In this paper, we study the Schrödinger Bridge Problem (SBP), which is central to entropic optimal transport. For general reference processes and begin-endpoint distributions, we propose a forward-reverse iterative Monte Carlo procedure to approximate the Schrödinger potentials in a nonparametric way. In particular, we use kernel based Monte Carlo regression in the context of Picard iteration of a corresponding fixed point problem as considered in [5]. By preserving in the iteration positivity and contractivity in a Hilbert metric sense, we develop a provably convergent algorithm. Furthermore, we provide convergence rates for the potential estimates and prove their optimality. Finally, as an application, we propose a non-nested Monte Carlo procedure for the final dimensional distributions of the Schrödinger Bridge process, based on the constructed potentials and the forward-reverse simulation method for conditional diffusions developed in [2].

This study considers the object localization problem and proposes a novel multiparticle Kalman filter to solve it in complex and symmetric environments. Two well-known classes of filtering algorithms to solve the localization problem are Kalman filter-based methods and particle filter-based methods. We consider these classes, demonstrate their complementary properties, and propose a novel filtering algorithm that takes the best from two classes. We evaluate the multiparticle Kalman filter in symmetric and noisy environments. Such environments are especially challenging for both classes of classical methods. We compare the proposed approach with the particle filter since only this method is feasible if the initial state is unknown. In the considered challenging environments, our method outperforms the particle filter in terms of both localization error and runtime.

Gaze-tracking is a novel way of interacting with computers which allows new scenarios, such as enabling people with motor-neuron disabilities to control their computers or doctors to interact with patient information without touching screen or keyboard. Further, there are emerging applications of gaze-tracking in interactive gaming, user experience research, human attention analysis and behavioral studies. Accurate estimation of the gaze may involve accounting for head-pose, head-position, eye rotation, distance from the object as well as operating conditions such as illumination, occlusion, background noise and various biological aspects of the user. Commercially available gaze-trackers utilize specialized sensor assemblies that usually consist of an infrared light source and camera. There are several challenges in the universal proliferation of gaze-tracking as accessibility technologies, specifically its affordability, reliability, and ease-of-use. In this paper, we try to address these challenges through the development of a hardware-agnostic gaze-tracker. We present a deep neural network architecture as an appearance-based method for constrained gaze-tracking that utilizes facial imagery captured on an ordinary RGB camera ubiquitous in all modern computing devices. Our system achieved an error of 1.8073cm on GazeCapture dataset without any calibration or device specific fine-tuning. This research shows promise that one day soon any computer, tablet, or phone will be controllable using just your eyes due to the prediction capabilities of deep neutral networks.

Logistic regression is commonly used for modeling dichotomous outcomes. In the classical setting, where the number of observations is much larger than the number of parameters, properties of the maximum likelihood estimator in logistic regression are well understood. Recently, Sur and Candes have studied logistic regression in the high-dimensional regime, where the number of observations and parameters are comparable, and show, among other things, that the maximum likelihood estimator is biased. In the high-dimensional regime the underlying parameter vector is often structured (sparse, block-sparse, finite-alphabet, etc.) and so in this paper we study regularized logistic regression (RLR), where a convex regularizer that encourages the desired structure is added to the negative of the log-likelihood function. An advantage of RLR is that it allows parameter recovery even for instances where the (unconstrained) maximum likelihood estimate does not exist. We provide a precise analysis of the performance of RLR via the solution of a system of six nonlinear equations, through which any performance metric of interest (mean, mean-squared error, probability of support recovery, etc.) can be explicitly computed. Our results generalize those of Sur and Candes and we provide a detailed study for the cases of $\ell_2^2$-RLR and sparse ($\ell_1$-regularized) logistic regression. In both cases, we obtain explicit expressions for various performance metrics and can find the values of the regularizer parameter that optimizes the desired performance. The theory is validated by extensive numerical simulations across a range of parameter values and problem instances.

Using a low-dimensional parametrization of signals is a generic and powerful way to enhance performance in signal processing and statistical inference. A very popular and widely explored type of dimensionality reduction is sparsity; another type is generative modelling of signal distributions. Generative models based on neural networks, such as GANs or variational auto-encoders, are particularly performant and are gaining on applicability. In this paper we study spiked matrix models, where a low-rank matrix is observed through a noisy channel. This problem with sparse structure of the spikes has attracted broad attention in the past literature. Here, we replace the sparsity assumption by generative modelling, and investigate the consequences on statistical and algorithmic properties. We analyze the Bayes-optimal performance under specific generative models for the spike. In contrast with the sparsity assumption, we do not observe regions of parameters where statistical performance is superior to the best known algorithmic performance. We show that in the analyzed cases the approximate message passing algorithm is able to reach optimal performance. We also design enhanced spectral algorithms and analyze their performance and thresholds using random matrix theory, showing their superiority to the classical principal component analysis. We complement our theoretical results by illustrating the performance of the spectral algorithms when the spikes come from real datasets.

Support vector machine (SVM) is one of the most widely used classification methods. In this paper, we consider soft margin support vector machine used on data points with independent features, where the sample size $n$ and the feature dimension $p$ grows to $\infty$ in a fixed ratio $p/n\rightarrow \delta$. We propose a set of equations that exactly characterizes the asymptotic behavior of support vector machine. In particular, we give exact formula for (1) the variability of the optimal coefficients, (2) proportion of data points lying on the margin boundary (i.e. number of support vectors), (3) the final objective function value, and (4) expected misclassification error on new data points, which in particular implies exact formula for the optimal tuning parameter given a data generating mechanism. The global null case is considered first, where the label $y\in\{+1,-1\}$ is independent of the feature $x$. Then the signaled case is considered, where the label $y\in\{+1,-1\}$ is allowed to have a general dependence on the feature $x$ through a linear combination $a_0^Tx$. These results for the non-smooth hinge loss serve as an analogue to the recent results in \citet{sur2018modern} for smooth logistic loss. Our approach is based on heuristic leave-one-out calculations.

Structured prediction provides a general framework to deal with supervised problems where the outputs have semantically rich structure. While classical approaches consider finite, albeit potentially huge, output spaces, in this paper we discuss how structured prediction can be extended to a continuous scenario. Specifically, we study a structured prediction approach to manifold-valued regression. We characterize a class of problems for which the considered approach is statistically consistent and study how geometric optimization can be used to compute the corresponding estimator. Promising experimental results on both simulated and real data complete our study.