Graph convolutional networks (GCNs) have achieved great success in graph representation learning by extracting high-level features from nodes and their topology. Since GCNs generally follow a message-passing mechanism, each node aggregates information from its first-order neighbour to update its representation. As a result, the representations of nodes with edges between them should be positively correlated and thus can be considered positive samples. However, there are more non-neighbour nodes in the whole graph, which provide diverse and useful information for the representation update. Two non-adjacent nodes usually have different representations, which can be seen as negative samples. Besides the node representations, the structural information of the graph is also crucial for learning. In this paper, we used quality-diversity decomposition in determinant point processes (DPP) to obtain diverse negative samples. When defining a distribution on diverse subsets of all non-neighbouring nodes, we incorporate both graph structure information and node representations. Since the DPP sampling process requires matrix eigenvalue decomposition, we propose a new shortest-path-base method to improve computational efficiency. Finally, we incorporate the obtained negative samples into the graph convolution operation. The ideas are evaluated empirically in experiments on node classification tasks. These experiments show that the newly proposed methods not only improve the overall performance of standard representation learning but also significantly alleviate over-smoothing problems.
Predicting discrete events in time and space has many scientific applications, such as predicting hazardous earthquakes and outbreaks of infectious diseases. History-dependent spatio-temporal Hawkes processes are often used to mathematically model these point events. However, previous approaches have faced numerous challenges, particularly when attempting to forecast one or multiple future events. In this work, we propose a new neural architecture for simultaneous multi-event forecasting of spatio-temporal point processes, utilizing transformers, augmented with normalizing flows and probabilistic layers. Our network makes batched predictions of complex history-dependent spatio-temporal distributions of future discrete events, achieving state-of-the-art performance on a variety of benchmark datasets including the South California Earthquakes, Citibike, Covid-19, and Hawkes synthetic pinwheel datasets. More generally, we illustrate how our network can be applied to any dataset of discrete events with associated markers, even when no underlying physics is known.
This paper develops a novel weak multilevel Monte-Carlo (MLMC) approximation scheme for L\'evy-driven Stochastic Differential Equations (SDEs). The scheme is based on the state space discretization (via a continuous-time Markov chain approximation) of the pure-jump component of the driving L\'evy process and is particularly suited if the multidimensional driver is given by a L\'evy copula. The multilevel version of the algorithm requires a new coupling of the approximate L\'evy drivers in the consecutive levels of the scheme, which is defined via a coupling of the corresponding Poisson point processes. The multilevel scheme is weak in the sense that the bound on the level variances is based on the coupling alone without requiring strong convergence. Moreover, the coupling is natural for the proposed discretization of jumps and is easy to simulate. The approximation scheme and its multilevel analogous are applied to examples taken from mathematical finance, including the pricing of credit, equity and interest rate derivatives.
Discrete Determinantal Point Processes (DPPs) have a wide array of potential applications for subsampling datasets. They are however held back in some cases by the high cost of sampling. In the worst-case scenario, the sampling cost scales as O(n^3) where n is the number of elements of the ground set. A popular workaround to this prohibitive cost is to sample DPPs defined by low-rank kernels. In such cases, the cost of standard sampling algorithms scales as O(np^2 + nm^2) where m is the (average) number of samples of the DPP (usually m << n) and p the rank of the kernel used to define the DPP (m \leq p \leq n). The first term, O(np^2), comes from a SVD-like step. We focus here on the second term of this cost, O(nm^2), and show that it can be brought down to O(nm + m^3 log m) without loss on the sampling's exactness. In practice, we observe very substantial speedups compared to the classical algorithm as soon as n > 1000. The algorithm described here is a close variant of the standard algorithm for sampling continuous DPPs, and uses rejection sampling. In the specific case of projection DPPs, we also show that any additional sample can be drawn in time O(m^3 log m). Finally, an interesting by-product of the analysis is that a realisation from a DPP is typically contained in a subset of size O(m log m) formed using leverage score i.i.d. sampling.
Temporal Point Processes (TPP) are probabilistic generative frameworks. They model discrete event sequences localized in continuous time. Generally, real-life events reveal descriptive information, known as marks. Marked TPPs model time and marks of the event together for practical relevance. Conditioned on past events, marked TPPs aim to learn the joint distribution of the time and the mark of the next event. For simplicity, conditionally independent TPP models assume time and marks are independent given event history. They factorize the conditional joint distribution of time and mark into the product of individual conditional distributions. This structural limitation in the design of TPP models hurt the predictive performance on entangled time and mark interactions. In this work, we model the conditional inter-dependence of time and mark to overcome the limitations of conditionally independent models. We construct a multivariate TPP conditioning the time distribution on the current event mark in addition to past events. Besides the conventional intensity-based models for conditional joint distribution, we also draw on flexible intensity-free TPP models from the literature. The proposed TPP models outperform conditionally independent and dependent models in standard prediction tasks. Our experimentation on various datasets with multiple evaluation metrics highlights the merit of the proposed approach.
We propose a new sparse Granger-causal learning framework for temporal event data. We focus on a specific class of point processes called the Hawkes process. We begin by pointing out that most of the existing sparse causal learning algorithms for the Hawkes process suffer from a singularity in maximum likelihood estimation. As a result, their sparse solutions can appear only as numerical artifacts. In this paper, we propose a mathematically well-defined sparse causal learning framework based on a cardinality-regularized Hawkes process, which remedies the pathological issues of existing approaches. We leverage the proposed algorithm for the task of instance-wise causal event analysis, where sparsity plays a critical role. We validate the proposed framework with two real use-cases, one from the power grid and the other from the cloud data center management domain.
The event camera is a novel bio-inspired vision sensor. When the brightness change exceeds the preset threshold, the sensor generates events asynchronously. The number of valid events directly affects the performance of event-based tasks, such as reconstruction, detection, and recognition. However, when in low-brightness or slow-moving scenes, events are often sparse and accompanied by noise, which poses challenges for event-based tasks. To solve these challenges, we propose an event temporal up-sampling algorithm1 to generate more effective and reliable events. The main idea of our algorithm is to generate up-sampling events on the event motion trajectory. First, we estimate the event motion trajectory by contrast maximization algorithm and then up-sampling the events by temporal point processes. Experimental results show that up-sampling events can provide more effective information and improve the performance of downstream tasks, such as improving the quality of reconstructed images and increasing the accuracy of object detection.
In this work, we identify open research opportunities in applying Neural Temporal Point Process (NTPP) models to industry scale customer behavior data by carefully reproducing NTPP models published up to date on known literature benchmarks as well as applying NTPP models to a novel, real world consumer behavior dataset that is twice as large as the largest publicly available NTPP benchmark. We identify the following challenges. First, NTPP models, albeit their generative nature, remain vulnerable to dataset imbalances and cannot forecast rare events. Second, NTPP models based on stochastic differential equations, despite their theoretical appeal and leading performance on literature benchmarks, do not scale easily to large industry-scale data. The former is in light of previously made observations on deep generative models. Additionally, to combat a cold-start problem, we explore a novel addition to NTPP models - a parametrization based on static user features.
This work introduces DADAO: the first decentralized, accelerated, asynchronous, primal, first-order algorithm to minimize a sum of $L$-smooth and $\mu$-strongly convex functions distributed over a given network of size $n$. Our key insight is based on modeling the local gradient updates and gossip communication procedures with separate independent Poisson Point Processes. This allows us to decouple the computation and communication steps, which can be run in parallel, while making the whole approach completely asynchronous. This leads to communication acceleration compared to synchronous approaches. Our new method employs primal gradients and does not use a multi-consensus inner loop nor other ad-hoc mechanisms such as Error Feedback, Gradient Tracking, or a Proximal operator. By relating the inverse of the smallest positive eigenvalue of the Laplacian matrix $\chi_1$ and the maximal resistance $\chi_2\leq \chi_1$ of the graph to a sufficient minimal communication rate between the nodes of the network, we show that our algorithm requires $\mathcal{O}(n\sqrt{\frac{L}{\mu}}\log(\frac{1}{\epsilon}))$ local gradients and only $\mathcal{O}(n\sqrt{\chi_1\chi_2}\sqrt{\frac{L}{\mu}}\log(\frac{1}{\epsilon}))$ communications to reach a precision $\epsilon$, up to logarithmic terms. Thus, we simultaneously obtain an accelerated rate for both computations and communications, leading to an improvement over state-of-the-art works, our simulations further validating the strength of our relatively unconstrained method.
A determinantal point process (DPP) is an elegant model that assigns a probability to every subset of a collection of $n$ items. While conventionally a DPP is parameterized by a symmetric kernel matrix, removing this symmetry constraint, resulting in nonsymmetric DPPs (NDPPs), leads to significant improvements in modeling power and predictive performance. Recent work has studied an approximate Markov chain Monte Carlo (MCMC) sampling algorithm for NDPPs restricted to size-$k$ subsets (called $k$-NDPPs). However, the runtime of this approach is quadratic in $n$, making it infeasible for large-scale settings. In this work, we develop a scalable MCMC sampling algorithm for $k$-NDPPs with low-rank kernels, thus enabling runtime that is sublinear in $n$. Our method is based on a state-of-the-art NDPP rejection sampling algorithm, which we enhance with a novel approach for efficiently constructing the proposal distribution. Furthermore, we extend our scalable $k$-NDPP sampling algorithm to NDPPs without size constraints. Our resulting sampling method has polynomial time complexity in the rank of the kernel, while the existing approach has runtime that is exponential in the rank. With both a theoretical analysis and experiments on real-world datasets, we verify that our scalable approximate sampling algorithms are orders of magnitude faster than existing sampling approaches for $k$-NDPPs and NDPPs.
A large fraction of data generated via human activities such as online purchases, health records, spatial mobility etc. can be represented as a sequence of events over a continuous-time. Learning deep learning models over these continuous-time event sequences is a non-trivial task as it involves modeling the ever-increasing event timestamps, inter-event time gaps, event types, and the influences between different events within and across different sequences. In recent years neural enhancements to marked temporal point processes (MTPP) have emerged as a powerful framework to model the underlying generative mechanism of asynchronous events localized in continuous time. However, most existing models and inference methods in the MTPP framework consider only the complete observation scenario i.e. the event sequence being modeled is completely observed with no missing events -- an ideal setting that is rarely applicable in real-world applications. A recent line of work which considers missing events while training MTPP utilizes supervised learning techniques that require additional knowledge of missing or observed label for each event in a sequence, which further restricts its practicability as in several scenarios the details of missing events is not known apriori. In this work, we provide a novel unsupervised model and inference method for learning MTPP in presence of event sequences with missing events. Specifically, we first model the generative processes of observed events and missing events using two MTPP, where the missing events are represented as latent random variables. Then, we devise an unsupervised training method that jointly learns both the MTPP by means of variational inference. Such a formulation can effectively impute the missing data among the observed events and can identify the optimal position of missing events in a sequence.
The maximum a posteriori (MAP) inference for determinantal point processes (DPPs) is crucial for selecting diverse items in many machine learning applications. Although DPP MAP inference is NP-hard, the greedy algorithm often finds high-quality solutions, and many researchers have studied its efficient implementation. One classical and practical method is the lazy greedy algorithm, which is applicable to general submodular function maximization, while a recent fast greedy algorithm based on the Cholesky factorization is more efficient for DPP MAP inference. This paper presents how to combine the ideas of "lazy" and "fast", which have been considered incompatible in the literature. Our lazy and fast greedy algorithm achieves almost the same time complexity as the current best one and runs faster in practice. The idea of "lazy + fast" is extendable to other greedy-type algorithms. We also give a fast version of the double greedy algorithm for unconstrained DPP MAP inference. Experiments validate the effectiveness of our acceleration ideas.
Networks and temporal point processes serve as fundamental building blocks for modeling complex dynamic relational data in various domains. We propose the latent space Hawkes (LSH) model, a novel generative model for continuous-time networks of relational events, using a latent space representation for nodes. We model relational events between nodes using mutually exciting Hawkes processes with baseline intensities dependent upon the distances between the nodes in the latent space and sender and receiver specific effects. We demonstrate that our proposed LSH model can replicate many features observed in real temporal networks including reciprocity and transitivity, while also achieving superior prediction accuracy and providing more interpretable fits than existing models.
Temporal networks allow representing connections between objects while incorporating the temporal dimension. While static network models can capture unchanging topological regularities, they often fail to model the effects associated with the causal generative process of the network that occurs in time. Hence, exploiting the temporal aspect of networks has been the focus of many recent studies. In this context, we propose a new framework for generative models of continuous-time temporal networks. We assume that the activation of the edges in a temporal network is driven by a specified temporal point process. This approach allows to directly model the waiting time between events while incorporating time-varying history-based features as covariates in the predictions. Coupled with a thinning algorithm designed for the simulation of point processes, SimHawNet enables simulation of the evolution of temporal networks in continuous time. Finally, we introduce a comprehensive evaluation framework to assess the performance of such an approach, in which we demonstrate that SimHawNet successfully simulates the evolution of networks with very different generative processes and achieves performance comparable to the state of the art, while being significantly faster.
There has been a recent surge in learning generative models for graphs. While impressive progress has been made on static graphs, work on generative modeling of temporal graphs is at a nascent stage with significant scope for improvement. First, existing generative models do not scale with either the time horizon or the number of nodes. Second, existing techniques are transductive in nature and thus do not facilitate knowledge transfer. Finally, due to relying on one-to-one node mapping from source to the generated graph, existing models leak node identity information and do not allow up-scaling/down-scaling the source graph size. In this paper, we bridge these gaps with a novel generative model called TIGGER. TIGGER derives its power through a combination of temporal point processes with auto-regressive modeling enabling both transductive and inductive variants. Through extensive experiments on real datasets, we establish TIGGER generates graphs of superior fidelity, while also being up to 3 orders of magnitude faster than the state-of-the-art.
Recent developments in predictive modeling using marked temporal point processes (MTPP) have enabled an accurate characterization of several real-world applications involving continuous-time event sequences (CTESs). However, the retrieval problem of such sequences remains largely unaddressed in literature. To tackle this, we propose NEUROSEQRET which learns to retrieve and rank a relevant set of continuous-time event sequences for a given query sequence, from a large corpus of sequences. More specifically, NEUROSEQRET first applies a trainable unwarping function on the query sequence, which makes it comparable with corpus sequences, especially when a relevant query-corpus pair has individually different attributes. Next, it feeds the unwarped query sequence and the corpus sequence into MTPP guided neural relevance models. We develop two variants of the relevance model which offer a tradeoff between accuracy and efficiency. We also propose an optimization framework to learn binary sequence embeddings from the relevance scores, suitable for the locality-sensitive hashing leading to a significant speedup in returning top-K results for a given query sequence. Our experiments with several datasets show the significant accuracy boost of NEUROSEQRET beyond several baselines, as well as the efficacy of our hashing mechanism.
Continual learning needs to overcome catastrophic forgetting of the past. Memory replay of representative old training samples has been shown as an effective solution, and achieves the state-of-the-art (SOTA) performance. However, existing work is mainly built on a small memory buffer containing a few original data, which cannot fully characterize the old data distribution. In this work, we propose memory replay with data compression (MRDC) to reduce the storage cost of old training samples and thus increase their amount that can be stored in the memory buffer. Observing that the trade-off between the quality and quantity of compressed data is highly nontrivial for the efficacy of memory replay, we propose a novel method based on determinantal point processes (DPPs) to efficiently determine an appropriate compression quality for currently-arrived training samples. In this way, using a naive data compression algorithm with a properly selected quality can largely boost recent strong baselines by saving more compressed data in a limited storage space. We extensively validate this across several benchmarks of class-incremental learning and in a realistic scenario of object detection for autonomous driving.
We consider a sequential decision making problem where the agent faces the environment characterized by the stochastic discrete events and seeks an optimal intervention policy such that its long-term reward is maximized. This problem exists ubiquitously in social media, finance and health informatics but is rarely investigated by the conventional research in reinforcement learning. To this end, we present a novel framework of the model-based reinforcement learning where the agent's actions and observations are asynchronous stochastic discrete events occurring in continuous-time. We model the dynamics of the environment by Hawkes process with external intervention control term and develop an algorithm to embed such process in the Bellman equation which guides the direction of the value gradient. We demonstrate the superiority of our method in both synthetic simulator and real-world problem.
Recent years have seen an increased interest in the application of methods and techniques commonly associated with machine learning and artificial intelligence to spatial statistics. Here, in a celebration of the ten-year anniversary of the journal Spatial Statistics, we bring together normalizing flows, commonly used for density function estimation in machine learning, and spherical point processes, a topic of particular interest to the journal's readership, to present a new approach for modeling non-homogeneous Poisson process intensity functions on the sphere. The central idea of this framework is to build, and estimate, a flexible bijective map that transforms the underlying intensity function of interest on the sphere into a simpler, reference, intensity function, also on the sphere. Map estimation can be done efficiently using automatic differentiation and stochastic gradient descent, and uncertainty quantification can be done straightforwardly via nonparametric bootstrap. We investigate the viability of the proposed method in a simulation study, and illustrate its use in a proof-of-concept study where we model the intensity of cyclone events in the North Pacific Ocean. Our experiments reveal that normalizing flows present a flexible and straightforward way to model intensity functions on spheres, but that their potential to yield a good fit depends on the architecture of the bijective map, which can be difficult to establish in practice.
A determinantal point process (DPP) on a collection of $M$ items is a model, parameterized by a symmetric kernel matrix, that assigns a probability to every subset of those items. Recent work shows that removing the kernel symmetry constraint, yielding nonsymmetric DPPs (NDPPs), can lead to significant predictive performance gains for machine learning applications. However, existing work leaves open the question of scalable NDPP sampling. There is only one known DPP sampling algorithm, based on Cholesky decomposition, that can directly apply to NDPPs as well. Unfortunately, its runtime is cubic in $M$, and thus does not scale to large item collections. In this work, we first note that this algorithm can be transformed into a linear-time one for kernels with low-rank structure. Furthermore, we develop a scalable sublinear-time rejection sampling algorithm by constructing a novel proposal distribution. Additionally, we show that imposing certain structural constraints on the NDPP kernel enables us to bound the rejection rate in a way that depends only on the kernel rank. In our experiments we compare the speed of all of these samplers for a variety of real-world tasks.