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Graph Convolutional Networks (GCNs) are one of the most popular architectures that are used to solve classification problems accompanied by graphical information. We present a rigorous theoretical understanding of the effects of graph convolutions in multi-layer networks. We study these effects through the node classification problem of a non-linearly separable Gaussian mixture model coupled with a stochastic block model. First, we show that a single graph convolution expands the regime of the distance between the means where multi-layer networks can classify the data by a factor of at least $1/\sqrt[4]{\mathbb{E}{\rm deg}}$, where $\mathbb{E}{\rm deg}$ denotes the expected degree of a node. Second, we show that with a slightly stronger graph density, two graph convolutions improve this factor to at least $1/\sqrt[4]{n}$, where $n$ is the number of nodes in the graph. Finally, we provide both theoretical and empirical insights into the performance of graph convolutions placed in different combinations among the layers of a network, concluding that the performance is mutually similar for all combinations of the placement. We present extensive experiments on both synthetic and real-world data that illustrate our results.

We present a sheaf-theoretic construction of shape space -- the space of all shapes. We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transform (PHT). Recent results that build on fundamental work of Schapira have shown that this transform is injective, thus making the PHT a good summary object for each shape. Our homotopy sheaf result allows us to "glue" PHTs of different shapes together to build up the PHT of a larger shape. In the case where our shape is a polyhedron we prove a generalized nerve lemma for the PHT. Finally, by re-examining the sampling result of Smale-Niyogi-Weinberger, we show that we can reliably approximate the PHT of a manifold by a polyhedron up to arbitrary precision.

There currently are two main approaches to reproducing visual appearance using Machine Learning (ML): The first is training models that generalize over different instances of a problem, e.g., different images from a dataset. Such models learn priors over the data corpus and use this knowledge to provide fast inference with little input, often as a one-shot operation. However, this generality comes at the cost of fidelity, as such methods often struggle to achieve the final quality required. The second approach does not train a model that generalizes across the data, but overfits to a single instance of a problem, e.g., a flash image of a material. This produces detailed and high-quality results, but requires time-consuming training and is, as mere non-linear function fitting, unable to exploit previous experience. Techniques such as fine-tuning or auto-decoders combine both approaches but are sequential and rely on per-exemplar optimization. We suggest to combine both techniques end-to-end using meta-learning: We over-fit onto a single problem instance in an inner loop, while also learning how to do so efficiently in an outer-loop that builds intuition over many optimization runs. We demonstrate this concept to be versatile and efficient, applying it to RGB textures, Bi-directional Reflectance Distribution Functions (BRDFs), or Spatially-varying BRDFs (svBRDFs).

We study the problem of testing whether a function $f: \mathbb{R}^n \to \mathbb{R}$ is a polynomial of degree at most $d$ in the \emph{distribution-free} testing model. Here, the distance between functions is measured with respect to an unknown distribution $\mathcal{D}$ over $\mathbb{R}^n$ from which we can draw samples. In contrast to previous work, we do not assume that $\mathcal{D}$ has finite support. We design a tester that given query access to $f$, and sample access to $\mathcal{D}$, makes $(d/\varepsilon)^{O(1)}$ many queries to $f$, accepts with probability $1$ if $f$ is a polynomial of degree $d$, and rejects with probability at least $2/3$ if every degree-$d$ polynomial $P$ disagrees with $f$ on a set of mass at least $\varepsilon$ with respect to $\mathcal{D}$. Our result also holds under mild assumptions when we receive only a polynomial number of bits of precision for each query to $f$, or when $f$ can only be queried on rational points representable using a logarithmic number of bits. Along the way, we prove a new stability theorem for multivariate polynomials that may be of independent interest.

Modern optimization strategies such as evolutionary algorithms, ant colony algorithms, Bayesian optimization techniques, etc.~come with several parameters that steer their behavior during the optimization process. To obtain high-performing algorithm instances, automated algorithm configuration techniques have been developed. One of the most popular tools is irace, which evaluates configurations in sequential races, making use of iterated statistical tests to discard poorly performing configurations. At the end of the race, a set of elite configurations are selected from those survivor configurations which were not discarded, using greedy truncation selection. We study two alternative selection methods: one keeps the best survivor and selects the remaining configurations uniformly at random from the set of survivors while the other applies entropy to maximize the diversity of the elites. These methods are tested for tuning ant colony optimization algorithms for traveling salesperson problems and the quadratic assignment problem and tuning an exact tree search solver for satisfiability problems. The experimental results show improvement on the tested benchmarks compared to the default selection of irace. In addition, the obtained results indicate that non-elitist can obtain diverse algorithm configurations, which encourages us to explore a wider range of solutions to understand the behavior of algorithms.

In this paper, we investigate the problem of Semantic Segmentation for agricultural aerial imagery. We observe that the existing methods used for this task are designed without considering two characteristics of the aerial data: (i) the top-down perspective implies that the model cannot rely on a fixed semantic structure of the scene, because the same scene may be experienced with different rotations of the sensor; (ii) there can be a strong imbalance in the distribution of semantic classes because the relevant objects of the scene may appear at extremely different scales (e.g., a field of crops and a small vehicle). We propose a solution to these problems based on two ideas: (i) we use together a set of suitable augmentation and a consistency loss to guide the model to learn semantic representations that are invariant to the photometric and geometric shifts typical of the top-down perspective (Augmentation Invariance); (ii) we use a sampling method (Adaptive Sampling) that selects the training images based on a measure of pixel-wise distribution of classes and actual network confidence. With an extensive set of experiments conducted on the Agriculture-Vision dataset, we demonstrate that our proposed strategies improve the performance of the current state-of-the-art method.

This paper proposes ResTv2, a simpler, faster, and stronger multi-scale vision Transformer for visual recognition. ResTv2 simplifies the EMSA structure in ResTv1 (i.e., eliminating the multi-head interaction part) and employs an upsample operation to reconstruct the lost medium- and high-frequency information caused by the downsampling operation. In addition, we explore different techniques for better apply ResTv2 backbones to downstream tasks. We found that although combining EMSAv2 and window attention can greatly reduce the theoretical matrix multiply FLOPs, it may significantly decrease the computation density, thus causing lower actual speed. We comprehensively validate ResTv2 on ImageNet classification, COCO detection, and ADE20K semantic segmentation. Experimental results show that the proposed ResTv2 can outperform the recently state-of-the-art backbones by a large margin, demonstrating the potential of ResTv2 as solid backbones. The code and models will be made publicly available at \url{//github.com/wofmanaf/ResT}

In a sports competition, a team might lose a powerful incentive to exert full effort if its final rank does not depend on the outcome of the matches still to be played. Therefore, the organiser should reduce the probability of such a situation to the extent possible. Our paper provides a classification scheme to identify these weakly (where one team is indifferent) or strongly (where both teams are indifferent) stakeless games. A statistical model is estimated to simulate the UEFA Champions League groups and compare the candidate schedules used in the 2021/22 season according to the competitiveness of the matches played in the last round(s). The option followed in four of the eight groups is found to be optimal under a wide set of parameters. Minimising the number of strongly stakeless matches is verified to be a likely goal in the computer draw of the fixture that remains hidden from the public.

This manuscript portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.

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.