Confounder selection may be efficiently conducted using penalized regression methods when causal effects are estimated from observational data with many variables. An outcome-adaptive lasso was proposed to build a model for the propensity score that can be employed in conjunction with other variable selection methods for the outcome model to apply the augmented inverse propensity weighted (AIPW) estimator. However, researchers may not know which method is optimal to use for outcome model when applying the AIPW estimator with the outcome-adaptive lasso. This study provided hints on readily implementable penalized regression methods that should be adopted for the outcome model as a counterpart of the outcome-adaptive lasso. We evaluated the bias and variance of the AIPW estimators using the propensity score (PS) model and an outcome model based on penalized regression methods under various conditions by analyzing a clinical trial example and numerical experiments; the estimates and standard errors of the AIPW estimators were almost identical in an example with over 5000 participants. The AIPW estimators using penalized regression methods with the oracle property performed well in terms of bias and variance in numerical experiments with smaller sample sizes. Meanwhile, the bias of the AIPW estimator using the ordinary lasso for the PS and outcome models was considerably larger.
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This contribution introduces a model order reduction approach for an advection-reaction problem with a parametrized reaction function. The underlying discretization uses an ultraweak formulation with an $L^2$-like trial space and an 'optimal' test space as introduced by Demkowicz et al. This ensures the stability of the discretization and in addition allows for a symmetric reformulation of the problem in terms of a dual solution which can also be interpreted as the normal equations of an adjoint least-squares problem. Classic model order reduction techniques can then be applied to the space of dual solutions which also immediately gives a reduced primal space. We show that the necessary computations do not require the reconstruction of any primal solutions and can instead be performed entirely on the space of dual solutions. We prove exponential convergence of the Kolmogorov $N$-width and show that a greedy algorithm produces quasi-optimal approximation spaces for both the primal and the dual solution space. Numerical experiments based on the benchmark problem of a catalytic filter confirm the applicability of the proposed method.
Weakly Supervised Semantic Segmentation (WSSS) employs weak supervision, such as image-level labels, to train the segmentation model. Despite the impressive achievement in recent WSSS methods, we identify that introducing weak labels with high mean Intersection of Union (mIoU) does not guarantee high segmentation performance. Existing studies have emphasized the importance of prioritizing precision and reducing noise to improve overall performance. In the same vein, we propose ORANDNet, an advanced ensemble approach tailored for WSSS. ORANDNet combines Class Activation Maps (CAMs) from two different classifiers to increase the precision of pseudo-masks (PMs). To further mitigate small noise in the PMs, we incorporate curriculum learning. This involves training the segmentation model initially with pairs of smaller-sized images and corresponding PMs, gradually transitioning to the original-sized pairs. By combining the original CAMs of ResNet-50 and ViT, we significantly improve the segmentation performance over the single-best model and the naive ensemble model, respectively. We further extend our ensemble method to CAMs from AMN (ResNet-like) and MCTformer (ViT-like) models, achieving performance benefits in advanced WSSS models. It highlights the potential of our ORANDNet as a final add-on module for WSSS models.
A major outstanding problem when interfacing with spinal motor neurons is how to accurately compensate for non-stationary effects in the signal during source separation routines, particularly when they cannot be estimated in advance. This forces current systems to instead use undifferentiated bulk signal, which limits the potential degrees of freedom for control. In this study we propose a potential solution, using an unsupervised learning algorithm to blindly correct for the effects of latent processes which drive the signal non-stationarities. We implement this methodology within the theoretical framework of a quasilinear version of independent component analysis (ICA). The proposed design, HarmonICA, sidesteps the identifiability problems of nonlinear ICA, allowing for equivalent predictability to linear ICA whilst retaining the ability to learn complex nonlinear relationships between non-stationary latents and their effects on the signal. We test HarmonICA on both invasive and non-invasive recordings both simulated and real, demonstrating an ability to blindly compensate for the non-stationary effects specific to each, and thus to significantly enhance the quality of a source separation routine.
In the study of extremes, the presence of asymptotic independence signifies that extreme events across multiple variables are probably less likely to occur together. Although well-understood in a bivariate context, the concept remains relatively unexplored when addressing the nuances of joint occurrence of extremes in higher dimensions. In this paper, we propose a notion of mutual asymptotic independence to capture the behavior of joint extremes in dimensions larger than two and contrast it with the classical notion of (pairwise) asymptotic independence. Furthermore, we define $k$-wise asymptotic independence which lies in between pairwise and mutual asymptotic independence. The concepts are compared using examples of Archimedean, Gaussian and Marshall-Olkin copulas among others. Notably, for the popular Gaussian copula, we provide explicit conditions on the correlation matrix for mutual asymptotic independence to hold; moreover, we are able to compute exact tail orders for various tail events.
A non-linear complex system governed by multi-spatial and multi-temporal physics scales cannot be fully understood with a single diagnostic, as each provides only a partial view and much information is lost during data extraction. Combining multiple diagnostics also results in imperfect projections of the system's physics. By identifying hidden inter-correlations between diagnostics, we can leverage mutual support to fill in these gaps, but uncovering these inter-correlations analytically is too complex. We introduce a groundbreaking machine learning methodology to address this issue. Our multimodal approach generates super resolution data encompassing multiple physics phenomena, capturing detailed structural evolution and responses to perturbations previously unobservable. This methodology addresses a critical problem in fusion plasmas: the Edge Localized Mode (ELM), a plasma instability that can severely damage reactor walls. One method to stabilize ELM is using resonant magnetic perturbation to trigger magnetic islands. However, low spatial and temporal resolution of measurements limits the analysis of these magnetic islands due to their small size, rapid dynamics, and complex interactions within the plasma. With super-resolution diagnostics, we can experimentally verify theoretical models of magnetic islands for the first time, providing unprecedented insights into their role in ELM stabilization. This advancement aids in developing effective ELM suppression strategies for future fusion reactors like ITER and has broader applications, potentially revolutionizing diagnostics in fields such as astronomy, astrophysics, and medical imaging.
This paper studies the influence of probabilism and non-determinism on some quantitative aspect X of the execution of a system modeled as a Markov decision process (MDP). To this end, the novel notion of demonic variance is introduced: For a random variable X in an MDP M, it is defined as 1/2 times the maximal expected squared distance of the values of X in two independent execution of M in which also the non-deterministic choices are resolved independently by two distinct schedulers. It is shown that the demonic variance is between 1 and 2 times as large as the maximal variance of X in M that can be achieved by a single scheduler. This allows defining a non-determinism score for M and X measuring how strongly the difference of X in two executions of M can be influenced by the non-deterministic choices. Properties of MDPs M with extremal values of the non-determinism score are established. Further, the algorithmic problems of computing the maximal variance and the demonic variance are investigated for two random variables, namely weighted reachability and accumulated rewards. In the process, also the structure of schedulers maximizing the variance and of scheduler pairs realizing the demonic variance is analyzed.
Triple periodic minimal surfaces (TPMS) have garnered significant interest due to their structural efficiency and controllable geometry, making them suitable for a wide range of applications. This paper investigates the relationships between porosity and persistence entropy with the shape factor of TPMS. We propose conjectures suggesting that these relationships are polynomial in nature, derived through the application of machine learning techniques. This study exemplifies the integration of machine learning methodologies in pure mathematical research. Besides the conjectures, we provide the mathematical models that might have the potential implications for the design and modeling of TPMS structures in various practical applications.
We describe a simple deterministic near-linear time approximation scheme for uncapacitated minimum cost flow in undirected graphs with real edge weights, a problem also known as transshipment. Specifically, our algorithm takes as input a (connected) undirected graph $G = (V, E)$, vertex demands $b \in \mathbb{R}^V$ such that $\sum_{v \in V} b(v) = 0$, positive edge costs $c \in \mathbb{R}_{>0}^E$, and a parameter $\varepsilon > 0$. In $O(\varepsilon^{-2} m \log^{O(1)} n)$ time, it returns a flow $f$ such that the net flow out of each vertex is equal to the vertex's demand and the cost of the flow is within a $(1 + \varepsilon)$ factor of optimal. Our algorithm is combinatorial and has no running time dependency on the demands or edge costs. With the exception of a recent result presented at STOC 2022 for polynomially bounded edge weights, all almost- and near-linear time approximation schemes for transshipment relied on randomization to embed the problem instance into low-dimensional space. Our algorithm instead deterministically approximates the cost of routing decisions that would be made if the input were subject to a random tree embedding. To avoid computing the $\Omega(n^2)$ vertex-vertex distances that an approximation of this kind suggests, we also take advantage of the clustering method used in the well-known Thorup-Zwick distance oracle.
With climate extremes' rising frequency and intensity, robust analytical tools are crucial to predict their impacts on terrestrial ecosystems. Machine learning techniques show promise but require well-structured, high-quality, and curated analysis-ready datasets. Earth observation datasets comprehensively monitor ecosystem dynamics and responses to climatic extremes, yet the data complexity can challenge the effectiveness of machine learning models. Despite recent progress in deep learning to ecosystem monitoring, there is a need for datasets specifically designed to analyse compound heatwave and drought extreme impact. Here, we introduce the DeepExtremeCubes database, tailored to map around these extremes, focusing on persistent natural vegetation. It comprises over 40,000 spatially sampled small data cubes (i.e. minicubes) globally, with a spatial coverage of 2.5 by 2.5 km. Each minicube includes (i) Sentinel-2 L2A images, (ii) ERA5-Land variables and generated extreme event cube covering 2016 to 2022, and (iii) ancillary land cover and topography maps. The paper aims to (1) streamline data accessibility, structuring, pre-processing, and enhance scientific reproducibility, and (2) facilitate biosphere dynamics forecasting in response to compound extremes.
Generalization theory has been established for sparse deep neural networks under high-dimensional regime. Beyond generalization, parameter estimation is also important since it is crucial for variable selection and interpretability of deep neural networks. Current theoretical studies concerning parameter estimation mainly focus on two-layer neural networks, which is due to the fact that the convergence of parameter estimation heavily relies on the regularity of the Hessian matrix, while the Hessian matrix of deep neural networks is highly singular. To avoid the unidentifiability of deep neural networks in parameter estimation, we propose to conduct nonparametric estimation of partial derivatives with respect to inputs. We first show that model convergence of sparse deep neural networks is guaranteed in that the sample complexity only grows with the logarithm of the number of parameters or the input dimension when the $\ell_{1}$-norm of parameters is well constrained. Then by bounding the norm and the divergence of partial derivatives, we establish that the convergence rate of nonparametric estimation of partial derivatives scales as $\mathcal{O}(n^{-1/4})$, a rate which is slower than the model convergence rate $\mathcal{O}(n^{-1/2})$. To the best of our knowledge, this study combines nonparametric estimation and parametric sparse deep neural networks for the first time. As nonparametric estimation of partial derivatives is of great significance for nonlinear variable selection, the current results show the promising future for the interpretability of deep neural networks.
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