A High-dimensional and sparse (HiDS) matrix is frequently encountered in a big data-related application like an e-commerce system or a social network services system. To perform highly accurate representation learning on it is of great significance owing to the great desire of extracting latent knowledge and patterns from it. Latent factor analysis (LFA), which represents an HiDS matrix by learning the low-rank embeddings based on its observed entries only, is one of the most effective and efficient approaches to this issue. However, most existing LFA-based models perform such embeddings on a HiDS matrix directly without exploiting its hidden graph structures, thereby resulting in accuracy loss. To address this issue, this paper proposes a graph-incorporated latent factor analysis (GLFA) model. It adopts two-fold ideas: 1) a graph is constructed for identifying the hidden high-order interaction (HOI) among nodes described by an HiDS matrix, and 2) a recurrent LFA structure is carefully designed with the incorporation of HOI, thereby improving the representa-tion learning ability of a resultant model. Experimental results on three real-world datasets demonstrate that GLFA outperforms six state-of-the-art models in predicting the missing data of an HiDS matrix, which evidently supports its strong representation learning ability to HiDS data.
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We study the scaling limits of stochastic gradient descent (SGD) with constant step-size in the high-dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite-dimensional functions) of SGD as the dimension goes to infinity. Our approach allows one to choose the summary statistics that are tracked, the initialization, and the step-size. It yields both ballistic (ODE) and diffusive (SDE) limits, with the limit depending dramatically on the former choices. Interestingly, we find a critical scaling regime for the step-size below which the effective ballistic dynamics matches gradient flow for the population loss, but at which, a new correction term appears which changes the phase diagram. About the fixed points of this effective dynamics, the corresponding diffusive limits can be quite complex and even degenerate. We demonstrate our approach on popular examples including estimation for spiked matrix and tensor models and classification via two-layer networks for binary and XOR-type Gaussian mixture models. These examples exhibit surprising phenomena including multimodal timescales to convergence as well as convergence to sub-optimal solutions with probability bounded away from zero from random (e.g., Gaussian) initializations.
Deep neural networks have been successfully applied to a broad range of problems where overparametrization yields weight matrices which are partially random. A comparison of weight matrix singular vectors to the Porter-Thomas distribution suggests that there is a boundary between randomness and learned information in the singular value spectrum. Inspired by this finding, we introduce an algorithm for noise filtering, which both removes small singular values and reduces the magnitude of large singular values to counteract the effect of level repulsion between the noise and the information part of the spectrum. For networks trained in the presence of label noise, we indeed find that the generalization performance improves significantly due to noise filtering.
In experiments that study social phenomena, such as peer influence or herd immunity, the treatment of one unit may influence the outcomes of others. Such "interference between units" violates traditional approaches for causal inference, so that additional assumptions are often imposed to model or limit the underlying social mechanism. For binary outcomes, we propose an approach that does not require such assumptions, allowing for interference that is both unmodeled and strong, with confidence intervals derived using only the randomization of treatment. However, the estimates will have wider confidence intervals and weaker causal implications than those attainable under stronger assumptions. The approach allows for the usage of regression, matching, or weighting, as may best fit the application at hand. Inference is done by bounding the distribution of the estimation error over all possible values of the unknown counterfactual, using an integer program. Examples are shown using using a vaccination trial and two experiments investigating social influence.
Factorization of matrices where the rank of the two factors diverges linearly with their sizes has many applications in diverse areas such as unsupervised representation learning, dictionary learning or sparse coding. We consider a setting where the two factors are generated from known component-wise independent prior distributions, and the statistician observes a (possibly noisy) component-wise function of their matrix product. In the limit where the dimensions of the matrices tend to infinity, but their ratios remain fixed, we expect to be able to derive closed form expressions for the optimal mean squared error on the estimation of the two factors. However, this remains a very involved mathematical and algorithmic problem. A related, but simpler, problem is extensive-rank matrix denoising, where one aims to reconstruct a matrix with extensive but usually small rank from noisy measurements. In this paper, we approach both these problems using high-temperature expansions at fixed order parameters. This allows to clarify how previous attempts at solving these problems failed at finding an asymptotically exact solution. We provide a systematic way to derive the corrections to these existing approximations, taking into account the structure of correlations particular to the problem. Finally, we illustrate our approach in detail on the case of extensive-rank matrix denoising. We compare our results with known optimal rotationally-invariant estimators, and show how exact asymptotic calculations of the minimal error can be performed using extensive-rank matrix integrals.
Exploratory factor analysis (EFA) has been widely used to learn the latent structure underlying multivariate data. Rotation and regularised estimation are two classes of methods in EFA that are widely used to find interpretable loading matrices. This paper proposes a new family of oblique rotations based on component-wise $L^p$ loss functions $(0 < p\leq 1)$ that is closely related to an $L^p$ regularised estimator. Model selection and post-selection inference procedures are developed based on the proposed rotation. When the true loading matrix is sparse, the proposed method tends to outperform traditional rotation and regularised estimation methods, in terms of statistical accuracy and computational cost. Since the proposed loss functions are non-smooth, an iteratively reweighted gradient projection algorithm is developed for solving the optimisation problem. Theoretical results are developed that establish the statistical consistency of the estimation, model selection, and post-selection inference. The proposed method is evaluated and compared with regularised estimation and traditional rotation methods via simulation studies. It is further illustrated by an application to big-five personality assessment.
Hierarchical matrices provide a powerful representation for significantly reducing the computational complexity associated with dense kernel matrices. For general kernel functions, interpolation-based methods are widely used for the efficient construction of hierarchical matrices. In this paper, we present a fast hierarchical data reduction (HiDR) procedure with $O(n)$ complexity for the memory-efficient construction of hierarchical matrices with nested bases where $n$ is the number of data points. HiDR aims to reduce the given data in a hierarchical way so as to obtain $O(1)$ representations for all nearfield and farfield interactions. Based on HiDR, a linear complexity $\mathcal{H}^2$ matrix construction algorithm is proposed. The use of data-driven methods enables {better efficiency than other general-purpose methods} and flexible computation without accessing the kernel function. Experiments demonstrate significantly improved memory efficiency of the proposed data-driven method compared to interpolation-based methods over a wide range of kernels. Though the method is not optimized for any special kernel, benchmark experiments for the Coulomb kernel show that the proposed general-purpose algorithm offers competitive performance for hierarchical matrix construction compared to several state-of-the-art algorithms for the Coulomb kernel.
In this work, we study the representation space of contextualized embeddings and gain insight into the hidden topology of large language models. We show there exists a network of latent states that summarize linguistic properties of contextualized representations. Instead of seeking alignments to existing well-defined annotations, we infer this latent network in a fully unsupervised way using a structured variational autoencoder. The induced states not only serve as anchors that mark the topology (neighbors and connectivity) of the representation manifold but also reveal the internal mechanism of encoding sentences. With the induced network, we: (1). decompose the representation space into a spectrum of latent states which encode fine-grained word meanings with lexical, morphological, syntactic and semantic information; (2). show state-state transitions encode rich phrase constructions and serve as the backbones of the latent space. Putting the two together, we show that sentences are represented as a traversal over the latent network where state-state transition chains encode syntactic templates and state-word emissions fill in the content. We demonstrate these insights with extensive experiments and visualizations.
Dialogue systems are a popular Natural Language Processing (NLP) task as it is promising in real-life applications. It is also a complicated task since many NLP tasks deserving study are involved. As a result, a multitude of novel works on this task are carried out, and most of them are deep learning-based due to the outstanding performance. In this survey, we mainly focus on the deep learning-based dialogue systems. We comprehensively review state-of-the-art research outcomes in dialogue systems and analyze them from two angles: model type and system type. Specifically, from the angle of model type, we discuss the principles, characteristics, and applications of different models that are widely used in dialogue systems. This will help researchers acquaint these models and see how they are applied in state-of-the-art frameworks, which is rather helpful when designing a new dialogue system. From the angle of system type, we discuss task-oriented and open-domain dialogue systems as two streams of research, providing insight into the hot topics related. Furthermore, we comprehensively review the evaluation methods and datasets for dialogue systems to pave the way for future research. Finally, some possible research trends are identified based on the recent research outcomes. To the best of our knowledge, this survey is the most comprehensive and up-to-date one at present in the area of dialogue systems and dialogue-related tasks, extensively covering the popular frameworks, topics, and datasets.
This paper focuses on two fundamental tasks of graph analysis: community detection and node representation learning, which capture the global and local structures of graphs, respectively. In the current literature, these two tasks are usually independently studied while they are actually highly correlated. We propose a probabilistic generative model called vGraph to learn community membership and node representation collaboratively. Specifically, we assume that each node can be represented as a mixture of communities, and each community is defined as a multinomial distribution over nodes. Both the mixing coefficients and the community distribution are parameterized by the low-dimensional representations of the nodes and communities. We designed an effective variational inference algorithm which regularizes the community membership of neighboring nodes to be similar in the latent space. Experimental results on multiple real-world graphs show that vGraph is very effective in both community detection and node representation learning, outperforming many competitive baselines in both tasks. We show that the framework of vGraph is quite flexible and can be easily extended to detect hierarchical communities.
Graph convolutional neural networks have recently shown great potential for the task of zero-shot learning. These models are highly sample efficient as related concepts in the graph structure share statistical strength allowing generalization to new classes when faced with a lack of data. However, multi-layer architectures, which are required to propagate knowledge to distant nodes in the graph, dilute the knowledge by performing extensive Laplacian smoothing at each layer and thereby consequently decrease performance. In order to still enjoy the benefit brought by the graph structure while preventing dilution of knowledge from distant nodes, we propose a Dense Graph Propagation (DGP) module with carefully designed direct links among distant nodes. DGP allows us to exploit the hierarchical graph structure of the knowledge graph through additional connections. These connections are added based on a node's relationship to its ancestors and descendants. A weighting scheme is further used to weigh their contribution depending on the distance to the node to improve information propagation in the graph. Combined with finetuning of the representations in a two-stage training approach our method outperforms state-of-the-art zero-shot learning approaches.
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