeled data, we define a DP distributed random measure for each label, and the resulting model generates an unbounded number of topics for each label. We apply DP-MRM on single-labeled and multi-labeled corpora of documents and compare the performance on label prediction with MedLDA, LDA-SVM, and Labeled-LDA. We further enhance the model by incorporating ddCRP and modeling multi-labeled images for image segmentation and object labeling, comparing the performance with nCuts and rddCRP.

We present local discriminative Gaussian (LDG) dimensionality reduction, a supervised dimensionality reduction technique for classification. The LDG objective function is an approximation to the leave-one-out training error of a local quadratic discriminant analysis classifier, and thus acts locally to each training point in order to find a mapping where similar data can be discriminated from dissimilar data. While other state-of-the-art linear dimensionality reduction methods require gradient descent or iterative solution approaches, LDG is solved with a single eigen-decomposition. Thus, it scales better for datasets with a large number of feature dimensions or training examples. We also adapt LDG to the transfer learning setting, and show that it achieves good performance when the test data distribution differs from that of the training data.

Random projections have been applied in many machine learning algorithms. However, whether margin is preserved after random projection is non-trivial and not well studied. In this paper we analyse margin distortion after random projection, and give the conditions of margin preservation for binary classification problems. We also extend our analysis to margin for multiclass problems, and provide theoretical bounds on multiclass margin on the projected data.

In real supervised learning scenarios, it is not uncommon that the training and test sample follow different probability distributions, thus rendering the necessity to correct the sampling bias. Focusing on a particular covariate shift problem, we derive high probability confidence bounds for the kernel mean matching (KMM) estimator, whose convergence rate turns out to depend on some regularity measure of the regression function and also on some capacity measure of the kernel. By comparing KMM with the natural plug-in estimator, we establish the superiority of the former hence provide concrete evidence/understanding to the effectiveness of KMM under covariate shift.

We present a comprehensive framework for structured sparse coding and modeling extending the recent ideas of using learnable fast regressors to approximate exact sparse codes. For this purpose, we develop a novel block-coordinate proximal splitting method for the iterative solution of hierarchical sparse coding problems, and show an efficient feed forward architecture derived from its iteration. This architecture faithfully approximates the exact structured sparse codes with a fraction of the complexity of the standard optimization methods. We also show that by using different training objective functions, learnable sparse encoders are no longer restricted to be mere approximants of the exact sparse code for a pre-given dictionary, as in earlier formulations, but can be rather used as full-featured sparse encoders or even modelers. A simple implementation shows several orders of magnitude speedup compared to the state-of-the-art at minimal performance degradation, making the proposed framework suitable for real time and large-scale applications.

Convex regression is a promising area for bridging statistical estimation and deterministic convex optimization. New piecewise linear convex regression methods are fast and scalable, but can have instability when used to approximate constraints or objective functions for optimization. Ensemble methods, like bagging, smearing and random partitioning, can alleviate this problem and maintain the theoretical properties of the underlying estimator. We empirically examine the performance of ensemble methods for prediction and optimization, and then apply them to device modeling and constraint approximation for geometric programming based circuit design.

Reconstruction based subspace clustering methods compute a self reconstruction matrix over the samples and use it for spectral clustering to obtain the final clustering result. Their success largely relies on the assumption that the underlying subspaces are independent, which, however, does not always hold in the applications with increasing number of subspaces. In this paper, we propose a novel reconstruction based subspace clustering model without making the subspace independence assumption. In our model, certain properties of the reconstruction matrix are explicitly characterized using the latent cluster indicators, and the affinity matrix used for spectral clustering can be directly built from the posterior of the latent cluster indicators instead of the reconstruction matrix. Experimental results on both synthetic and real-world datasets show that the proposed model can outperform the state-of-the-art methods.

The kernel method is a potential approach to analyzing structured data such as sequences, trees, and graphs; however, unordered trees have not been investigated extensively. Kimura et al. (2011) proposed a kernel function for unordered trees on the basis of their subpaths, which are vertical substructures of trees responsible for hierarchical information in them. Their kernel exhibits practically good performance in terms of accuracy and speed; however, linear-time computation is not guaranteed theoretically, unlike the case of the other unordered tree kernel proposed by Vishwanathan and Smola (2003). In this paper, we propose a theoretically guaranteed linear-time kernel computation algorithm that is practically fast, and we present an efficient prediction algorithm whose running time depends only on the size of the input tree. Experimental results show that the proposed algorithms are quite efficient in practice.

In recent years, total variation (TV) and Euler's elastica (EE) have been successfully applied to image processing tasks such as denoising and inpainting. This paper investigates how to extend TV and EE to the supervised learning settings on high dimensional data. The supervised learning problem can be formulated as an energy functional minimization under Tikhonov regularization scheme, where the energy is composed of a squared loss and a total variation smoothing (or Euler's elastica smoothing). Its solution via variational principles leads to an Euler-Lagrange PDE. However, the PDE is always high-dimensional and cannot be directly solved by common methods. Instead, radial basis functions are utilized to approximate the target function, reducing the problem to finding the linear coefficients of basis functions. We apply the proposed methods to supervised learning tasks (including binary classification, multi-class classification, and regression) on benchmark data sets. Extensive experiments have demonstrated promising results of the proposed methods.

We study the stability vis a vis adversarial noise of matrix factorization algorithm for matrix completion. In particular, our results include: (I) we bound the gap between the solution matrix of the factorization method and the ground truth in terms of root mean square error; (II) we treat the matrix factorization as a subspace fitting problem and analyze the difference between the solution subspace and the ground truth; (III) we analyze the prediction error of individual users based on the subspace stability. We apply these results to the problem of collaborative filtering under manipulator attack, which leads to useful insights and guidelines for collaborative filtering system design.