The Euler characteristic transform (ECT) is a signature from topological data analysis (TDA) which summarises shapes embedded in Euclidean space. Compared with other TDA methods, the ECT is fast to compute and it is a sufficient statistic for a broad class of shapes. However, small perturbations of a shape can lead to large distortions in its ECT. In this paper, we propose a new metric on compact one-dimensional shapes and prove that the ECT is stable with respect to this metric. Crucially, our result uses curvature, rather than the size of a triangulation of an underlying shape, to control stability. We further construct a computationally tractable statistical estimator of the ECT based on the theory of Gaussian processes. We use our stability result to prove that our estimator is consistent on shapes perturbed by independent ambient noise; i.e., the estimator converges to the true ECT as the sample size increases.
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Despite several advances in recent years, learning causal structures represented by directed acyclic graphs (DAGs) remains a challenging task in high dimensional settings when the graphs to be learned are not sparse. In this paper, we propose to exploit a low rank assumption regarding the (weighted) adjacency matrix of a DAG causal model to help address this problem. We utilize existing low rank techniques to adapt causal structure learning methods to take advantage of this assumption and establish several useful results relating interpretable graphical conditions to the low rank assumption. Specifically, we show that the maximum rank is highly related to hubs, suggesting that scale-free networks, which are frequently encountered in practice, tend to be low rank. Our experiments demonstrate the utility of the low rank adaptations for a variety of data models, especially with relatively large and dense graphs. Moreover, with a validation procedure, the adaptations maintain a superior or comparable performance even when graphs are not restricted to be low rank.
Few-weight codes over finite chain rings are associated with combinatorial objects such as strongly regular graphs (SRGs), strongly walk-regular graphs (SWRGs) and finite geometries, and are also widely used in data storage systems and secret sharing schemes. The first objective of this paper is to characterize all possible parameters of Plotkin-optimal two-homogeneous weight regular projective codes over finite chain rings, as well as their weight distributions. We show the existence of codes with these parameters by constructing an infinite family of two-homogeneous weight codes. The parameters of their Gray images have the same weight distribution as that of the two-weight codes of type SU1 in the sense of Calderbank and Kantor (Bull Lond Math Soc 18: 97-122, 1986). Further, we also construct three-homogeneous weight regular projective codes over finite chain rings combined with some known results. Finally, we study applications of our constructed codes in secret sharing schemes and graph theory. In particular, infinite families of SRGs and SWRGs with non-trivial parameters are obtained.
Local search is a powerful heuristic in optimization and computer science, the complexity of which was studied in the white box and black box models. In the black box model, we are given a graph $G = (V,E)$ and oracle access to a function $f : V \to \mathbb{R}$. The local search problem is to find a vertex $v$ that is a local minimum, i.e. with $f(v) \leq f(u)$ for all $(u,v) \in E$, using as few queries as possible. The query complexity is well understood on the grid and the hypercube, but much less is known beyond. We show the query complexity of local search on $d$-regular expanders with constant degree is $\Omega\left(\frac{\sqrt{n}}{\log{n}}\right)$, where $n$ is the number of vertices. This matches within a logarithmic factor the upper bound of $O(\sqrt{n})$ for constant degree graphs from Aldous (1983), implying that steepest descent with a warm start is an essentially optimal algorithm for expanders. The best lower bound known from prior work was $\Omega\left(\frac{\sqrt[8]{n}}{\log{n}}\right)$, shown by Santha and Szegedy (2004) for quantum and randomized algorithms. We obtain this result by considering a broader framework of graph features such as vertex congestion and separation number. We show that for each graph, the randomized query complexity of local search is $\Omega\left(\frac{n^{1.5}}{g}\right)$, where $g$ is the vertex congestion of the graph; and $\Omega\left(\sqrt[4]{\frac{s}{\Delta}}\right)$, where $s$ is the separation number and $\Delta$ is the maximum degree. For separation number the previous bound was $\Omega\left(\sqrt[8]{\frac{s}{\Delta}} /\log{n}\right)$, given by Santha and Szegedy for quantum and randomized algorithms. We also show a variant of the relational adversary method from Aaronson (2006), which is asymptotically at least as strong as the version in Aaronson (2006) for all randomized algorithms and strictly stronger for some problems.
Adversarial robustness is a critical property in a variety of modern machine learning applications. While it has been the subject of several recent theoretical studies, many important questions related to adversarial robustness are still open. In this work, we study a fundamental question regarding Bayes optimality for adversarial robustness. We provide general sufficient conditions under which the existence of a Bayes optimal classifier can be guaranteed for adversarial robustness. Our results can provide a useful tool for a subsequent study of surrogate losses in adversarial robustness and their consistency properties. This manuscript is the extended and corrected version of the paper \emph{On the Existence of the Adversarial Bayes Classifier} published in NeurIPS 2021. There were two errors in theorem statements in the original paper -- one in the definition of pseudo-certifiable robustness and the other in the measurability of $A^\e$ for arbitrary metric spaces. In this version we correct the errors. Furthermore, the results of the original paper did not apply to some non-strictly convex norms and here we extend our results to all possible norms.
Training Generative adversarial networks (GANs) stably is a challenging task. The generator in GANs transform noise vectors, typically Gaussian distributed, into realistic data such as images. In this paper, we propose a novel approach for training GANs with images as inputs, but without enforcing any pairwise constraints. The intuition is that images are more structured than noise, which the generator can leverage to learn a more robust transformation. The process can be made efficient by identifying closely related datasets, or a ``friendly neighborhood'' of the target distribution, inspiring the moniker, Spider GAN. To define friendly neighborhoods leveraging proximity between datasets, we propose a new measure called the signed inception distance (SID), inspired by the polyharmonic kernel. We show that the Spider GAN formulation results in faster convergence, as the generator can discover correspondence even between seemingly unrelated datasets, for instance, between Tiny-ImageNet and CelebA faces. Further, we demonstrate cascading Spider GAN, where the output distribution from a pre-trained GAN generator is used as the input to the subsequent network. Effectively, transporting one distribution to another in a cascaded fashion until the target is learnt -- a new flavor of transfer learning. We demonstrate the efficacy of the Spider approach on DCGAN, conditional GAN, PGGAN, StyleGAN2 and StyleGAN3. The proposed approach achieves state-of-the-art Frechet inception distance (FID) values, with one-fifth of the training iterations, in comparison to their baseline counterparts on high-resolution small datasets such as MetFaces, Ukiyo-E Faces and AFHQ-Cats.
Optimal transport (OT) is a powerful geometric tool used to compare and align probability measures following the least effort principle. Among many successful applications of OT in machine learning (ML), domain adaptation (DA) -- a field of study where the goal is to transfer a classifier from one labelled domain to another similar, yet different unlabelled or scarcely labelled domain -- has been historically among the most investigated ones. This success is due to the ability of OT to provide both a meaningful discrepancy measure to assess the similarity of two domains' distributions and a mapping that can project source domain data onto the target one. In this paper, we propose a principally new OT-based approach applied to DA that uses the closed-form solution of the OT problem given by an affine mapping and learns an embedding space for which this solution is optimal and computationally less complex. We show that our approach works in both homogeneous and heterogeneous DA settings and outperforms or is on par with other famous baselines based on both traditional OT and OT in incomparable spaces. Furthermore, we show that our proposed method vastly reduces computational complexity.
We describe a proof-of-concept development and application of a phase averaging technique to the nonlinear rotating shallow water equations on the sphere, discretised using compatible finite element methods. Phase averaging consists of averaging the nonlinearity over phase shifts in the exponential of the linear wave operator. Phase averaging aims to capture the slow dynamics in a solution that is smoother in time (in transformed variables) so that larger timesteps may be taken. We overcome the two key technical challenges that stand in the way of studying the phase averaging and advancing its implementation: 1) we have developed a stable matrix exponential specific to finite elements and 2) we have developed a parallel finite averaging proceedure. Following Peddle et al (2019), we consider finite width phase averaging windows, since the equations have a finite timescale separation. In our numerical implementation, the averaging integral is replaced by a Riemann sum, where each term can be evaluated in parallel. This creates an opportunity for parallelism in the timestepping method, which we use here to compute our solutions. Here, we focus on the stability and accuracy of the numerical solution. We confirm there is an optimal averaging window, in agreement with theory. Critically, we observe that the combined time discretisation and averaging error is much smaller than the time discretisation error in a semi-implicit method applied to the same spatial discretisation. An evaluation of the parallel aspects will follow in later work.
We study kernel methods in machine learning from the perspective of feature subspace. We establish a one-to-one correspondence between feature subspaces and kernels and propose an information-theoretic measure for kernels. In particular, we construct a kernel from Hirschfeld--Gebelein--R\'{e}nyi maximal correlation functions, coined the maximal correlation kernel, and demonstrate its information-theoretic optimality. We use the support vector machine (SVM) as an example to illustrate a connection between kernel methods and feature extraction approaches. We show that the kernel SVM on maximal correlation kernel achieves minimum prediction error. Finally, we interpret the Fisher kernel as a special maximal correlation kernel and establish its optimality.
Entities involve important concepts with concrete meanings and play important roles in numerous linguistic tasks. Entities have different forms in different tasks and researchers treat those forms as different concepts. In this paper, we are curious to know whether there are some common characteristics connecting those different forms of entities. Specifically, we investigate the underlying distributions of entities from different types and different languages, trying to figure out some common properties behind those diverse entities. We find from twelve datasets about different types of entities and eighteen datasets about different languages of entities that although these entities are dramatically diverse from each in many aspects, their length-frequencies can be well characterized by Marshall-Olkin power-law (MOPL) distributions, and these distributions possess defined means and finite variances. Our experiments show that while not all the entities are drawn from the same underlying population, those entities under same types tend to be drawn from the same distribution. Our experiments also show that Marshall-Olkin power-law models characterize the length-frequencies of entities much better than pure power-law models and log-normal models.
Causality can be described in terms of a structural causal model (SCM) that carries information on the variables of interest and their mechanistic relations. For most processes of interest the underlying SCM will only be partially observable, thus causal inference tries to leverage any exposed information. Graph neural networks (GNN) as universal approximators on structured input pose a viable candidate for causal learning, suggesting a tighter integration with SCM. To this effect we present a theoretical analysis from first principles that establishes a novel connection between GNN and SCM while providing an extended view on general neural-causal models. We then establish a new model class for GNN-based causal inference that is necessary and sufficient for causal effect identification. Our empirical illustration on simulations and standard benchmarks validate our theoretical proofs.
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