Over the past decade, machine learning research has made great strides and significant impact in several fields. Its success is greatly attributed to the development of effective machine learning algorithms like deep neural networks (a.k.a. deep learning), availability of large-scale databases and access to specialized hardware like Graphic Processing Units. When designing and training machine learning systems, researchers often assume access to large quantities of data that capture different possible variations. Variations in the data is needed to incorporate desired invariance and robustness properties in the machine learning system, especially in the case of deep learning algorithms. However, it is very difficult to gather such data in a real-world setting. For example, in certain medical/healthcare applications, it is very challenging to have access to data from all possible scenarios or with the necessary amount of variations as required to train the system. Additionally, the over-parameterized and unconstrained nature of deep neural networks can cause them to be poorly trained and in many cases over-confident which, in turn, can hamper their reliability and generalizability. This dissertation is a compendium of my research efforts to address the above challenges. I propose building invariant feature representations by wedding concepts from topological data analysis and Riemannian geometry, that automatically incorporate the desired invariance properties for different computer vision applications. I discuss how deep learning can be used to address some of the common challenges faced when working with topological data analysis methods. I describe alternative learning strategies based on unsupervised learning and transfer learning to address issues like dataset shifts and limited training data. Finally, I discuss my preliminary work on applying simple orthogonal constraints on deep learning feature representations to help develop more reliable and better calibrated models.

There has been tremendous technological advancement in the past two decades. Faster computers and improved sensing devices have broadened the research scope in computer vision. With these developments, the task of assessing the quality of human actions, is considered an important problem that needs to be tackled. Movement quality assessment finds wide range of application in motor control, health-care, rehabilitation and physical therapy. Home-based interactive physical therapy requires the ability to monitor, inform and assess the quality of everyday movements. Obtaining labeled data from trained therapists/experts is the main limitation, since it is both expensive and time consuming.

Motivated by recent studies in motor control and therapy, in this thesis an existing computational framework is used to assess balance impairment and disease severity in people suffering from Parkinson's disease. The framework uses high-dimensional shape descriptors of the reconstructed phase space, of the subjects' center of pressure (CoP) tracings while performing dynamical postural shifts. The performance of the framework is evaluated using a dataset collected from 43 healthy and 17 Parkinson's disease impaired subjects, and outperforms other methods, such as dynamical shift indices and use of chaotic invariants, in assessment of balance impairment.

In this thesis, an unsupervised method is also proposed that measures movement quality assessment of simple actions like sit-to-stand and dynamic posture shifts by modeling the deviation of a given movement from an ideal movement path in the configuration space, i.e. the quality of movement is directly related to similarity to the ideal trajectory, between the start and end pose. The S^1xS^1 configuration space was used to model the interaction of two joint angles in sit-to-stand actions, and the R^2 space was used to model the subject's CoP while performing dynamic posture shifts for application in movement quality estimation.