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Regularization is used in many different areas of optimization when solutions are sought which not only minimize a given function, but also possess a certain degree of regularity. Popular applications are image denoising, sparse regression…

A theoretical and experimental analysis related to the identification of vertices of unknown shapes is presented. Shapes are seen as real functions of their closed boundary. Unlike traditional approaches, which see curvature as the rate of…

The predictive performance of supervised learning algorithms depends on the quality of labels. In a typical label collection process, multiple annotators provide subjective noisy estimates of the "truth" under the influence of their varying…

As a powerful statistical image modeling technique, sparse representation has been successfully used in various image restoration applications. The success of sparse representation owes to the development of l1-norm optimization techniques,…

We provide a general framework for studying recurrent neural networks (RNNs) trained by injecting noise into hidden states. Specifically, we consider RNNs that can be viewed as discretizations of stochastic differential equations driven by…

Spectral clustering is a leading and popular technique in unsupervised data analysis. Two of its major limitations are scalability and generalization of the spectral embedding (i.e., out-of-sample-extension). In this paper we introduce a…

The truncated singular value decomposition may be used to find the solution of linear discrete ill-posed problems in conjunction with Tikhonov regularization and requires the estimation of a regularization parameter that balances between…

Spectral clustering is one of the most popular clustering methods. However, the high computational cost due to the involved eigen-decomposition procedure can immediately hinder its applications in large-scale tasks. In this paper we use…

Representing data residing on a graph as a linear combination of building block signals can enable efficient and insightful visual or statistical analysis of the data, and such representations prove useful as regularizers in signal…

The large capacity of neural networks enables them to learn complex functions. To avoid overfitting, networks however require a lot of training data that can be expensive and time-consuming to collect. A common practical approach to…

We consider the problem of minimizing an objective function that is the sum of a convex function and a group sparsity-inducing regularizer. Problems that integrate such regularizers arise in modern machine learning applications, often for…

Although much of the focus of statistical works on networks has been on static networks, multiple networks are currently becoming more common among network data sets. Usually, a number of network data sets, which share some form of…

Many important problems are characterized by the eigenvalues of a large matrix. For example, the difficulty of many optimization problems, such as those arising from the fitting of large models in statistics and machine learning, can be…

Deep Neural Networks have achieved remarkable success relying on the developing high computation capability of GPUs and large-scale datasets with increasing network depth and width in image recognition, object detection and many other…

Spectral clustering uses the global information embedded in eigenvectors of an inter-item similarity matrix to correctly identify clusters of irregular shape, an ability lacking in commonly used approaches such as k-means and agglomerative…

Large language models and deep neural networks achieve strong performance but suffer from reliability issues and high computational cost. This thesis proposes a unified framework based on spectral geometry and random matrix theory to…

Underpinning the success of deep learning is effective regularizations that allow a variety of priors in data to be modeled. For example, robustness to adversarial perturbations, and correlations between multiple modalities. However, most…

Image classification is a challenging problem for computer in reality. Large numbers of methods can achieve satisfying performances with sufficient labeled images. However, labeled images are still highly limited for certain image…

Detecting and recovering a low-rank signal in a noisy data matrix is a fundamental task in data analysis. Typically, this task is addressed by inspecting and manipulating the spectrum of the observed data, e.g., thresholding the singular…

Various forms of regularization in learning tasks strive for different notions of simplicity. This paper presents a spectral regularization technique, which attaches a unique inductive bias to sequence modeling based on an intuitive concept…