We discuss structure-preserving time discretization for nonlinear port-Hamiltonian systems with state-dependent mass matrix. Such systems occur, for instance, in the context of structure-preserving nonlinear model order reduction for port-Hamiltonian systems and, in this context, structure-preserving time discretization is crucial for preserving some of the properties of the time-continuous reduced-order model. For this purpose, we introduce a new class of time discretization schemes which is based on so-called discrete gradient pairs and leads to an exact power balance on the time-discrete level. Moreover, for the special case of a pointwise symmetric and positive definite mass matrix, we present an explicit construction of a discrete gradient pair. Finally, we illustrate the theoretical findings by means of a numerical example, where the time-continuous system is a nonlinear reduced-order model for an advection-diffusion problem.
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Group equivariance ensures consistent responses to group transformations of the input, leading to more robust models and enhanced generalization capabilities. However, this property can lead to overly constrained models if the symmetries considered in the group differ from those observed in data. While common methods address this by determining the appropriate level of symmetry at the dataset level, they are limited to supervised settings and ignore scenarios in which multiple levels of symmetry co-exist in the same dataset. For instance, pictures of cars and planes exhibit different levels of rotation, yet both are included in the CIFAR-10 dataset. In this paper, we propose a method able to detect the level of symmetry of each input without the need for labels. To this end, we derive a sufficient and necessary condition to learn the distribution of symmetries in the data. Using the learned distribution, we generate pseudo-labels that allow us to learn the levels of symmetry of each input in a self-supervised manner. We validate the effectiveness of our approach on synthetic datasets with different per-class levels of symmetries e.g. MNISTMultiple, in which digits are uniformly rotated within a class-dependent interval. We demonstrate that our method can be used for practical applications such as the generation of standardized datasets in which the symmetries are not present, as well as the detection of out-of-distribution symmetries during inference. By doing so, both the generalization and robustness of non-equivariant models can be improved. Our code is publicly available at //github.com/aurban0/ssl-sym.
High-dimensional datasets often contain multiple meaningful clusterings in different subspaces. For example, objects can be clustered either by color, weight, or size, revealing different interpretations of the given dataset. A variety of approaches are able to identify such non-redundant clusterings. However, most of these methods require the user to specify the expected number of subspaces and clusters for each subspace. Stating these values is a non-trivial problem and usually requires detailed knowledge of the input dataset. In this paper, we propose a framework that utilizes the Minimum Description Length Principle (MDL) to detect the number of subspaces and clusters per subspace automatically. We describe an efficient procedure that greedily searches the parameter space by splitting and merging subspaces and clusters within subspaces. Additionally, an encoding strategy is introduced that allows us to detect outliers in each subspace. Extensive experiments show that our approach is highly competitive to state-of-the-art methods.
Existing out-of-distribution (OOD) methods have shown great success on balanced datasets but become ineffective in long-tailed recognition (LTR) scenarios where 1) OOD samples are often wrongly classified into head classes and/or 2) tail-class samples are treated as OOD samples. To address these issues, current studies fit a prior distribution of auxiliary/pseudo OOD data to the long-tailed in-distribution (ID) data. However, it is difficult to obtain such an accurate prior distribution given the unknowingness of real OOD samples and heavy class imbalance in LTR. A straightforward solution to avoid the requirement of this prior is to learn an outlier class to encapsulate the OOD samples. The main challenge is then to tackle the aforementioned confusion between OOD samples and head/tail-class samples when learning the outlier class. To this end, we introduce a novel calibrated outlier class learning (COCL) approach, in which 1) a debiased large margin learning method is introduced in the outlier class learning to distinguish OOD samples from both head and tail classes in the representation space and 2) an outlier-class-aware logit calibration method is defined to enhance the long-tailed classification confidence. Extensive empirical results on three popular benchmarks CIFAR10-LT, CIFAR100-LT, and ImageNet-LT demonstrate that COCL substantially outperforms state-of-the-art OOD detection methods in LTR while being able to improve the classification accuracy on ID data. Code is available at //github.com/mala-lab/COCL.
Reliable automatic hate speech (HS) detection systems must adapt to the in-flow of diverse new data to curtail hate speech. However, hate speech detection systems commonly lack generalizability in identifying hate speech dissimilar to data used in training, impeding their robustness in real-world deployments. In this work, we propose a hate speech generalization framework that leverages emotion knowledge in a multitask architecture to improve the generalizability of hate speech detection in a cross-domain setting. We investigate emotion corpora with varying emotion categorical scopes to determine the best corpus scope for supplying emotion knowledge to foster generalized hate speech detection. We further assess the relationship between using pretrained Transformers models adapted for hate speech and its effect on our emotion-enriched hate speech generalization model. We perform extensive experiments on six publicly available datasets sourced from different online domains and show that our emotion-enriched HS detection generalization method demonstrates consistent generalization improvement in cross-domain evaluation, increasing generalization performance up to 18.1% and average cross-domain performance up to 8.5%, according to the F1 measure.
Exact Bayesian inference on state-space models~(SSM) is in general untractable, and unfortunately, basic Sequential Monte Carlo~(SMC) methods do not yield correct approximations for complex models. In this paper, we propose a mixed inference algorithm that computes closed-form solutions using belief propagation as much as possible, and falls back to sampling-based SMC methods when exact computations fail. This algorithm thus implements automatic Rao-Blackwellization and is even exact for Gaussian tree models.
Many functions characterising physical systems are additively separable. This is the case, for instance, of mechanical Hamiltonian functions in physics, population growth equations in biology, and consumer preference and utility functions in economics. We consider the scenario in which a surrogate of a function is to be tested for additive separability. The detection that the surrogate is additively separable can be leveraged to improve further learning. Hence, it is beneficial to have the ability to test for such separability in surrogates. The mathematical approach is to test if the mixed partial derivative of the surrogate is zero; or empirically, lower than a threshold. We present and comparatively and empirically evaluate the eight methods to compute the mixed partial derivative of a surrogate function.
Latent variable models (LVMs) represent observed variables by parameterized functions of latent variables. Prominent examples of LVMs for unsupervised learning are probabilistic PCA or probabilistic SC which both assume a weighted linear summation of the latents to determine the mean of a Gaussian distribution for the observables. In many cases, however, observables do not follow a Gaussian distribution. For unsupervised learning, LVMs which assume specific non-Gaussian observables have therefore been considered. Already for specific choices of distributions, parameter optimization is challenging and only a few previous contributions considered LVMs with more generally defined observable distributions. Here, we consider LVMs that are defined for a range of different distributions, i.e., observables can follow any (regular) distribution of the exponential family. The novel class of LVMs presented is defined for binary latents, and it uses maximization in place of summation to link the latents to observables. To derive an optimization procedure, we follow an EM approach for maximum likelihood parameter estimation. We show that a set of very concise parameter update equations can be derived which feature the same functional form for all exponential family distributions. The derived generic optimization can consequently be applied to different types of metric data as well as to different types of discrete data. Also, the derived optimization equations can be combined with a recently suggested variational acceleration which is likewise generically applicable to the LVMs considered here. So, the combination maintains generic and direct applicability of the derived optimization procedure, but, crucially, enables efficient scalability. We numerically verify our analytical results and discuss some potential applications such as learning of variance structure, noise type estimation and denoising.
Krylov subspace, which is generated by multiplying a given vector by the matrix of a linear transformation and its successive powers, has been extensively studied in classical optimization literature to design algorithms that converge quickly for large linear inverse problems. For example, the conjugate gradient method (CG), one of the most popular Krylov subspace methods, is based on the idea of minimizing the residual error in the Krylov subspace. However, with the recent advancement of high-performance diffusion solvers for inverse problems, it is not clear how classical wisdom can be synergistically combined with modern diffusion models. In this study, we propose a novel and efficient diffusion sampling strategy that synergistically combine the diffusion sampling and Krylov subspace methods. Specifically, we prove that if the tangent space at a denoised sample by Tweedie's formula forms a Krylov subspace, then the CG initialized with the denoised data ensures the data consistency update to remain in the tangent space. This negates the need to compute the manifold-constrained gradient (MCG), leading to a more efficient diffusion sampling method. Our method is applicable regardless of the parametrization and setting (i.e., VE, VP). Notably, we achieve state-of-the-art reconstruction quality on challenging real-world medical inverse imaging problems, including multi-coil MRI reconstruction and 3D CT reconstruction. Moreover, our proposed method achieves more than 80 times faster inference time than the previous state-of-the-art method.
We study the problem of designing hard negative sampling distributions for unsupervised contrastive representation learning. We propose and analyze a novel min-max framework that seeks a representation which minimizes the maximum (worst-case) generalized contrastive learning loss over all couplings (joint distributions between positive and negative samples subject to marginal constraints) and prove that the resulting min-max optimum representation will be degenerate. This provides the first theoretical justification for incorporating additional regularization constraints on the couplings. We re-interpret the min-max problem through the lens of Optimal Transport (OT) theory and utilize regularized transport couplings to control the degree of hardness of negative examples. Through experiments we demonstrate that the negative samples generated from our designed negative distribution are more similar to the anchor than those generated from the baseline negative distribution. We also demonstrate that entropic regularization yields negative sampling distributions with parametric form similar to that in a recent state-of-the-art negative sampling design and has similar performance in multiple datasets. Utilizing the uncovered connection with OT, we propose a new ground cost for designing the negative distribution and show improved performance of the learned representation on downstream tasks compared to the representation learned when using squared Euclidean cost.
Image segmentation is an important component of many image understanding systems. It aims to group pixels in a spatially and perceptually coherent manner. Typically, these algorithms have a collection of parameters that control the degree of over-segmentation produced. It still remains a challenge to properly select such parameters for human-like perceptual grouping. In this work, we exploit the diversity of segments produced by different choices of parameters. We scan the segmentation parameter space and generate a collection of image segmentation hypotheses (from highly over-segmented to under-segmented). These are fed into a cost minimization framework that produces the final segmentation by selecting segments that: (1) better describe the natural contours of the image, and (2) are more stable and persistent among all the segmentation hypotheses. We compare our algorithm's performance with state-of-the-art algorithms, showing that we can achieve improved results. We also show that our framework is robust to the choice of segmentation kernel that produces the initial set of hypotheses.
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