The Human Gut Microbiome (GM) modulates a variety of structural, metabolic, and protective functions to benefit the host. A few recent studies also support the role of the gut microbiome in the regulation of bone health. The relationship between GM and bone health was analyzed based on the data collected from a group of twenty-three adolescent boys and girls who participated in a controlled feeding study, during which two different doses (0 g/d fiber and 12 g/d fiber) of Soluble Corn Fiber (SCF) were added to their diet. This analysis was performed by predicting measures of Bone Mineral Density (BMD) and Bone Mineral Content (BMC) which are indicators of bone strength, using the GM sequence of proportions of 178 microbes collected from 23 subjects, by building a machine learning regression model. The model developed was evaluated by calculating performance metrics such as Root Mean Squared Error, Pearson’s correlation coefficient, and Spearman’s rank correlation coefficient, using cross-validation. A noticeable correlation was observed between the GM and bone health, and it was observed that the overall prediction correlation was higher with SCF intervention (r ~ 0.51). The genera of microbes that played an important role in this relationship were identified. Eubacterium (g), Bacteroides (g), Megamonas (g), Acetivibrio (g), Faecalibacterium (g), and Paraprevotella (g) were some of the microbes that showed an increase in proportion with SCF intervention.

According to the Center for Disease Control and Prevention report around 29,668 United States residents aged greater than 65 years had died as a result of a fall in 2016. Other injuries like wrist fractures, hip fractures, and head injuries occur as a result of a fall. Certain groups of people are more prone to experience falls than others, one of which being individuals with stroke. The two most common issues with individuals with strokes are ankle weakness and foot drop, both of which contribute to falls. To mitigate this issue, the most popular clinical remedy given to these users is thermoplastic Ankle Foot Orthosis. These AFO's help improving gait velocity, stride length, and cadence. However, studies have shown that a continuous restraint on the ankle harms the compensatory stepping response and forward propulsion. It has been shown in previous studies that compensatory stepping and forward propulsion are crucial for the user's ability to recover from postural perturbations. Hence, there is a need for active devices that can supply a plantarflexion during the push-off and dorsiflexion during the swing phase of gait. Although advancements in the orthotic research have shown major improvements in supporting the ankle joint for rehabilitation, there is a lack of available active devices that can help impaired users in daily activities. In this study, our primary focus is to build an unobtrusive, cost-effective, and easy to wear active device for gait rehabilitation and fall prevention in individuals who are at risk. The device will be using a double-acting cylinder that can be easily incorporated into the user's footwear using a novel custom-designed powered ankle brace. The device will use Inertial Measurement Units to measure kinematic parameters of the lower body and a custom control algorithm to actuate the device based on the measurements. The study can be used to advance the field of gait assistance, rehabilitation, and potentially fall prevention of individuals with lower-limb impairments through the use of Active Ankle Foot Orthosis.

Hyperspectral unmixing is an important remote sensing task with applications including material identification and analysis. Characteristic spectral features make many pure materials identifiable from their visible-to-infrared spectra, but quantifying their presence within a mixture is a challenging task due to nonlinearities and factors of variation. In this thesis, physics-based approaches are incorporated into an end-to-end spectral unmixing algorithm via differentiable programming. First, sparse regularization and constraints are implemented by adding differentiable penalty terms to a cost function to avoid unrealistic predictions. Secondly, a physics-based dispersion model is introduced to simulate realistic spectral variation, and an efficient method to fit the parameters is presented. Then, this dispersion model is utilized as a generative model within an analysis-by-synthesis spectral unmixing algorithm. Further, a technique for inverse rendering using a convolutional neural network to predict parameters of the generative model is introduced to enhance performance and speed when training data are available. Results achieve state-of-the-art on both infrared and visible-to-near-infrared (VNIR) datasets as compared to baselines, and show promise for the synergy between physics-based models and deep learning in hyperspectral unmixing in the future.

Modern digital applications have significantly increased the leakage of private and sensitive personal data. While worst-case measures of leakage such as Differential Privacy (DP) provide the strongest guarantees, when utility matters, average-case information-theoretic measures can be more relevant. However, most such information-theoretic measures do not have clear operational meanings. This dissertation addresses this challenge.

This work introduces a tunable leakage measure called maximal $\alpha$-leakage which quantifies the maximal gain of an adversary in inferring any function of a data set. The inferential capability of the adversary is modeled by a class of loss functions, namely, $\alpha$-loss. The choice of $\alpha$ determines specific adversarial actions ranging from refining a belief for $\alpha =1$ to guessing the best posterior for $\alpha = \infty$, and for the two specific values maximal $\alpha$-leakage simplifies to mutual information and maximal leakage, respectively. Maximal $\alpha$-leakage is proved to have a composition property and be robust to side information.

There is a fundamental disjoint between theoretical measures of information leakages and their applications in practice. This issue is addressed in the second part of this dissertation by proposing a data-driven framework for learning Censored and Fair Universal Representations (CFUR) of data. This framework is formulated as a constrained minimax optimization of the expected $\alpha$-loss where the constraint ensures a measure of the usefulness of the representation. The performance of the CFUR framework with $\alpha=1$ is evaluated on publicly accessible data sets; it is shown that multiple sensitive features can be effectively censored to achieve group fairness via demographic parity while ensuring accuracy for several \textit{a priori} unknown downstream tasks.

Finally, focusing on worst-case measures, novel information-theoretic tools are used to refine the existing relationship between two such measures, $(\epsilon,\delta)$-DP and R\'enyi-DP. Applying these tools to the moments accountant framework, one can track the privacy guarantee achieved by adding Gaussian noise to Stochastic Gradient Descent (SGD) algorithms. Relative to state-of-the-art, for the same privacy budget, this method allows about 100 more SGD rounds for training deep learning models.

Deep neural networks (DNNs) have had tremendous success in a variety of

statistical learning applications due to their vast expressive power. Most

applications run DNNs on the cloud on parallelized architectures. There is a need

for for efficient DNN inference on edge with low precision hardware and analog

accelerators. To make trained models more robust for this setting, quantization and

analog compute noise are modeled as weight space perturbations to DNNs and an

information theoretic regularization scheme is used to penalize the KL-divergence

between perturbed and unperturbed models. This regularizer has similarities to

both natural gradient descent and knowledge distillation, but has the advantage of

explicitly promoting the network to and a broader minimum that is robust to

weight space perturbations. In addition to the proposed regularization,

KL-divergence is directly minimized using knowledge distillation. Initial validation

on FashionMNIST and CIFAR10 shows that the information theoretic regularizer

and knowledge distillation outperform existing quantization schemes based on the

straight through estimator or L2 constrained quantization.

The problem of multiple object tracking seeks to jointly estimate the time-varying cardinality and trajectory of each object. There are numerous challenges that are encountered in tracking multiple objects including a time-varying number of measurements, under varying constraints, and environmental conditions. In this thesis, the proposed statistical methods integrate the use of physical-based models with Bayesian nonparametric methods to address the main challenges in a tracking problem. In particular, Bayesian nonparametric methods are exploited to efficiently and robustly infer object identity and learn time-dependent cardinality; together with Bayesian inference methods, they are also used to associate measurements to objects and estimate the trajectory of objects. These methods differ from the current methods to the core as the existing methods are mainly based on random finite set theory.

The first contribution proposes dependent nonparametric models such as the dependent Dirichlet process and the dependent Pitman-Yor process to capture the inherent time-dependency in the problem at hand. These processes are used as priors for object state distributions to learn dependent information between previous and current time steps. Markov chain Monte Carlo sampling methods exploit the learned information to sample from posterior distributions and update the estimated object parameters.

The second contribution proposes a novel, robust, and fast nonparametric approach based on a diffusion process over infinite random trees to infer information on object cardinality and trajectory. This method follows the hierarchy induced by objects entering and leaving a scene and the time-dependency between unknown object parameters. Markov chain Monte Carlo sampling methods integrate the prior distributions over the infinite random trees with time-dependent diffusion processes to update object states.

The third contribution develops the use of hierarchical models to form a prior for statistically dependent measurements in a single object tracking setup. Dependency among the sensor measurements provides extra information which is incorporated to achieve the optimal tracking performance. The hierarchical Dirichlet process as a prior provides the required flexibility to do inference. Bayesian tracker is integrated with the hierarchical Dirichlet process prior to accurately estimate the object trajectory.

The fourth contribution proposes an approach to model both the multiple dependent objects and multiple dependent measurements. This approach integrates the dependent Dirichlet process modeling over the dependent object with the hierarchical Dirichlet process modeling of the measurements to fully capture the dependency among both object and measurements. Bayesian nonparametric models can successfully associate each measurement to the corresponding object and exploit dependency among them to more accurately infer the trajectory of objects. Markov chain Monte Carlo methods amalgamate the dependent Dirichlet process with the hierarchical Dirichlet process to infer the object identity and object cardinality.

Simulations are exploited to demonstrate the improvement in multiple object tracking performance when compared to approaches that are developed based on random finite set theory.

Non-line-of-sight (NLOS) imaging of objects not visible to either the camera or illumina-

tion source is a challenging task with vital applications including surveillance and robotics.

Recent NLOS reconstruction advances have been achieved using time-resolved measure-

ments. Acquiring these time-resolved measurements requires expensive and specialized

detectors and laser sources. In work proposes a data-driven approach for NLOS 3D local-

ization requiring only a conventional camera and projector. The localisation is performed

using a voxelisation and a regression problem. Accuracy of greater than 90% is achieved

in localizing a NLOS object to a 5cm × 5cm × 5cm volume in real data. By adopting

the regression approach an object of width 10cm to localised to approximately 1.5cm. To

generalize to line-of-sight (LOS) scenes with non-planar surfaces, an adaptive lighting al-

gorithm is adopted. This algorithm, based on radiosity, identifies and illuminates scene

patches in the LOS which most contribute to the NLOS light paths, and can factor in sys-

tem power constraints. Improvements ranging from 6%-15% in accuracy with a non-planar

LOS wall using adaptive lighting is reported, demonstrating the advantage of combining

the physics of light transport with active illumination for data-driven NLOS imaging.