Theoretically understanding stochastic gradient descent (SGD) in overparameterized models has led to the development of several optimization algorithms that are widely used in practice today. Recent work by Zou et al. [2021] provides sharp rates for SGD optimization in linear regression using constant learning rate, both with and without tail iterate averaging, based on a bias-variance decomposition of the risk. In our work, we provide a simplified analysis recovering the same bias and variance bounds provided in [Zou et al., 2021] based on simple linear algebra tools, bypassing the requirement to manipulate operators on positive semi-definite (PSD) matrices. We believe our work makes the analysis of SGD on linear regression very accessible and will be helpful in further analyzing mini-batching and learning rate scheduling, leading to improvements in the training of realistic models. We use bolded small letters for vectors and bolded capital letters for matrices, and we use 1 for a vector of ones.

Ferry quays experience rapid deterioration due to their exposure to harsh maritime environments and ferry impacts. Vibration-based structural health monitoring offers a valuable approach to assessing structural integrity and understanding the structural implications of these impacts. However, practical limitations often restrict sensor placement at critical locations. Consequently, virtual sensing techniques become essential for establishing a Digital Twin and estimating the structural response. This study investigates the application of the Gaussian Process Latent Force Model (GPLFM) for virtual sensing on the Magerholm ferry quay, combining in-operation experimental data collected during a ferry impact with a detailed physics-based model. The proposed Physics-Encoded Machine Learning model integrates a reduced-order structural model with a data-driven GPLFM representing the unknown impact forces via their modal contributions. Significant challenges are addressed for the development of the Digital Twin of the ferry quay, including unknown impact characteristics (location, direction, intensity), time-varying boundary conditions, and sparse sensor configurations. Results show that the GPLFM provides accurate acceleration response estimates at most locations, even under simplifying modeling assumptions such as linear time-invariant behavior during the impact phase. Lower accuracy was observed at locations in the impact zone. A numerical study was conducted to explore an optimal real-world sensor placement strategy using a Backward Sequential Sensor Placement approach. Sensitivity analyses were conducted to examine the influence of sensor types, sampling frequencies, and incorrectly assumed damping ratios. The results suggest that the GP latent forces can help accommodate modeling and measurement uncertainties, maintaining acceptable estimation accuracy across scenarios.

We propose a data-driven method to learn the time-dependent probability density of a multivariate stochastic process from sample paths, assuming that the initial probability density is known and can be evaluated. Our method uses a novel time-dependent binary classifier trained using a contrastive estimation-based objective that trains the classifier to discriminate between realizations of the stochastic process at two nearby time instants. Significantly, the proposed method explicitly models the time-dependent probability distribution, which means that it is possible to obtain the value of the probability density within the time horizon of interest. Additionally, the input before the final activation in the time-dependent classifier is a second-order approximation to the partial derivative, with respect to time, of the logarithm of the density. We apply the proposed approach to approximate the time-dependent probability density functions for systems driven by stochastic excitations. We also use the proposed approach to synthesize new samples of a random vector from a given set of its realizations. In such applications, we generate sample paths necessary for training using stochastic interpolants. Subsequently, new samples are generated using gradient-based Markov chain Monte Carlo methods because automatic differentiation can efficiently provide the necessary gradient. Further, we demonstrate the utility of an explicit approximation to the time-dependent probability density function through applications in unsupervised outlier detection. Through several numerical experiments, we show that the proposed method accurately reconstructs complex time-dependent, multi-modal, and near-degenerate densities, scales effectively to moderately high-dimensional problems, and reliably detects rare events among real-world data.

Clustering and estimating cluster means are core problems in statistics and machine learning, with k-means and Expectation Maximization (EM) being two widely used algorithms. In this work, we provide a theoretical explanation for the failure of k-means in high-dimensional settings with high noise and limited sample sizes, using a simple Gaussian Mixture Model (GMM). We identify regimes where, with high probability, almost every partition of the data becomes a fixed point of the k-means algorithm. This study is motivated by challenges in the analysis of more complex cases, such as masked GMMs, and those arising from applications in Cryo-Electron Microscopy.

There has been much recent interest in understanding the continuum from adversarial to stochastic settings in online learning, with various frameworks including smoothed settings proposed to bridge this gap. We consider the more general and flexible framework of distributionally constrained adversaries in which instances are drawn from distributions chosen by an adversary within some constrained distribution class [RST11]. Compared to smoothed analysis, we consider general distributional classes which allows for a fine-grained understanding of learning settings between fully stochastic and fully adversarial for which a learner can achieve non-trivial regret. We give a characterization for which distribution classes are learnable in this context against both oblivious and adaptive adversaries, providing insights into the types of interplay between the function class and distributional constraints on adversaries that enable learnability. In particular, our results recover and generalize learnability for known smoothed settings. Further, we show that for several natural function classes including linear classifiers, learning can be achieved without any prior knowledge of the distribution class -- in other words, a learner can simultaneously compete against any constrained adversary within learnable distribution classes.

In this paper, we propose an accelerated version for the Sinkhorn algorithm, which is the reference method for computing the solution to Entropic Optimal Transport. Its main draw-back is the exponential slow-down of convergence as the regularization weakens $\varepsilon \rightarrow 0$. Thanks to spectral insights on the behavior of the Hessian, we propose to mitigate the problem via an original spectral warm-start strategy. This leads to faster convergence compared to the reference method, as also demonstrated in our numerical experiments.

We study the problem of clustering $T$ trajectories of length $H$, each generated by one of $K$ unknown ergodic Markov chains over a finite state space of size $S$. The goal is to accurately group trajectories according to their underlying generative model. We begin by deriving an instance-dependent, high-probability lower bound on the clustering error rate, governed by the weighted KL divergence between the transition kernels of the chains. We then present a novel two-stage clustering algorithm. In Stage~I, we apply spectral clustering using a new injective Euclidean embedding for ergodic Markov chains -- a contribution of independent interest that enables sharp concentration results. Stage~II refines the initial clusters via a single step of likelihood-based reassignment. Our method achieves a near-optimal clustering error with high probability, under the conditions $H = \tildeΩ(γ_{\mathrm{ps}}^{-1} (S^2 \vee π_{\min}^{-1}))$ and $TH = \tildeΩ(γ_{\mathrm{ps}}^{-1} S^2 )$, where $π_{\min}$ is the minimum stationary probability of a state across the $K$ chains and $γ_{\mathrm{ps}}$ is the minimum pseudo-spectral gap. These requirements provide significant improvements, if not at least comparable, to the state-of-the-art guarantee (Kausik et al., 2023), and moreover, our algorithm offers a key practical advantage: unlike existing approach, it requires no prior knowledge of model-specific quantities (e.g., separation between kernels or visitation probabilities). We conclude by discussing the inherent gap between our upper and lower bounds, providing insights into the unique structure of this clustering problem.

Deep learning models often achieve high performance by inadvertently learning spurious correlations between targets and non-essential features. For example, an image classifier may identify an object via its background that spuriously correlates with it. This prediction behavior, known as spurious bias, severely degrades model performance on data that lacks the learned spurious correlations. Existing methods on spurious bias mitigation typically require a variety of data groups with spurious correlation annotations called group labels. However, group labels require costly human annotations and often fail to capture subtle spurious biases such as relying on specific pixels for predictions. In this paper, we propose a novel post hoc spurious bias mitigation framework without requiring group labels. Our framework, termed ShortcutProbe, identifies prediction shortcuts that reflect potential non-robustness in predictions in a given model's latent space. The model is then retrained to be invariant to the identified prediction shortcuts for improved robustness. We theoretically analyze the effectiveness of the framework and empirically demonstrate that it is an efficient and practical tool for improving a model's robustness to spurious bias on diverse datasets.

--This study presents TSLFormer, a light and robust word-level T urkish Sign Language (TID) recognition model that treats sign gestures as ordered, string-like language. In contrast to working with raw RGB or depth videos, our method only works with 3D joint positions--articulation points--extracted using Google's Mediapipe library, which focuses on the hand and torso skeletal locations. This creates efficient input dimensionality reduction with significant preservation of important semantic information of the gesture. Our approach revisits sign language recognition as sequence-to-sequence translation, drawing inspiration from sign languages' linguistic nature and transformer's success at natural language translation. Since TSLFormer adapts the transformers' self-attention mechanism, it effectively represents the temporal co-occurrence of a sign sequence, stressing significant movement habits over time as words are referenced in a sentence. Experimented and validated on the AUTSL dataset holding over 36,000 sign samples of over 226 different words, the TSLFormer achieves competitive performance and with minimal computational demands. From the experimentation, rich spatiotemporal understanding of signs is evidenced, and using only joint landmarks, it is possible within any real-time, mobile, and assistive technology facilitating communication between hearing-impaired members. Sign language is an essential communication method for the hearing impaired to express ideas and sentiments through hand gestures, facial expressions, and body movement. Unlike spoken languages, which employ auditory and verbal modalities, sign language utilizes visual and spatial modalities to express meaning. However, despite the limited number of sign language proficient individuals, communication gaps still exist to hinder inclusion--particularly in social interaction on a daily basis and in employment, educational, and healthcare environments.

Graph autoencoders (GAE) and variational graph autoencoders (VGAE) emerged as two powerful groups of unsupervised node embedding methods, with various applications to graph-based machine learning problems such as link prediction and community detection. Nonetheless, at the beginning of this Ph.D. project, GAE and VGAE models were also suffering from key limitations, preventing them from being adopted in the industry. In this thesis, we present several contributions to improve these models, with the general aim of facilitating their use to address industrial-level problems involving graph representations. Firstly, we propose two strategies to overcome the scalability issues of previous GAE and VGAE models, permitting to effectively train these models on large graphs with millions of nodes and edges. These strategies leverage graph degeneracy and stochastic subgraph decoding techniques, respectively. Besides, we introduce Gravity-Inspired GAE and VGAE, providing the first extensions of these models for directed graphs, that are ubiquitous in industrial applications. We also consider extensions of GAE and VGAE models for dynamic graphs. Furthermore, we argue that GAE and VGAE models are often unnecessarily complex, and we propose to simplify them by leveraging linear encoders. Lastly, we introduce Modularity-Aware GAE and VGAE to improve community detection on graphs, while jointly preserving good performances on link prediction. In the last part of this thesis, we evaluate our methods on several graphs extracted from the music streaming service Deezer. We put the emphasis on graph-based music recommendation problems. In particular, we show that our methods can improve the detection of communities of similar musical items to recommend to users, that they can effectively rank similar artists in a cold start setting, and that they permit modeling the music genre perception across cultures.