Visualizing very large matrices involves many formidable problems. Various popular solutions to these problems involve sampling, clustering, projection, or feature selection to reduce the size and complexity of the original task. An important aspect of these methods is how to preserve relative distances between points in the higher-dimensional space after reducing rows and columns to fit in a lower dimensional space. This aspect is important because conclusions based on faulty visual reasoning can be harmful. Judging dissimilar points as similar or similar points as dissimilar on the basis of a visualization can lead to false conclusions. To ameliorate this bias and to make visualizations of very large datasets feasible, we introduce two new algorithms that respectively select a subset of rows and columns of a rectangular matrix. This selection is designed to preserve relative distances as closely as possible. We compare our matrix sketch to more traditional alternatives on a variety of artificial and real datasets.
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In image generation, generative models can be evaluated naturally by visually inspecting model outputs. However, this is not always the case for graph generative models (GGMs), making their evaluation challenging. Currently, the standard process for evaluating GGMs suffers from three critical limitations: i) it does not produce a single score which makes model selection challenging, ii) in many cases it fails to consider underlying edge and node features, and iii) it is prohibitively slow to perform. In this work, we mitigate these issues by searching for scalar, domain-agnostic, and scalable metrics for evaluating and ranking GGMs. To this end, we study existing GGM metrics and neural-network-based metrics emerging from generative models of images that use embeddings extracted from a task-specific network. Motivated by the power of certain Graph Neural Networks (GNNs) to extract meaningful graph representations without any training, we introduce several metrics based on the features extracted by an untrained random GNN. We design experiments to thoroughly test metrics on their ability to measure the diversity and fidelity of generated graphs, as well as their sample and computational efficiency. Depending on the quantity of samples, we recommend one of two random-GNN-based metrics that we show to be more expressive than pre-existing metrics. While we focus on applying these metrics to GGM evaluation, in practice this enables the ability to easily compute the dissimilarity between any two sets of graphs regardless of domain. Our code is released at: //github.com/uoguelph-mlrg/GGM-metrics.
The image deblurring problem consists of reconstructing images from blur and noise contaminated available data. In this AMS Notices article, we provide an overview of some well known numerical linear algebra techniques that are use for solving this problem. In particular, we start by carefully describing how to represent images, the process of blurring an image and modeling different kind of added noise. Then, we present regularization methods such as Tikhonov (on the standard and general form), Total Variation and other variations with sparse and edge preserving properties. Additionally, we briefly overview some of the main matrix structures for the blurring operator and finalize presenting multilevel methods that preserve such structures. Numerical examples are used to illustrate the techniques described.
The concept of traditional farming is changing rapidly with the introduction of smart technologies like the Internet of Things (IoT). Under the concept of smart agriculture, precision agriculture is gaining popularity to enable Decision Support System (DSS)-based farming management that utilizes widespread IoT sensors and wireless connectivity to enable automated detection and optimization of resources. Undoubtedly the success of the system would be impacted on crop productivity, where failure would impact severely. Like many other cyber-physical systems, one of the growing challenges to avoid system adversity is to ensure the system's security, privacy, and trust. But what are the vulnerabilities, threats, and security issues we should consider while deploying precision agriculture? This paper has conducted a holistic threat modeling on component levels of precision agriculture's standard infrastructure using popular threat intelligence tools STRIDE to identify common security issues. Our modeling identifies a noticing of fifty-eight potential security threats to consider. This presentation systematically presented them and advised general mitigation suggestions to support cyber security in precision agriculture.
In 1954, Alston S. Householder published Principles of Numerical Analysis, one of the first modern treatments on matrix decomposition that favored a (block) LU decomposition-the factorization of a matrix into the product of lower and upper triangular matrices. And now, matrix decomposition has become a core technology in machine learning, largely due to the development of the back propagation algorithm in fitting a neural network. The sole aim of this survey is to give a self-contained introduction to concepts and mathematical tools in numerical linear algebra and matrix analysis in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of the Euclidean space, Hermitian space, Hilbert space, and things in the complex domain. We refer the reader to literature in the field of linear algebra for a more detailed introduction to the related fields.
We employ a toolset -- dubbed Dr. Frankenstein -- to analyse the similarity of representations in deep neural networks. With this toolset, we aim to match the activations on given layers of two trained neural networks by joining them with a stitching layer. We demonstrate that the inner representations emerging in deep convolutional neural networks with the same architecture but different initializations can be matched with a surprisingly high degree of accuracy even with a single, affine stitching layer. We choose the stitching layer from several possible classes of linear transformations and investigate their performance and properties. The task of matching representations is closely related to notions of similarity. Using this toolset, we also provide a novel viewpoint on the current line of research regarding similarity indices of neural network representations: the perspective of the performance on a task.
Distance metric learning based on triplet loss has been applied with success in a wide range of applications such as face recognition, image retrieval, speaker change detection and recently recommendation with the CML model. However, as we show in this article, CML requires large batches to work reasonably well because of a too simplistic uniform negative sampling strategy for selecting triplets. Due to memory limitations, this makes it difficult to scale in high-dimensional scenarios. To alleviate this problem, we propose here a 2-stage negative sampling strategy which finds triplets that are highly informative for learning. Our strategy allows CML to work effectively in terms of accuracy and popularity bias, even when the batch size is an order of magnitude smaller than what would be needed with the default uniform sampling. We demonstrate the suitability of the proposed strategy for recommendation and exhibit consistent positive results across various datasets.
In Natural Language Processing (NLP), we often need to extract information from tree topology. Sentence structure can be represented via a dependency tree or a constituency tree structure. For this reason, a variant of LSTMs, named Tree-LSTM, was proposed to work on tree topology. In this paper, we design a generalized attention framework for both dependency and constituency trees by encoding variants of decomposable attention inside a Tree-LSTM cell. We evaluated our models on a semantic relatedness task and achieved notable results compared to Tree-LSTM based methods with no attention as well as other neural and non-neural methods and good results compared to Tree-LSTM based methods with attention.
We show that for the problem of testing if a matrix $A \in F^{n \times n}$ has rank at most $d$, or requires changing an $\epsilon$-fraction of entries to have rank at most $d$, there is a non-adaptive query algorithm making $\widetilde{O}(d^2/\epsilon)$ queries. Our algorithm works for any field $F$. This improves upon the previous $O(d^2/\epsilon^2)$ bound (SODA'03), and bypasses an $\Omega(d^2/\epsilon^2)$ lower bound of (KDD'14) which holds if the algorithm is required to read a submatrix. Our algorithm is the first such algorithm which does not read a submatrix, and instead reads a carefully selected non-adaptive pattern of entries in rows and columns of $A$. We complement our algorithm with a matching query complexity lower bound for non-adaptive testers over any field. We also give tight bounds of $\widetilde{\Theta}(d^2)$ queries in the sensing model for which query access comes in the form of $\langle X_i, A\rangle:=tr(X_i^\top A)$; perhaps surprisingly these bounds do not depend on $\epsilon$. We next develop a novel property testing framework for testing numerical properties of a real-valued matrix $A$ more generally, which includes the stable rank, Schatten-$p$ norms, and SVD entropy. Specifically, we propose a bounded entry model, where $A$ is required to have entries bounded by $1$ in absolute value. We give upper and lower bounds for a wide range of problems in this model, and discuss connections to the sensing model above.
Enhancing Network Embedding with Auxiliary Information: An Explicit Matrix Factorization Perspective
Recent advances in the field of network embedding have shown the low-dimensional network representation is playing a critical role in network analysis. However, most of the existing principles of network embedding do not incorporate auxiliary information such as content and labels of nodes flexibly. In this paper, we take a matrix factorization perspective of network embedding, and incorporate structure, content and label information of the network simultaneously. For structure, we validate that the matrix we construct preserves high-order proximities of the network. Label information can be further integrated into the matrix via the process of random walk sampling to enhance the quality of embedding in an unsupervised manner, i.e., without leveraging downstream classifiers. In addition, we generalize the Skip-Gram Negative Sampling model to integrate the content of the network in a matrix factorization framework. As a consequence, network embedding can be learned in a unified framework integrating network structure and node content as well as label information simultaneously. We demonstrate the efficacy of the proposed model with the tasks of semi-supervised node classification and link prediction on a variety of real-world benchmark network datasets.
This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.
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