Group sparsity in Machine Learning (ML) encourages simpler, more interpretable models with fewer active parameter groups. This work aims to incorporate structured group sparsity into the shared parameters of a Multi-Task Learning (MTL) framework, to develop parsimonious models that can effectively address multiple tasks with fewer parameters while maintaining comparable or superior performance to a dense model. Sparsifying the model during training helps decrease the model's memory footprint, computation requirements, and prediction time during inference. We use channel-wise l1/l2 group sparsity in the shared layers of the Convolutional Neural Network (CNN). This approach not only facilitates the elimination of extraneous groups (channels) but also imposes a penalty on the weights, thereby enhancing the learning of all tasks. We compare the outcomes of single-task and multi-task experiments under group sparsity on two publicly available MTL datasets, NYU-v2 and CelebAMask-HQ. We also investigate how changing the sparsification degree impacts both the performance of the model and the sparsity of groups.
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A comprehensive mathematical model of the multiphysics flow of blood and Cerebrospinal Fluid (CSF) in the brain can be expressed as the coupling of a poromechanics system and Stokes' equations: the first describes fluids filtration through the cerebral tissue and the tissue's elastic response, while the latter models the flow of the CSF in the brain ventricles. This model describes the functioning of the brain's waste clearance mechanism, which has been recently discovered to play an essential role in the progress of neurodegenerative diseases. To model the interactions between different scales in the porous medium, we propose a physically consistent coupling between Multi-compartment Poroelasticity (MPE) equations and Stokes' equations. In this work, we introduce a numerical scheme for the discretization of such coupled MPE-Stokes system, employing a high-order discontinuous Galerkin method on polytopal grids to efficiently account for the geometric complexity of the domain. We analyze the stability and convergence of the space semidiscretized formulation, we prove a-priori error estimates, and we present a temporal discretization based on a combination of Newmark's $\beta$-method for the elastic wave equation and the $\theta$-method for the other equations of the model. Numerical simulations carried out on test cases with manufactured solutions validate the theoretical error estimates. We also present numerical results on a two-dimensional slice of a patient-specific brain geometry reconstructed from diagnostic images, to test in practice the advantages of the proposed approach.
We propose a Monte Carlo method to efficiently find, count, and sample abstract triangulations of a given manifold M. The method is based on a biased random walk through all possible triangulations of M (in the Pachner graph), constructed by combining (bi-stellar) moves with suitable chosen accept/reject probabilities (Metropolis-Hastings). Asymptotically, the method guarantees that samples of triangulations are drawn at random from a chosen probability. This enables us not only to sample (rare) triangulations of particular interest but also to estimate the (extremely small) probability of obtaining them when isomorphism types of triangulations are sampled uniformly at random. We implement our general method for surface triangulations and 1-vertex triangulations of 3-manifolds. To showcase its usefulness, we present a number of experiments: (a) we recover asymptotic growth rates for the number of isomorphism types of simplicial triangulations of the 2-dimensional sphere; (b) we experimentally observe that the growth rate for the number of isomorphism types of 1-vertex triangulations of the 3-dimensional sphere appears to be singly exponential in the number of their tetrahedra; and (c) we present experimental evidence that a randomly chosen isomorphism type of 1-vertex n-tetrahedra 3-sphere triangulation, for n tending to infinity, almost surely shows a fixed edge-degree distribution which decays exponentially for large degrees, but shows non-monotonic behaviour for small degrees.
Box-supervised polyp segmentation attracts increasing attention for its cost-effective potential. Existing solutions often rely on learning-free methods or pretrained models to laboriously generate pseudo masks, triggering Dice constraint subsequently. In this paper, we found that a model guided by the simplest box-filled masks can accurately predict polyp locations/sizes, but suffers from shape collapsing. In response, we propose two innovative learning fashions, Improved Box-dice (IBox) and Contrastive Latent-Anchors (CLA), and combine them to train a robust box-supervised model IBoxCLA. The core idea behind IBoxCLA is to decouple the learning of location/size and shape, allowing for focused constraints on each of them. Specifically, IBox transforms the segmentation map into a proxy map using shape decoupling and confusion-region swapping sequentially. Within the proxy map, shapes are disentangled, while locations/sizes are encoded as box-like responses. By constraining the proxy map instead of the raw prediction, the box-filled mask can well supervise IBoxCLA without misleading its shape learning. Furthermore, CLA contributes to shape learning by generating two types of latent anchors, which are learned and updated using momentum and segmented polyps to steadily represent polyp and background features. The latent anchors facilitate IBoxCLA to capture discriminative features within and outside boxes in a contrastive manner, yielding clearer boundaries. We benchmark IBoxCLA on five public polyp datasets. The experimental results demonstrate the competitive performance of IBoxCLA compared to recent fully-supervised polyp segmentation methods, and its superiority over other box-supervised state-of-the-arts with a relative increase of overall mDice and mIoU by at least 6.5% and 7.5%, respectively.
Remotely sensed data are dominated by mixed Land Use and Land Cover (LULC) types. Spectral unmixing is a technique to extract information from mixed pixels into their constituent LULC types and corresponding abundance fractions. Traditionally, solving this task has relied on either classical methods that require prior knowledge of endmembers or machine learning methods that avoid explicit endmembers calculation, also known as blind spectral unmixing (BSU). Most BSU studies based on Deep Learning (DL) focus on one time-step hyperspectral data, yet its acquisition remains quite costly compared with multispectral data. To our knowledge, here we provide the first study on BSU of LULC classes using multispectral time series data with DL models. We further boost the performance of a Long-Short Term Memory (LSTM)-based model by incorporating geographic plus topographic (geo-topographic) and climatic ancillary information. Our experiments show that combining spectral-temporal input data together with geo-topographic and climatic information substantially improves the abundance estimation of LULC classes in mixed pixels. To carry out this study, we built a new labeled dataset of the region of Andalusia (Spain) with monthly multispectral time series of pixels for the year 2013 from MODIS at 460m resolution, for two hierarchical levels of LULC classes, named Andalusia MultiSpectral MultiTemporal Unmixing (Andalusia-MSMTU). This dataset provides, at the pixel level, a multispectral time series plus ancillary information annotated with the abundance of each LULC class inside each pixel. The dataset and code are available to the public.
This paper presents a new weak Galerkin (WG) method for elliptic interface problems on general curved polygonal partitions. The method's key innovation lies in its ability to transform the complex interface jump condition into a more manageable Dirichlet boundary condition, simplifying the theoretical analysis significantly. The numerical scheme is designed by using locally constructed weak gradient on the curved polygonal partitions. We establish error estimates of optimal order for the numerical approximation in both discrete $H^1$ and $L^2$ norms. Additionally, we present various numerical results that serve to illustrate the robust numerical performance of the proposed WG interface method.
One of the main challenges for interpreting black-box models is the ability to uniquely decompose square-integrable functions of non-mutually independent random inputs into a sum of functions of every possible subset of variables. However, dealing with dependencies among inputs can be complicated. We propose a novel framework to study this problem, linking three domains of mathematics: probability theory, functional analysis, and combinatorics. We show that, under two reasonable assumptions on the inputs (non-perfect functional dependence and non-degenerate stochastic dependence), it is always possible to decompose uniquely such a function. This ``canonical decomposition'' is relatively intuitive and unveils the linear nature of non-linear functions of non-linearly dependent inputs. In this framework, we effectively generalize the well-known Hoeffding decomposition, which can be seen as a particular case. Oblique projections of the black-box model allow for novel interpretability indices for evaluation and variance decomposition. Aside from their intuitive nature, the properties of these novel indices are studied and discussed. This result offers a path towards a more precise uncertainty quantification, which can benefit sensitivity analyses and interpretability studies, whenever the inputs are dependent. This decomposition is illustrated analytically, and the challenges to adopting these results in practice are discussed.
We present a coordination method for multiple mobile manipulators to sort objects in clutter. We consider the object rearrangement problem in which the objects must be sorted into different groups in a particular order. In clutter, the order constraints could not be easily satisfied since some objects occlude other objects so the occluded ones are not directly accessible to the robots. Those objects occluding others need to be moved more than once to make the occluded objects accessible. Such rearrangement problems fall into the class of nonmonotone rearrangement problems which are computationally intractable. While the nonmonotone problems with order constraints are harder, involving with multiple robots requires another computation for task allocation. The proposed method first finds a sequence of objects to be sorted using a search such that the order constraint in each group is satisfied. The search can solve nonmonotone instances that require temporal relocation of some objects to access the next object to be sorted. Once a complete sorting sequence is found, the objects in the sequence are assigned to multiple mobile manipulators using a greedy allocation method. We develop four versions of the method with different search strategies. In the experiments, we show that our method can find a sorting sequence quickly (e.g., 4.6 sec with 20 objects sorted into five groups) even though the solved instances include hard nonmonotone ones. The extensive tests and the experiments in simulation show the ability of the method to solve the real-world sorting problem using multiple mobile manipulators.
Controlling spurious oscillations is crucial for designing reliable numerical schemes for hyperbolic conservation laws. This paper proposes a novel, robust, and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method on general meshes, motivated by the damping technique in [Lu, Liu, and Shu, SIAM J. Numer. Anal., 59:1299-1324, 2021]. The OEDG method incorporates an OE procedure after each Runge-Kutta stage, devised by alternately evolving conventional semidiscrete DG scheme and a damping equation. A novel damping operator is carefully designed to possess scale-invariant and evolution-invariant properties. We rigorously prove optimal error estimates of the fully discrete OEDG method for linear scalar conservation laws. This might be the first generic fully-discrete error estimates for nonlinear DG schemes with automatic oscillation control mechanism. The OEDG method exhibits many notable advantages. It effectively eliminates spurious oscillations for challenging problems across various scales and wave speeds, without problem-specific parameters. It obviates the need for characteristic decomposition in hyperbolic systems. It retains key properties of conventional DG method, such as conservation, optimal convergence rates, and superconvergence. Moreover, it remains stable under normal CFL condition. The OE procedure is non-intrusive, facilitating integration into existing DG codes as an independent module. Its implementation is easy and efficient, involving only simple multiplications of modal coefficients by scalars. The OEDG approach provides new insights into the damping mechanism for oscillation control. It reveals the role of damping operator as a modal filter and establishes close relations between the damping and spectral viscosity techniques. Extensive numerical results confirm the theoretical analysis and validate the effectiveness and advantages of the OEDG method.
Discovering causal relationships from observational data is a fundamental yet challenging task. Invariant causal prediction (ICP, Peters et al., 2016) is a method for causal feature selection which requires data from heterogeneous settings and exploits that causal models are invariant. ICP has been extended to general additive noise models and to nonparametric settings using conditional independence tests. However, the latter often suffer from low power (or poor type I error control) and additive noise models are not suitable for applications in which the response is not measured on a continuous scale, but reflects categories or counts. Here, we develop transformation-model (TRAM) based ICP, allowing for continuous, categorical, count-type, and uninformatively censored responses (these model classes, generally, do not allow for identifiability when there is no exogenous heterogeneity). As an invariance test, we propose TRAM-GCM based on the expected conditional covariance between environments and score residuals with uniform asymptotic level guarantees. For the special case of linear shift TRAMs, we also consider TRAM-Wald, which tests invariance based on the Wald statistic. We provide an open-source R package 'tramicp' and evaluate our approach on simulated data and in a case study investigating causal features of survival in critically ill patients.
In large-scale systems there are fundamental challenges when centralised techniques are used for task allocation. The number of interactions is limited by resource constraints such as on computation, storage, and network communication. We can increase scalability by implementing the system as a distributed task-allocation system, sharing tasks across many agents. However, this also increases the resource cost of communications and synchronisation, and is difficult to scale. In this paper we present four algorithms to solve these problems. The combination of these algorithms enable each agent to improve their task allocation strategy through reinforcement learning, while changing how much they explore the system in response to how optimal they believe their current strategy is, given their past experience. We focus on distributed agent systems where the agents' behaviours are constrained by resource usage limits, limiting agents to local rather than system-wide knowledge. We evaluate these algorithms in a simulated environment where agents are given a task composed of multiple subtasks that must be allocated to other agents with differing capabilities, to then carry out those tasks. We also simulate real-life system effects such as networking instability. Our solution is shown to solve the task allocation problem to 6.7% of the theoretical optimal within the system configurations considered. It provides 5x better performance recovery over no-knowledge retention approaches when system connectivity is impacted, and is tested against systems up to 100 agents with less than a 9% impact on the algorithms' performance.
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