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Machine learning is at the heart of managing the real-world problems associated with massive data. With the success of neural networks on such large-scale problems, more research in machine learning is being conducted now than ever before.…
Function approximation based on data drawn randomly from an unknown distribution is an important problem in machine learning. The manifold hypothesis assumes that the data is sampled from an unknown submanifold of a high dimensional…
Manifold learning is a popular and quickly-growing subfield of machine learning based on the assumption that one's observed data lie on a low-dimensional manifold embedded in a higher-dimensional space. This thesis presents a mathematical…
The knowledge that data lies close to a particular submanifold of the ambient Euclidean space may be useful in a number of ways. For instance, one may want to automatically mark any point far away from the submanifold as an outlier or to…
Learning mappings of data on manifolds is an important topic in contemporary machine learning, with applications in astrophysics, geophysics, statistical physics, medical diagnosis, biochemistry, 3D object analysis. This paper studies the…
Meta-Learning is a subarea of Machine Learning that aims to take advantage of prior knowledge to learn faster and with fewer data [1]. There are different scenarios where meta-learning can be applied, and one of the most common is algorithm…
A central problem in machine learning is often formulated as follows: Given a dataset $\{(x_j, y_j)\}_{j=1}^M$, which is a sample drawn from an unknown probability distribution, the goal is to construct a functional model $f$ such that…
Recent literature has shown that symbolic data, such as text and graphs, is often better represented by points on a curved manifold, rather than in Euclidean space. However, geometrical operations on manifolds are generally more complicated…
Meta-learning, or learning-to-learn, seeks to design algorithms that can utilize previous experience to rapidly learn new skills or adapt to new environments. Representation learning -- a key tool for performing meta-learning -- learns a…
The low-dimensional manifold hypothesis posits that the data found in many applications, such as those involving natural images, lie (approximately) on low-dimensional manifolds embedded in a high-dimensional Euclidean space. In this…
Classification is a core topic in functional data analysis. A large number of functional classifiers have been proposed in the literature, most of which are based on functional principal component analysis or functional regression. In…
Meta-learning, or "learning to learn," is a subfield of machine learning where the goal is to develop models and algorithms that can learn from various tasks and improve their learning process over time. Unlike traditional machine learning…
The problem of extending a function $f$ defined on a training data $\mathcal{C}$ on an unknown manifold $\mathbb{X}$ to the entire manifold and a tubular neighborhood of this manifold is considered in this paper. For $\mathbb{X}$ embedded…
In much of the literature on function approximation by deep networks, the function is assumed to be defined on some known domain, such as a cube or a sphere. In practice, the data might not be dense on these domains, and therefore, the…
As the application of deep learning has expanded to real-world problems with insufficient volume of training data, transfer learning recently has gained much attention as means of improving the performance in such small-data regime.…
Deep learning is the mainstream technique for many machine learning tasks, including image recognition, machine translation, speech recognition, and so on. It has outperformed conventional methods in various fields and achieved great…
Transfer learning involves taking information and insight from one problem domain and applying it to a new problem domain. Although widely used in practice, theory for transfer learning remains less well-developed. To address this, we prove…
The availability of abundant labeled data in recent years led the researchers to introduce a methodology called transfer learning, which utilizes existing data in situations where there are difficulties in collecting new annotated data.…
Autoencoders, which consist of an encoder and a decoder, are widely used in machine learning for dimension reduction of high-dimensional data. The encoder embeds the input data manifold into a lower-dimensional latent space, while the…
Training a classifier with high mean accuracy from a manifold-distributed dataset can be challenging. This problem is compounded further when there are only few labels available for training. For transfer learning to work, both the source…