Manifold Based Deep Learning: Advances and Machine Learning Applications

Abstract

Manifolds are topological spaces that are locally Euclidean and find applications in dimensionality reduction, subspace learning, visual domain adaptation, clustering, and more. In this dissertation, we propose a framework for linear dimensionality reduction called the proxy matrix optimization (PMO) that uses the Grassmann manifold for optimizing over orthogonal matrix manifolds. PMO is an iterative and flexible method that finds the lower-dimensional projections for various linear dimensionality reduction methods by changing the objective function. PMO is suitable for Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), Canonical Correlation Analysis (CCA), Maximum Autocorrelation Factors (MAF), and Locality Preserving Projections (LPP). We extend PMO to incorporate robust Lp-norm versions of PCA and LDA, which uses fractional p-norms making them more robust to noisy data and outliers. The PMO method is designed to be realized as a layer in a neural network for maximum benefit. In order to do so, the incremental versions of PCA, LDA, and LPP are included in the PMO framework for problems where the data is not all available at once. Next, we explore the topic of domain shift in visual domain adaptation by combining concepts from spherical manifolds and deep learning. We investigate domain shift, which quantifies how well a model trained on a source domain adapts to a similar target domain with a metric called Spherical Optimal Transport (SpOT). We adopt the spherical manifold along with an orthogonal projection loss to obtain the features from the source and target domains. We then use the optimal transport with the cosine distance between the features as a way to measure the gap between the domains. We show, in our experiments with domain adaptation datasets, that SpOT does better than existing measures for quantifying domain shift and demonstrates a better correlation with the gain of transfer across domains