We propose a novel approach to dimensionality reduction combining techniques of metric geometry and distributed persistent homology, in the form of a gradient-descent based method called DIPOLE. DIPOLE is a dimensionality-reduction post-processing step that corrects an initial embedding by minimizing a loss functional with both a local, metric term and a global, topological term. By fixing an initial embedding method (we use Isomap), DIPOLE can also be viewed as a full dimensionality-reduction pipeline. This framework is based on the strong theoretical and computational properties of distributed persistent homology and comes with the guarantee of almost sure convergence. We observe that DIPOLE outperforms popular methods like UMAP, t-SNE, and Isomap on a number of popular datasets, both visually and in terms of precise quantitative metrics.
Stochastic Gradient Descent (SGD) has proven to be remarkably effective in optimizing deep neural networks that employ ever-larger numbers of parameters. Yet, improving the efficiency of large-scale optimization remains a vital and highly active area of research. Recent work has shown that deep neural networks can be optimized in randomly-projected subspaces of much smaller dimensionality than their native parameter space. While such training is promising for more efficient and scalable optimization schemes, its practical application is limited by inferior optimization performance. Here, we im…
Dimensionality reduction techniques are powerful tools for data preprocessing and visualization which typically come with few guarantees concerning the topological correctness of an embedding. The interleaving distance between the persistent homology of Vietoris-Rips filtrations can be used to identify a scale at which topological features such as clusters or holes in an embedding and original data set are in correspondence. We show how optimization seeking to minimize the interleaving distance can be incorporated into dimensionality reduction algorithms, and explicitly demonstrate its use in …
Datasets with non-trivial large scale topology can be hard to embed in low-dimensional Euclidean space with existing dimensionality reduction algorithms. We propose to model topologically complex datasets using vector bundles, in such a way that the base space accounts for the large scale topology, while the fibers account for the local geometry. This allows one to reduce the dimensionality of the fibers, while preserving the large scale topology. We formalize this point of view, and, as an application, we describe an algorithm which takes as input a dataset together with an initial representa…
The problem of learning a manifold structure on a dataset is framed in terms of a generative model, to which we use ideas behind autoencoders (namely adversarial/Wasserstein autoencoders) to fit deep neural networks. From a machine learning perspective, the resulting structure, an atlas of a manifold, may be viewed as a combination of dimensionality reduction and "fuzzy" clustering.
Goal: We consider the problem of automatically grouping logs of runs that failed for the same underlying reasons, so that they can be treated more effectively, and investigate the following questions: (1) Does an approach developed to identify problems in system logs generalize to identifying problems in continuous deployment logs? (2) How does dimensionality reduction affect the quality of automated log clustering? (3) How does the criterion used for merging clusters in the clustering algorithm affect clustering quality? Method: We replicate and extend earlier work on clustering system log fi…
Persistent Homology for High-dimensional Data Based on Spectral Methods
Hertie Institute for AI in Brain Health, University of Tübingen, Germany（图宾根大学希特研究所，德国）；Tübingen AI Center, Germany（图宾根人工智能中心，德国）；IWR, Heidelberg University, Germany（海德堡大学IWR研究所，德国）
Classes are not Clusters: Improving Label-based Evaluation of Dimensionality Reduction
Seoul National University（首尔国立大学）；University of California, Davis（加州大学戴维斯分校）；Qatar Computing Research Institute, Hamad Bin Khalifa University（哈马德·本·哈利法大学卡塔尔计算研究所）
Dimensionality reduction methods are unsupervised approaches which learn low-dimensional spaces where some properties of the initial space, typically the notion of "neighborhood", are preserved. Such methods usually require propagation on large k-NN graphs or complicated optimization solvers. On the other hand, self-supervised learning approaches, typically used to learn representations from scratch, rely on simple and more scalable frameworks for learning. In this paper, we propose TLDR, a dimensionality reduction method for generic input spaces that is porting the recent self-supervised lear…
Most dimensionality reduction methods employ frequency domain representations obtained from matrix diagonalization and may not be efficient for large datasets with relatively high intrinsic dimensions. To address this challenge, Correlated Clustering and Projection (CCP) offers a novel data domain strategy that does not need to solve any matrix. CCP partitions high-dimensional features into correlated clusters and then projects correlated features in each cluster into a one-dimensional representation based on sample correlations. Residue-Similarity (R-S) scores and indexes, the shape of data i…
In the last decade, information diffusion (also known as information cascade) on social networks has been massively investigated due to its application values in many fields. In recent years, many sequential models including those models based on recurrent neural networks have been broadly employed to predict information cascade. However, the user dependencies in a cascade sequence captured by sequential models are generally unidirectional and inconsistent with diffusion trees. For example, the true trigger of a successor may be a non-immediate predecessor rather than the immediate predecessor…
Nonnegative Matrix Factorization in Dimensionality Reduction: A Survey
Graduate University of Advanced Technology（高级技术研究生院）；Queensland University of Technology (QUT)（昆士兰科技大学）；Ardakan University（阿尔达坎大学）；Griffith University（格里菲斯大学）
Bielefeld University（比勒费尔德大学）；University of Groningen（格罗宁根大学）
The growing number of dimensionality reduction methods available for data visualization has recently inspired the development of quality assessment measures, in order to evaluate the resulting low-dimensional representation independently from a methods' inherent criteria. Several (existing) quality measures can be (re)formulated based on the so-called co-ranking matrix, which subsumes all rank errors (i.e. differences between the ranking of distances from every point to all others, comparing the low-dimensional representation to the original data). The measures are often based on the partionin…
The recent increase in dimensionality of data has thrown a great challenge to the existing dimensionality reduction methods in terms of their effectiveness. Dimensionality reduction has emerged as one of the significant preprocessing steps in machine learning applications and has been effective in removing inappropriate data, increasing learning accuracy, and improving comprehensibility. Feature redundancy exercises great influence on the performance of classification process. Towards the better classification performance, this paper addresses the usefulness of truncating the highly correlated…
Dimensionality Reduction and Nearest Neighbors for Improving Out-of-Distribution Detection in Medical Image Segmentation
The University of Texas MD Anderson Cancer Center（德克萨斯大学MD安德森癌症中心）；Rice University（莱斯大学）；Houston Methodist Hospital（休斯顿卫理公会医院）；Baylor College of Medicine（贝勒医学院）
In this paper, we investigate Dimensionality reduction (DR) maps in an information retrieval setting from a quantitative topology point of view. In particular, we show that no DR maps can achieve perfect precision and perfect recall simultaneously. Thus a continuous DR map must have imperfect precision. We further prove an upper bound on the precision of Lipschitz continuous DR maps. While precision is a natural measure in an information retrieval setting, it does not measure `how' wrong the retrieved data is. We therefore propose a new measure based on Wasserstein distance that comes with sim…
Recently, Graph Convolutional Networks (GCNs) and their variants have been receiving many research interests for learning graph-related tasks. While the GCNs have been successfully applied to this problem, some caveats inherited from classical deep learning still remain as open research topics in the context of the node classification problem. One such inherited caveat is that GCNs only consider the nodes that are a few propagations away from the labeled nodes to classify them. However, taking only a few propagation steps away nodes into account defeats the purpose of using the graph topologic…
Objective and interpretable metrics to evaluate current artificial intelligent systems are of great importance, not only to analyze the current state of such systems but also to objectively measure progress in the future. In this work, we focus on the evaluation of image generation tasks. We propose a novel approach, called Fuzzy Topology Impact (FTI), that determines both the quality and diversity of an image set using topology representations combined with fuzzy logic. When compared to current evaluation methods, FTI shows better and more stable performance on multiple experiments evaluating…
Manifold learning is used for dimensionality reduction, with the goal of finding a projection subspace to increase and decrease the inter- and intraclass variances, respectively. However, a bottleneck for subspace learning methods often arises from the high dimensionality of datasets. In this paper, a hierarchical approach is proposed to scale subspace learning methods, with the goal of improving classification in large datasets by a range of 3% to 10%. Different combinations of methods are studied. We assess the proposed method on five publicly available large datasets, for different eigen-va…
Probabilistic Foundations of Fuzzy Simplicial Sets for Nonlinear Dimensionality Reduction
Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）；Max Planck Institute for Human Cognitive and Brain Sciences（马克斯·普朗克人类认知与脑科学研究所）；Max Planck School of Cognition（马克斯·普朗克认知学校）；ScaDS.AI (Center for Scalable Data Analytics and AI)（ScaDS.AI（可扩展数据与人工智能中心））；Santa Fe Institute for the Sciences of Complexity（复杂性科学圣菲研究所）
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 wel…
Chongqing University（重庆大学）；University of Rhode Island（罗德岛大学）；Wuhan University（武汉大学）
The graph embedding (GE) methods have been widely applied for dimensionality reduction of hyperspectral imagery (HSI). However, a major challenge of GE is how to choose proper neighbors for graph construction and explore the spatial information of HSI data. In this paper, we proposed an unsupervised dimensionality reduction algorithm termed spatial-spectral manifold reconstruction preserving embedding (SSMRPE) for HSI classification. At first, a weighted mean filter (WMF) is employed to preprocess the image, which aims to reduce the influence of background noise. According to the spatial consi…
Black-box policy optimization is a class of reinforcement learning algorithms that explores and updates the policies at the parameter level. This class of algorithms is widely applied in robotics with movement primitives or non-differentiable policies. Furthermore, these approaches are particularly relevant where exploration at the action level could cause actuator damage or other safety issues. However, Black-box optimization does not scale well with the increasing dimensionality of the policy, leading to high demand for samples, which are expensive to obtain in real-world systems. In many pr…
Over the past few decades, we have witnessed a large family of algorithms that have been designed to provide different solutions to the problem of dimensionality reduction (DR). The DR is an essential tool to excavate the important information from the high-dimensional data by mapping the data to a low-dimensional subspace. Furthermore, for the diversity of varied high-dimensional data, the multi-view features can be utilized for improving the learning performance. However, many DR methods fail to integrating multiple views. Although the features from different views are extracted by different…
Benchmarking Dimensionality Reduction Techniques for Spatial Transcriptomics
Institute for Clarity in Documentation（清晰文档研究所）；Inria Paris-Rocquencourt（巴黎-罗克奎恩特研究所）；Rajiv Gandhi University（拉贾夫·甘地大学）；Tsinghua University（清华大学）；Palmer Research Laboratories（帕勒尔研究实验室）
Dimensionality-reduction techniques are a fundamental tool for extracting useful information from high-dimensional data sets. Because secant sets encode manifold geometry, they are a useful tool for designing meaningful data-reduction algorithms. In one such approach, the goal is to construct a projection that maximally avoids secant directions and hence ensures that distinct data points are not mapped too close together in the reduced space. This type of algorithm is based on a mathematical framework inspired by the constructive proof of Whitney's embedding theorem from differential topology.…
Aarhus University（奥胡斯大学）；Academic College of Tel-Aviv Yaffo（特拉维夫雅法学术学院）；UC Berkeley（加州大学伯克利分校）
The sparse Johnson-Lindenstrauss transform is one of the central techniques in dimensionality reduction. It supports embedding a set of $n$ points in $\mathbb{R}^d$ into $m=O(\varepsilon^{-2} \lg n)$ dimensions while preserving all pairwise distances to within $1 \pm \varepsilon$. Each input point $x$ is embedded to $Ax$, where $A$ is an $m \times d$ matrix having $s$ non-zeros per column, allowing for an embedding time of $O(s \|x\|_0)$. Since the sparsity of $A$ governs the embedding time, much work has gone into improving the sparsity $s$. The current state-of-the-art by Kane and Nelson (JA…
