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Many applications, such as optimization, uncertainty quantification and inverse problems, require repeatedly performing simulations of large-dimensional physical systems for different choices of parameters. This can be prohibitively expensive. In order to save computational cost, one can construct surrogate models by expressing the system in a low-dimensional basis, obtained from training data. This is referred to as model reduction. Past investigations have shown that, when performing model reduction of Hamiltonian systems, it is crucial to preserve the symplectic structure associated with the system in order to ensure long-term numerical stability. Up to this point structure-preserving reductions have largely been limited to linear transformations. We propose a new neural network architecture in the spirit of autoencoders, which are established tools for dimension reduction and feature extraction in data science, to obtain more general mappings. In order to train the network, a non-standard gradient descent approach is applied that leverages the differential-geometric structure emerging from the network design. The new architecture is shown to significantly outperform existing designs in accuracy.

The enormous amount of data to be represented using large graphs exceeds in some cases the resources of a conventional computer. Edges in particular can take up a considerable amount of memory as compared to the number of nodes. However, rigorous edge storage might not always be essential to be able to draw the needed conclusions. A similar problem takes records with many variables and attempts to extract the most discernible features. It is said that the ``dimension'' of this data is reduced. Following an approach with the same objective in mind, we can map a graph representation to a $k$-dimensional space and answer queries of neighboring nodes mainly by measuring Euclidean distances. The accuracy of our answers would decrease but would be compensated for by fuzzy logic which gives an idea about the likelihood of error. This method allows for reasonable representation in memory while maintaining a fair amount of useful information, and allows for concise embedding in $k$-dimensional Euclidean space as well as solving some problems without having to decompress the graph. Of particular interest is the case where $k=2$. Promising highly accurate experimental results are obtained and reported.

We characterize a number of well known systems of approximate inference as loss models: lax sections of 2-fibrations of statistical games, constructed by attaching internally-defined loss functions to Bayesian lenses. Our examples include the relative entropy, which constitutes a strict section, and whose chain rule is formalized by the horizontal composition of the 2-fibration. In order to capture this compositional structure, we first introduce the notion of 'copy-composition', alongside corresponding bicategories through which the composition of copy-discard categories factorizes. These bicategories are a variant of the Copara construction, and so we additionally introduce coparameterized Bayesian lenses, proving that coparameterized Bayesian updates compose optically, as in the non-coparameterized case.

Minimal and efficient graph representations are key to store, communicate, and sample the search space of graphs and networks while meeting user-defined criteria. In this paper, we investigate the feasibility of gradient-free optimization heuristics based on Differential Evolution to search for minimal integer representations of undirected graphs. The class of Differential Evolution algorithms are population-based gradient-free optimization heuristics having found a relevant attention in the nonconvex and nonlinear optimization communities. Our computational experiments using eight classes of Differential Evolution schemes and graph instances with varying degrees of sparsity have shown the merit of attaining minimal numbers for graph encoding/representation rendered by exploration-oriented strategies within few function evaluations. Our results have the potential to elucidate new number-based encoding and sample-based algorithms for graph representation, network design and optimization.

Causal abstraction (CA) theory establishes formal criteria for relating multiple structural causal models (SCMs) at different levels of granularity by defining maps between them. These maps have significant relevance for real-world challenges such as synthesizing causal evidence from multiple experimental environments, learning causally consistent representations at different resolutions, and linking interventions across multiple SCMs. In this work, we propose COTA, the first method to learn abstraction maps from observational and interventional data without assuming complete knowledge of the underlying SCMs. In particular, we introduce a multi-marginal Optimal Transport (OT) formulation that enforces do-calculus causal constraints, together with a cost function that relies on interventional information. We extensively evaluate COTA on synthetic and real world problems, and showcase its advantages over non-causal, independent and aggregated COTA formulations. Finally, we demonstrate the efficiency of our method as a data augmentation tool by comparing it against the state-of-the-art CA learning framework, which assumes fully specified SCMs, on a real-world downstream task.

Local explanation methods highlight the input tokens that have a considerable impact on the outcome of classifying the document at hand. For example, the Anchor algorithm applies a statistical analysis of the sensitivity of the classifier to changes in the token. Aggregating local explanations over a dataset provides a global explanation of the model. Such aggregation aims to detect words with the most impact, giving valuable insights about the model, like what it has learned in training and which adversarial examples expose its weaknesses. However, standard aggregation methods bear a high computational cost: a na\"ive implementation applies a costly algorithm to each token of each document, and hence, it is infeasible for a simple user running in the scope of a short analysis session. % We devise techniques for accelerating the global aggregation of the Anchor algorithm. Specifically, our goal is to compute a set of top-$k$ words with the highest global impact according to different aggregation functions. Some of our techniques are lossless and some are lossy. We show that for a very mild loss of quality, we are able to accelerate the computation by up to 30$\times$, reducing the computation from hours to minutes. We also devise and study a probabilistic model that accounts for noise in the Anchor algorithm and diminishes the bias toward words that are frequent yet low in impact.

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.

This work considers the question of how convenient access to copious data impacts our ability to learn causal effects and relations. In what ways is learning causality in the era of big data different from -- or the same as -- the traditional one? To answer this question, this survey provides a comprehensive and structured review of both traditional and frontier methods in learning causality and relations along with the connections between causality and machine learning. This work points out on a case-by-case basis how big data facilitates, complicates, or motivates each approach.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.

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.