Principal component analysis (PCA) is a commonly used pattern analysis method that maps high-dimensional data into a lower-dimensional space maximizing the data variance, that results in the promotion of separability of data. Inspired by the principle of PCA, a novel type of learning discriminative matrix information geometry (MIG) detectors in the unsupervised scenario are developed, and applied to signal detection in nonhomogeneous environments. Hermitian positive-definite (HPD) matrices can be used to model the sample data, while the clutter covariance matrix is estimated by the geometric mean of a set of secondary HPD matrices. We define a projection that maps the HPD matrices in a high-dimensional manifold to a low-dimensional and more discriminative one to increase the degree of separation of HPD matrices by maximizing the data variance. Learning a mapping can be formulated as a two-step mini-max optimization problem in Riemannian manifolds, which can be solved by the Riemannian gradient descent algorithm. Three discriminative MIG detectors are illustrated with respect to different geometric measures, i.e., the Log-Euclidean metric, the Jensen--Bregman LogDet divergence and the symmetrized Kullback--Leibler divergence. Simulation results show that performance improvements of the novel MIG detectors can be achieved compared with the conventional detectors and their state-of-the-art counterparts within nonhomogeneous environments.

Major depressive disorder (MDD) is one of the most common mental health conditions that has been intensively investigated for its association with brain atrophy and mortality. Recent studies reveal that the deviation between the predicted and the chronological age can be a marker of accelerated brain aging to characterize MDD. However, current conclusions are usually drawn based on structural MRI information collected from Caucasian participants. The universality of this biomarker needs to be further validated by subjects with different ethnic/racial backgrounds and by different types of data. Here we make use of the REST-meta-MDD, a large scale resting-state fMRI dataset collected from multiple cohort participants in China. We develop a stacking machine learning model based on 1101 healthy controls, which estimates a subject's chronological age from fMRI with promising accuracy. The trained model is then applied to 1276 MDD patients from 24 sites. We observe that MDD patients exhibit a $+4.43$ years ($\text{$p$} < 0.0001$, $\text{Cohen's $d$} = 0.35$, $\text{95\% CI}:1.86 - 3.91$) higher brain-predicted age difference (brain-PAD) compared to controls. In the MDD subgroup, we observe a statistically significant $+2.09$ years ($\text{$p$} < 0.05$, $\text{Cohen's $d$} = 0.134483$) brain-PAD in antidepressant users compared to medication-free patients. The statistical relationship observed is further checked by three different machine learning algorithms. The positive brain-PAD observed in participants in China confirms the presence of accelerated brain aging in MDD patients. The utilization of functional brain connectivity for age estimation verifies existing findings from a new dimension.

The Graph Convolutional Networks (GCNs) have achieved excellent results in node classification tasks, but the model's performance at low label rates is still unsatisfactory. Previous studies in Semi-Supervised Learning (SSL) for graph have focused on using network predictions to generate soft pseudo-labels or instructing message propagation, which inevitably contains the incorrect prediction due to the over-confident in the predictions. Our proposed Dual-Channel Consistency based Graph Convolutional Networks (DCC-GCN) uses dual-channel to extract embeddings from node features and topological structures, and then achieves reliable low-confidence and high-confidence samples selection based on dual-channel consistency. We further confirmed that the low-confidence samples obtained based on dual-channel consistency were low in accuracy, constraining the model's performance. Unlike previous studies ignoring low-confidence samples, we calibrate the feature embeddings of the low-confidence samples by using the neighborhood's high-confidence samples. Our experiments have shown that the DCC-GCN can more accurately distinguish between low-confidence and high-confidence samples, and can also significantly improve the accuracy of low-confidence samples. We conducted extensive experiments on the benchmark datasets and demonstrated that DCC-GCN is significantly better than state-of-the-art baselines at different label rates.

This thesis tackles the subject of spatio-temporal forecasting with deep learning. The motivating application at Electricity de France (EDF) is short-term solar energy forecasting with fisheye images. We explore two main research directions for improving deep forecasting methods by injecting external physical knowledge. The first direction concerns the role of the training loss function. We show that differentiable shape and temporal criteria can be leveraged to improve the performances of existing models. We address both the deterministic context with the proposed DILATE loss function and the probabilistic context with the STRIPE model. Our second direction is to augment incomplete physical models with deep data-driven networks for accurate forecasting. For video prediction, we introduce the PhyDNet model that disentangles physical dynamics from residual information necessary for prediction, such as texture or details. We further propose a learning framework (APHYNITY) that ensures a principled and unique linear decomposition between physical and data-driven components under mild assumptions, leading to better forecasting performances and parameter identification.

As technology advanced, collecting data via automatic collection devices become popular, thus we commonly face data sets with lengthy variables, especially when these data sets are collected without specific research goals beforehand. It has been pointed out in the literature that the difficulty of high-dimensional classification problems is intrinsically caused by too many noise variables useless for reducing classification error, which offer less benefits for decision-making, and increase complexity, and confusion in model-interpretation. A good variable selection strategy is therefore a must for using such kinds of data well; especially when we expect to use their results for the succeeding applications/studies, where the model-interpretation ability is essential. hus, the conventional classification measures, such as accuracy, sensitivity, precision, cannot be the only performance tasks. In this paper, we propose a probability-based nonparametric multiple-class classification method, and integrate it with the ability of identifying high impact variables for individual class such that we can have more information about its classification rule and the character of each class as well. The proposed method can have its prediction power approximately equal to that of the Bayes rule, and still retains the ability of "model-interpretation." We report the asymptotic properties of the proposed method, and use both synthesized and real data sets to illustrate its properties under different classification situations. We also separately discuss the variable identification, and training sample size determination, and summarize those procedures as algorithms such that users can easily implement them with different computing languages.

The Li-ion battery is a complex physicochemical system that generally takes applied current as input and terminal voltage as output. The mappings from current to voltage can be described by several kinds of models, such as accurate but inefficient physics-based models, and efficient but sometimes inaccurate equivalent circuit and black-box models. To realize accuracy and efficiency simultaneously in battery modeling, we propose to build a data-driven surrogate for a battery system while incorporating the underlying physics as constraints. In this work, we innovatively treat the functional mapping from current curve to terminal voltage as a composite of operators, which is approximated by the powerful deep operator network (DeepONet). Its learning capability is firstly verified through a predictive test for Li-ion concentration at two electrodes. In this experiment, the physics-informed DeepONet is found to be more robust than the purely data-driven DeepONet, especially in temporal extrapolation scenarios. A composite surrogate is then constructed for mapping current curve and solid diffusivity to terminal voltage with three operator networks, in which two parallel physics-informed DeepONets are firstly used to predict Li-ion concentration at two electrodes, and then based on their surface values, a DeepONet is built to give terminal voltage predictions. Since the surrogate is differentiable anywhere, it is endowed with the ability to learn from data directly, which was validated by using terminal voltage measurements to estimate input parameters. The proposed surrogate built upon operator networks possesses great potential to be applied in on-board scenarios, such as battery management system, since it integrates efficiency and accuracy by incorporating underlying physics, and also leaves an interface for model refinement through a totally differentiable model structure.

The data representation in a machine-learning model strongly influences its performance. This becomes even more important for quantum machine learning models implemented on noisy intermediate scale quantum (NISQ) devices. Encoding high dimensional data into a quantum circuit for a NISQ device without any loss of information is not trivial and brings a lot of challenges. While simple encoding schemes (like single qubit rotational gates to encode high dimensional data) often lead to information loss within the circuit, complex encoding schemes with entanglement and data re-uploading lead to an increase in the encoding gate count. This is not well-suited for NISQ devices. This work proposes 'incremental data-uploading', a novel encoding pattern for high dimensional data that tackles these challenges. We spread the encoding gates for the feature vector of a given data point throughout the quantum circuit with parameterized gates in between them. This encoding pattern results in a better representation of data in the quantum circuit with a minimal pre-processing requirement. We show the efficiency of our encoding pattern on a classification task using the MNIST and Fashion-MNIST datasets, and compare different encoding methods via classification accuracy and the effective dimension of the model.

Content-based collaborative filtering (CCF) predicts user-item interactions based on both users' interaction history and items' content information. Recently, pre-trained language models (PLM) have been used to extract high-quality item encodings for CCF. However, it is resource-intensive to train a PLM-based CCF model in an end-to-end (E2E) manner, since optimization involves back-propagating through every content encoding within a given user interaction sequence. To tackle this issue, we propose GRAM (GRadient Accumulation for Multi-modality in CCF), which exploits the fact that a given item often appears multiple times within a batch of interaction histories. Specifically, Single-step GRAM aggregates each item encoding's gradients for back-propagation, with theoretic equivalence to the standard E2E training. As an extension of Single-step GRAM, we propose Multi-step GRAM, which increases the gradient update latency, achieving a further speedup with drastically less GPU memory. GRAM significantly improves training efficiency (up to 146x) on five datasets from two task domains of Knowledge Tracing and News Recommendation. Our code is available at https://github.com/yoonseok312/GRAM.

Nonconvex regularization has been popularly used in low-rank matrix learning. However, extending it for low-rank tensor learning is still computationally expensive. To address this problem, we develop an efficient solver for use with a nonconvex extension of the overlapped nuclear norm regularizer. Based on the proximal average algorithm, the proposed algorithm can avoid expensive tensor folding/unfolding operations. A special "sparse plus low-rank" structure is maintained throughout the iterations, and allows fast computation of the individual proximal steps. Empirical convergence is further improved with the use of adaptive momentum. We provide convergence guarantees to critical points on smooth losses and also on objectives satisfying the Kurdyka-{\L}ojasiewicz condition. While the optimization problem is nonconvex and nonsmooth, we show that its critical points still have good statistical performance on the tensor completion problem. Experiments on various synthetic and real-world data sets show that the proposed algorithm is efficient in both time and space and more accurate than the existing state-of-the-art.

This paper considers learning of the graphical structure of a $p$-dimensional random vector $X \in R^p$ using both parametric and non-parametric methods. Unlike the previous works which observe $x$ directly, we consider the indirect observation scenario in which samples $y$ are collected via a sensing matrix $A \in R^{d\times p}$, and corrupted with some additive noise $w$, i.e, $Y = AX + W$. For the parametric method, we assume $X$ to be Gaussian, i.e., $x\in R^p\sim N(\mu, \Sigma)$ and $\Sigma \in R^{p\times p}$. For the first time, we show that the correct graphical structure can be correctly recovered under the indefinite sensing system ($d < p$) using insufficient samples ($n < p$). In particular, we show that for the exact recovery, we require dimension $d = \Omega(p^{0.8})$ and sample number $n = \Omega(p^{0.8}\log^3 p)$. For the nonparametric method, we assume a nonparanormal distribution for $X$ rather than Gaussian. Under mild conditions, we show that our graph-structure estimator can obtain the correct structure. We derive the minimum sample number $n$ and dimension $d$ as $n\gtrsim (deg)^4 \log^4 n$ and $d \gtrsim p + (deg\cdot\log(d-p))^{\beta/4}$, respectively, where deg is the maximum Markov blanket in the graphical model and $\beta > 0$ is some fixed positive constant. Additionally, we obtain a non-asymptotic uniform bound on the estimation error of the CDF of $X$ from indirect observations with inexact knowledge of the noise distribution. To the best of our knowledge, this bound is derived for the first time and may serve as an independent interest. Numerical experiments on both real-world and synthetic data are provided confirm the theoretical results.