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Conjectures have historically played an important role in the development of pure mathematics. We propose a systematic approach to finding abstract patterns in mathematical data, in order to generate conjectures about mathematical inequalities, using machine intelligence. We focus on strict inequalities of type f < g and associate them with a vector space. By geometerising this space, which we refer to as a conjecture space, we prove that this space is isomorphic to a Banach manifold. We develop a structural understanding of this conjecture space by studying linear automorphisms of this manifold and show that this space admits several free group actions. Based on these insights, we propose an algorithmic pipeline to generate novel conjectures using geometric gradient descent, where the metric is informed by the invariances of the conjecture space. As proof of concept, we give a toy algorithm to generate novel conjectures about the prime counting function and diameters of Cayley graphs of non-abelian simple groups. We also report private communications with colleagues in which some conjectures were proved, and highlight that some conjectures generated using this procedure are still unproven. Finally, we propose a pipeline of mathematical discovery in this space and highlight the importance of domain expertise in this pipeline.

We introduce the local information cost (LIC), which quantifies the amount of information that nodes in a network need to learn when solving a graph problem. We show that the local information cost presents a natural lower bound on the communication complexity of distributed algorithms. For the synchronous CONGEST $KT_1$ model, where each node has initial knowledge of its neighbors' IDs, we prove that $\Omega(\frac{\text{LIC}_\gamma(P)}{\log\tau \log n})$ bits are required for solving a graph problem $P$ with a $\tau$-round algorithm that errs with probability at most $\gamma$. Our result is the first lower bound that yields a general trade-off between communication and time for graph problems in the CONGEST $KT_1$ model. We demonstrate how to apply the local information cost by deriving a lower bound on the communication complexity of computing routing tables for all-pairs-shortest-paths (APSP) routing, as well as for computing a spanner with multiplicative stretch $2t-1$ that consists of at most $O(n^{1+\frac{1}{t} + \epsilon})$ edges, where $\epsilon = O( {1}/{t^2} )$. More concretely, we derive the following lower bounds in the CONGEST model under the $KT_1$ assumption: For constructing routing tables, we show that any $O(\text{poly}(n))$-time algorithm has a communication complexity of $\Omega( {n^2}/{\log^2 n} )$ bits. Our main result is for constructing graph spanners: We show that any $O(\text{poly}(n))$-time algorithm must send at least $\tilde\Omega(\tfrac{1}{t^2} n^{1+{1}/{2t}})$ bits. Previously, only a trivial lower bound of $\tilde \Omega(n)$ bits was known for these problems.

Hadwiger's Conjecture asserts that every $K_h$-minor-free graph is properly $(h-1)$-colourable. We prove the following improper analogue of Hadwiger's Conjecture: for fixed $h$, every $K_h$-minor-free graph is $(h-1)$-colourable with monochromatic components of bounded size. The number of colours is best possible regardless of the size of monochromatic components. It solves an open problem of Edwards, Kang, Kim, Oum and Seymour [\emph{SIAM J. Disc. Math.} 2015], and concludes a line of research initiated in 2007. Similarly, for fixed $t\geq s$, we show that every $K_{s,t}$-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is best possible, solving an open problem of van de Heuvel and Wood [\emph{J.~London Math.\ Soc.} 2018]. We actually prove a single theorem from which both of the above results are immediate corollaries. For an excluded apex minor, we strengthen the result as follows: for fixed $t\geq s\geq 3$, and for any fixed apex graph $X$, every $K_{s,t}$-subgraph-free $X$-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is again best possible.

We present for the first time a complete solution to the problem of proving the correctness of a concurrency control algorithm for collaborative text editors against the standard consistency model. The success of our approach stems from the use of comprehensive stringwise operational transformations, which appear to have escaped a formal treatment until now. Because these transformations sometimes lead to an increase in the number of operations as they are transformed, we cannot use inductive methods and adopt the novel idea of decreasing diagrams instead. We also base our algorithm on a client-server model rather than a peer-to-peer one, which leads to the correct application of operational transformations to both newly generated and pending operations. And lastly we solve the problem of latency, so that our algorithm works perfectly in practice. The result of these innovations is the first ever formally correct concurrency control algorithm for collaborative text editors together with a fast, fault tolerant and highly scalable implementation.

We study the causal bandit problem when the causal graph is unknown and develop an efficient algorithm for finding the parent node of the reward node using atomic interventions. We derive the exact equation for the expected number of interventions performed by the algorithm and show that under certain graphical conditions it could perform either logarithmically fast or, under more general assumptions, slower but still sublinearly in the number of variables. We formally show that our algorithm is optimal as it meets the universal lower bound we establish for any algorithm that performs atomic interventions. Finally, we extend our algorithm to the case when the reward node has multiple parents. Using this algorithm together with a standard algorithm from bandit literature leads to improved regret bounds.

We consider an inverse problem for a finite graph $(X,E)$ where we are given a subset of vertices $B\subset X$ and the distances $d_{(X,E)}(b_1,b_2)$ of all vertices $b_1,b_2\in B$. The distance of points $x_1,x_2\in X$ is defined as the minimal number of edges needed to connect two vertices, so all edges have length 1. The inverse problem is a discrete version of the boundary rigidity problem in Riemannian geometry or the inverse travel time problem in geophysics. We will show that this problem has unique solution under certain conditions and develop quantum computing methods to solve it. We prove the following uniqueness result: when $(X,E)$ is a tree and $B$ is the set of leaves of the tree, the graph $(X,E)$ can be uniquely determined in the class of all graphs having a fixed number of vertices. We present a quantum computing algorithm which produces a graph $(X,E)$, or one of those, which has a given number of vertices and the required distances between vertices in $B$. To this end we develop an algorithm that takes in a qubit representation of a graph and combine it with Grover's search algorithm. The algorithm can be implemented using only $O(|X|^2)$ qubits, the same order as the number of elements in the adjacency matrix of $(X,E)$. It also has a quadratic improvement in computational cost compared to standard classical algorithms. Finally, we consider applications in theory of computation, and show that a slight modification of the above inverse problem is NP-complete: all NP-problems can be reduced to a discrete inverse problem we consider.

For a constraint satisfaction problem (CSP), a robust satisfaction algorithm is one that outputs an assignment satisfying most of the constraints on instances that are near-satisfiable. It is known that the CSPs that admit efficient robust satisfaction algorithms are precisely those of bounded width, i.e., CSPs whose satisfiability can be checked by a simple local consistency algorithm (eg., 2-SAT or Horn-SAT in the Boolean case). While the exact satisfiability of a bounded width CSP can be checked by combinatorial algorithms, the robust algorithm is based on rounding a canonical Semidefinite programming(SDP) relaxation. In this work, we initiate the study of robust satisfaction algorithms for promise CSPs, which are a vast generalization of CSPs that have received much attention recently. The motivation is to extend the theory beyond CSPs, as well as to better understand the power of SDPs. We present robust SDP rounding algorithms under some general conditions, namely the existence of particular high-dimensional Boolean symmetries known as majority or alternating threshold polymorphisms. On the hardness front, we prove that the lack of such polymorphisms makes the PCSP hard for all pairs of symmetric Boolean predicates. Our method involves a novel method to argue SDP gaps via the absence of certain colorings of the sphere, with connections to sphere Ramsey theory. We conjecture that PCSPs with robust satisfaction algorithms are precisely those for which the feasibility of the canonical SDP implies (exact) satisfiability. We also give a precise algebraic condition, known as a minion characterization, of which PCSPs have the latter property.

In this work, we prove rigorous error estimates for a hybrid method introduced in [15] for solving the time-dependent radiation transport equation (RTE). The method relies on a splitting of the kinetic distribution function for the radiation into uncollided and collided components. A high-resolution method (in angle) is used to approximate the uncollided components and a low-resolution method is used to approximate the the collided component. After each time step, the kinetic distribution is reinitialized to be entirely uncollided. For this analysis, we consider a mono-energetic problem on a periodic domains, with constant material cross-sections of arbitrary size. To focus the analysis, we assume the uncollided equation is solved exactly and the collided part is approximated in angle via a spherical harmonic expansion ($\text{P}_N$ method). Using a non-standard set of semi-norms, we obtain estimates of the form $C(\varepsilon,\sigma,\Delta t)N^{-s}$ where $s\geq 1$ denotes the regularity of the solution in angle, $\varepsilon$ and $\sigma$ are scattering parameters, $\Delta t$ is the time-step before reinitialization, and $C$ is a complicated function of $\varepsilon$, $\sigma$, and $\Delta t$. These estimates involve analysis of the multiscale RTE that includes, but necessarily goes beyond, usual spectral analysis. We also compute error estimates for the monolithic $\text{P}_N$ method with the same resolution as the collided part in the hybrid. Our results highlight the benefits of the hybrid approach over the monolithic discretization in both highly scattering and streaming regimes.

Deep learning models on graphs have achieved remarkable performance in various graph analysis tasks, e.g., node classification, link prediction and graph clustering. However, they expose uncertainty and unreliability against the well-designed inputs, i.e., adversarial examples. Accordingly, various studies have emerged for both attack and defense addressed in different graph analysis tasks, leading to the arms race in graph adversarial learning. For instance, the attacker has poisoning and evasion attack, and the defense group correspondingly has preprocessing- and adversarial- based methods. Despite the booming works, there still lacks a unified problem definition and a comprehensive review. To bridge this gap, we investigate and summarize the existing works on graph adversarial learning tasks systemically. Specifically, we survey and unify the existing works w.r.t. attack and defense in graph analysis tasks, and give proper definitions and taxonomies at the same time. Besides, we emphasize the importance of related evaluation metrics, and investigate and summarize them comprehensively. Hopefully, our works can serve as a reference for the relevant researchers, thus providing assistance for their studies. More details of our works are available at //github.com/gitgiter/Graph-Adversarial-Learning.

Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.