Learning Models For Corrupted Multi-Dimensional Data: Fundamental Limits And Algorithms

Developing machine learning models for unstructured multi-dimensional datasets such as datasets with unreliable labels and noisy multi-dimensional signals with or without missing information have becoming a central necessity. We are not always fortunate enough to get noise-free datasets for developing classification and representation models. Though there is a number of techniques available to deal with noisy datasets, these methods do not exploit the multi-dimensional structures of the signals, which could be used to improve the overall classification and representation performance of the model. In this thesis, we develop a Kronecker-structure (K-S) subspace model that exploits the multi-dimensional structure of the signal. First, we study the classification performance of K-S subspace models in two asymptotic regimes when the signal dimensions go to infinity and when the noise power tends to zero. We characterize the misclassification probability in terms of diversity order and we drive an exact expression for the diversity order. We further derive a tighter bound on misclassification probability in terms of pairwise geometry of the subspaces. The proposed scheme is optimal in most of the signal dimension regimes except in one regime where the signal dimension is less than twice the subspace dimension, however, hitting such a signal dimension regime is very rare in practice. We empirically show that the classification performance of K-S subspace models agrees with the diversity order analysis. We also develop an algorithm, Kronecker- Structured Learning of Discriminative Dictionaries (K-SLD2), for fast and compact K-S subspace learning for better classification and representation of multidimensional signals. We show that the K-SLD2 algorithm balances compact signal representation and good classification performance on synthetic and real-world datasets. Next, we develop a scheme to detect whether a given multi-dimensional signal with missing information lies on a given K-S subspace. We find that under some mild incoherence conditions we must observe ��(��1 log ��1) number of rows and ��(��2 log ��2) number of columns in order to detect the K-S subspace. In order to account for unreliable labels in datasets we present Nonlinear, Noise- aware, Quasiclustering (NNAQC), a method for learning deep convolutional networks from datasets corrupted by unknown label noise. We append a nonlinear noise model to a standard convolutional network, which is learned in tandem with the parameters of the network. Further, we train the network using a loss function that encourages the clustering of training images. We argue that the non-linear noise model, while not rigorous as a probabilistic model, results in a more effective denoising operator during backpropagation. We evaluate the performance of NNAQC on artificially injected label noise to MNIST, CIFAR-10, CIFAR-100, and ImageNet datasets and on a large-scale Clothing1M dataset with inherent label noise. We show that on all these datasets, NNAQC provides significantly improved classification performance over the state of the art and is robust to the amount of label noise and the training samples

Regularized and Smooth Double Core Tensor Factorization for Heterogeneous Data

We introduce a general tensor model suitable for data analytic tasks for heterogeneous data sets, wherein there are joint low-rank structures within groups of observations, but also discriminative structures across different groups. To capture such complex structures, a double core tensor (DCOT) factorization model is introduced together with a family of smoothing loss functions. By leveraging the proposed smoothing function, the model accurately estimates the model factors, even in the presence of missing entries. A linearized ADMM method is employed to solve regularized versions of DCOT factorizations, that avoid large tensor operations and large memory storage requirements. Further, we establish theoretically its global convergence, together with consistency of the estimates of the model parameters. The effectiveness of the DCOT model is illustrated on several real-world examples including image completion, recommender systems, subspace clustering and detecting modules in heterogeneous Omics multi-modal data, since it provides more insightful decompositions than conventional tensor methods

Robust Kronecker component analysis

Abstract Dictionary learning and component analysis models are fundamental for learning compact representations that are relevant to a given task (feature extraction, dimensionality reduction, denoising, etc.). The model complexity is encoded by means of specific structure, such as sparsity, low-rankness, or nonnegativity. Unfortunately, approaches like K-SVD — that learn dictionaries for sparse coding via Singular Value Decomposition (SVD) — are hard to scale to high-volume and high-dimensional visual data, and fragile in the presence of outliers. Conversely, robust component analysis methods such as the Robust Principal Component Analysis (RPCA) are able to recover low-complexity (e.g., low-rank) representations from data corrupted with noise of unknown magnitude and support, but do not provide a dictionary that respects the structure of the data (e.g., images), and also involve expensive computations. In this paper, we propose a novel Kronecker-decomposable component analysis model, coined as Robust Kronecker Component Analysis (RKCA), that combines ideas from sparse dictionary learning and robust component analysis. RKCA has several appealing properties, including robustness to gross corruption; it can be used for low-rank modeling, and leverages separability to solve significantly smaller problems. We design an efficient learning algorithm by drawing links with a restricted form of tensor factorization, and analyze its optimality and low-rankness properties. The effectiveness of the proposed approach is demonstrated on real-world applications, namely background subtraction and image denoising and completion, by performing a thorough comparison with the current state of the art

Robust Kronecker component analysis

Dictionary learning and component analysis models are fundamental for learning compact representations relevant to a given task. The model complexity is encoded by means of structure, such as sparsity, low-rankness, or nonnegativity. Unfortunately, approaches like K-SVD that learn dictionaries for sparse coding via Singular Value Decomposition (SVD) are hard to scale, and fragile in the presence of outliers. Conversely, robust component analysis methods such as the Robust Principal Component Analysis (RPCA) are able to recover low-complexity representations from data corrupted with noise of unknown magnitude and support, but do not provide a dictionary that respects the structure of the data, and also involve expensive computations. In this paper, we propose a novel Kronecker-decomposable component analysis model, coined as Robust Kronecker Component Analysis (RKCA), that combines ideas from sparse dictionary learning and robust component analysis. RKCA has several appealing properties, including robustness to gross corruption; it can be used for low-rank modeling, and leverages separability to solve significantly smaller problems. We design an efficient learning algorithm by drawing links with tensor factorizations, and analyze its optimality and low-rankness properties. The effectiveness of the proposed approach is demonstrated on real-world applications, namely background subtraction and image denoising and completion, by performing a thorough comparison with the current state of the art