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In this paper, we set the mathematical foundations of the Dynamical Low-Rank Approximation (DLRA) method for stochastic differential equations (SDEs). DLRA aims at approximating the solution as a linear combination of a small number of basis vectors with random coefficients (low rank format) with the peculiarity that both the basis vectors and the random coefficients vary in time. While the formulation and properties of DLRA are now well understood for random/parametric equations, the same cannot be said for SDEs and this work aims to fill this gap. We start by rigorously formulating a Dynamically Orthogonal (DO) approximation (an instance of DLRA successfully used in applications) for SDEs, which we then generalize to define a parametrization independent DLRA for SDEs. We show local well-posedness of the DO equations and their equivalence with the DLRA formulation. We also characterize the explosion time of the DO solution by a loss of linear independence of the random coefficients defining the solution expansion and give sufficient conditions for global existence.

In this paper, we employ a Bayesian approach to assess the reliability of a critical component in the Mars Sample Return program, focusing on the Earth Entry System's risk of containment not assured upon reentry. Our study uses Gaussian Process modeling under a Bayesian regime to analyze the Earth Entry System's resilience against operational stress. This Bayesian framework allows for a detailed probabilistic evaluation of the risk of containment not assured, indicating the feasibility of meeting the mission's stringent safety goal of 0.999999 probability of success. The findings underscore the effectiveness of Bayesian methods for complex uncertainty quantification analyses of computer simulations, providing valuable insights for computational reliability analysis in a risk-averse setting.

In this work, we propose a simple yet effective method to tackle the problem of imbalanced multi-class semantic segmentation in deep learning systems. One of the key properties for a good training set is the balancing among the classes. When the input distribution is heavily imbalanced in the number of instances, the learning process could be hindered or difficult to carry on. To this end, we propose a Dynamic Label Injection (DLI) algorithm to impose a uniform distribution in the input batch. Our algorithm computes the current batch defect distribution and re-balances it by transferring defects using a combination of Poisson-based seamless image cloning and cut-paste techniques. A thorough experimental section on the Magnetic Tiles dataset shows better results of DLI compared to other balancing loss approaches also in the challenging weakly-supervised setup. The code is available at //github.com/covisionlab/dynamic-label-injection.git

In this paper, we propose a novel method for detecting DeepFakes, enhancing the generalization of detection through semantic decoupling. There are now multiple DeepFake forgery technologies that not only possess unique forgery semantics but may also share common forgery semantics. The unique forgery semantics and irrelevant content semantics may promote over-fitting and hamper generalization for DeepFake detectors. For our proposed method, after decoupling, the common forgery semantics could be extracted from DeepFakes, and subsequently be employed for developing the generalizability of DeepFake detectors. Also, to pursue additional generalizability, we designed an adaptive high-pass module and a two-stage training strategy to improve the independence of decoupled semantics. Evaluation on FF++, Celeb-DF, DFD, and DFDC datasets showcases our method's excellent detection and generalization performance. Code is available at: //github.com/leaffeall/DFS-GDD.

In this paper, we propose an information geometry (IG) framework to solve the standard linear regression problem. The proposed framework is an extension of the one for computing the mean of complex multivariate Gaussian distribution. By applying the proposed framework, the information geometry approach (IGA) and the approximate information geometry approach (AIGA) for basis pursuit de-noising (BPDN) in standard linear regression are derived. The framework can also be applied to other standard linear regression problems. With the transformations of natural and expectation parameters of Gaussian distributions, we then show the relationship between the IGA and the message passing (MP) algorithm. Finally, we prove that the AIGA is equivalent to the approximate message passing (AMP) algorithm. These intrinsic results offer a new perspective for the AMP algorithm, and clues for understanding and improving stochastic reasoning methods.

In this work, we explore the intersection of sparse coding theory and deep learning to enhance our understanding of feature extraction capabilities in advanced neural network architectures. We begin by introducing a novel class of Deep Sparse Coding (DSC) models and establish a thorough theoretical analysis of their uniqueness and stability properties. By applying iterative algorithms to these DSC models, we derive convergence rates for convolutional neural networks (CNNs) in their ability to extract sparse features. This provides a strong theoretical foundation for the use of CNNs in sparse feature learning tasks. We additionally extend this convergence analysis to more general neural network architectures, including those with diverse activation functions, as well as self-attention and transformer-based models. This broadens the applicability of our findings to a wide range of deep learning methods for deep sparse feature extraction. Inspired by the strong connection between sparse coding and CNNs, we also explore training strategies to encourage neural networks to learn more sparse features. Through numerical experiments, we demonstrate the effectiveness of these approaches, providing valuable insights for the design of efficient and interpretable deep learning models.

In this paper, we propose a data-driven method to learn interpretable topological features of biomolecular data and demonstrate the efficacy of parsimonious models trained on topological features in predicting the stability of synthetic mini proteins. We compare models that leverage automatically-learned structural features against models trained on a large set of biophysical features determined by subject-matter experts (SME). Our models, based only on topological features of the protein structures, achieved 92%-99% of the performance of SME-based models in terms of the average precision score. By interrogating model performance and feature importance metrics, we extract numerous insights that uncover high correlations between topological features and SME features. We further showcase how combining topological features and SME features can lead to improved model performance over either feature set used in isolation, suggesting that, in some settings, topological features may provide new discriminating information not captured in existing SME features that are useful for protein stability prediction.

In this paper, we introduce a low-cost and low-power tiny supervised on-device learning (ODL) core that can address the distributional shift of input data for human activity recognition. Although ODL for resource-limited edge devices has been studied recently, how exactly to provide the training labels to these devices at runtime remains an open-issue. To address this problem, we propose to combine an automatic data pruning with supervised ODL to reduce the number queries needed to acquire predicted labels from a nearby teacher device and thus save power consumption during model retraining. The data pruning threshold is automatically tuned, eliminating a manual threshold tuning. As a tinyML solution at a few mW for the human activity recognition, we design a supervised ODL core that supports our automatic data pruning using a 45nm CMOS process technology. We show that the required memory size for the core is smaller than the same-shaped multilayer perceptron (MLP) and the power consumption is only 3.39mW. Experiments using a human activity recognition dataset show that the proposed automatic data pruning reduces the communication volume by 55.7% and power consumption accordingly with only 0.9% accuracy loss.

In this paper, we introduce the Reinforced Mnemonic Reader for machine reading comprehension tasks, which enhances previous attentive readers in two aspects. First, a reattention mechanism is proposed to refine current attentions by directly accessing to past attentions that are temporally memorized in a multi-round alignment architecture, so as to avoid the problems of attention redundancy and attention deficiency. Second, a new optimization approach, called dynamic-critical reinforcement learning, is introduced to extend the standard supervised method. It always encourages to predict a more acceptable answer so as to address the convergence suppression problem occurred in traditional reinforcement learning algorithms. Extensive experiments on the Stanford Question Answering Dataset (SQuAD) show that our model achieves state-of-the-art results. Meanwhile, our model outperforms previous systems by over 6% in terms of both Exact Match and F1 metrics on two adversarial SQuAD datasets.

In this paper, we propose a conceptually simple and geometrically interpretable objective function, i.e. additive margin Softmax (AM-Softmax), for deep face verification. In general, the face verification task can be viewed as a metric learning problem, so learning large-margin face features whose intra-class variation is small and inter-class difference is large is of great importance in order to achieve good performance. Recently, Large-margin Softmax and Angular Softmax have been proposed to incorporate the angular margin in a multiplicative manner. In this work, we introduce a novel additive angular margin for the Softmax loss, which is intuitively appealing and more interpretable than the existing works. We also emphasize and discuss the importance of feature normalization in the paper. Most importantly, our experiments on LFW BLUFR and MegaFace show that our additive margin softmax loss consistently performs better than the current state-of-the-art methods using the same network architecture and training dataset. Our code has also been made available at //github.com/happynear/AMSoftmax