An official website of the United States government
Abstract Given a Banach space X and a real number α ≥ 1, we write: (1) D ( X ) ≤ α if, for any locally finite metric space A , all finite subsets of which admit bilipschitz embeddings into X with distortions ≤ C , the space A itself admits a bilipschitz embedding into X with distortion ≤ α ⋅ C ; (2) D ( X ) = α + if, for every ϵ > 0, the condition D ( X ) ≤ α + ϵ holds, while D ( X ) ≤ α does not; (3) D ( X ) ≤ α + if D ( X ) = α + or D ( X ) ≤ α. It is known that D ( X ) is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1) D ((⊕ n =1 ∞ X n ) p ) ≤ 1 + for every nested family of finite-dimensional Banach spaces { X n } n =1 ∞ and every 1 ≤ p ≤ ∞. (2) D ((⊕ n =1 ∞ ℓ ∞ n ) p ) = 1 + for 1 < p < ∞. (3) D ( X ) ≤ 4 + for every Banach space X with no nontrivial cotype. Statement (3) is a strengthening of the Baudier–Lancien result (2008).
more »
« less
Cluster Appearance Glyphs: A Methodology for Illustrating High-Dimensional Data Patterns in 2-D Data Layouts
Two-dimensional space embeddings such as Multi-Dimensional Scaling (MDS) are a popular means to gain insight into high-dimensional data relationships. However, in all but the simplest cases these embeddings suffer from significant distortions, which can lead to misinterpretations of the high-dimensional data. These distortions occur both at the global inter-cluster and the local intra-cluster levels. The former leads to misinterpretation of the distances between the various N-D cluster populations, while the latter hampers the appreciation of their individual shapes and composition, which we call cluster appearance. The distortion of cluster appearance incurred in the 2-D embedding is unavoidable since such low-dimensional embeddings always come at the loss of some of the intra-cluster variance. In this paper, we propose techniques to overcome these limitations by conveying the N-D cluster appearance via a framework inspired by illustrative design. Here we make use of Scagnostics which offers a set of intuitive feature descriptors to describe the appearance of 2-D scatterplots. We extend the Scagnostics analysis to N-D and then devise and test via crowd-sourced user studies a set of parameterizable texture patterns that map to the various Scagnostics descriptors. Finally, we embed these N-D Scagnostics-informed texture patterns into shapes derived from N-D statistics to yield what we call Cluster Appearance Glyphs. We demonstrate our framework with a dataset acquired to analyze program execution times in file systems.
more »
« less
- Award ID(s):
- 1650499
- PAR ID:
- 10317224
- Date Published:
- Journal Name:
- Information
- Volume:
- 13
- Issue:
- 1
- ISSN:
- 2078-2489
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Graphs are powerful representations for relations among objects, which have attracted plenty of attention in both academia and industry. A fundamental challenge for graph learning is how to train an effective Graph Neural Network (GNN) encoder without labels, which are expensive and time consuming to obtain. Contrastive Learning (CL) is one of the most popular paradigms to address this challenge, which trains GNNs by discriminating positive and negative node pairs. Despite the success of recent CL methods, there are still two under-explored problems. Firstly, how to reduce the semantic error introduced by random topology based data augmentations. Traditional CL defines positive and negative node pairs via the node-level topological proximity, which is solely based on the graph topology regardless of the semantic information of node attributes, and thus some semantically similar nodes could be wrongly treated as negative pairs. Secondly, how to effectively model the multiplexity of the real-world graphs, where nodes are connected by various relations and each relation could form a homogeneous graph layer. To solve these problems, we propose a novel multiplex heterogeneous graph prototypical contrastive leaning (X-GOAL) framework to extract node embeddings. X-GOAL is comprised of two components: the GOAL framework, which learns node embeddings for each homogeneous graph layer, and an alignment regularization, which jointly models different layers by aligning layer-specific node embeddings. Specifically, the GOAL framework captures the node-level information by a succinct graph transformation technique, and captures the cluster-level information by pulling nodes within the same semantic cluster closer in the embedding space. The alignment regularization aligns embeddings across layers at both node level and cluster level. We evaluate the proposed X-GOAL on a variety of real-world datasets and downstream tasks to demonstrate the effectiveness of the X-GOAL framework.more » « less
-
Knitting creates complex, soft fabrics with unique texture properties that can be used to create interactive objects.However, little work addresses the challenges of designing and using knitted textures computationally. We present KnitPick: a pipeline for interpreting hand-knitting texture patterns into KnitGraphs which can be output to machine and hand-knitting instructions. Using KnitPick, we contribute a measured and photographed data set of 472 knitted textures. Based on findings from this data set, we contribute two algorithms for manipulating KnitGraphs. KnitCarving shapes a graph while respecting a texture, and KnitPatching combines graphs with disparate textures while maintaining a consistent shape. KnitPick is the first system to bridge the gap between hand- and machine-knitting when creating complex knitted textures.more » « less
-
Abstract The t-distributed stochastic neighbor embedding (t-SNE) method is one of the leading techniques for data visualization and clustering. This method finds lower-dimensional embedding of data points while minimizing distortions in distances between neighboring data points. By construction, t-SNE discards information about large-scale structure of the data. We show that adding a global cost function to the t-SNE cost function makes it possible to cluster the data while preserving global intercluster data structure. We test the new global t-SNE (g-SNE) method on one synthetic and two real data sets on flower shapes and human brain cells. We find that significant and meaningful global structure exists in both the plant and human brain data sets. In all cases, g-SNE outperforms t-SNE and UMAP in preserving the global structure. Topological analysis of the clustering result makes it possible to find an appropriate trade-off of data distribution across scales. We find differences in how data are distributed across scales between the two subjects that were part of the human brain data set. Thus, by striving to produce both accurate clustering and positioning between clusters, the g-SNE method can identify new aspects of data organization across scales.more » « less
-
This paper introduces our Diversity Advanced Actor-Critic reinforcement learning (A2C) framework (DAAC) to improve the generalization and accuracy of Natural Language Processing (NLP). We show that the diversification of training samples alleviates overfitting and improves model generalization and accuracy. We quantify diversity on a set of samples using the max dispersion, convex hull volume, and graph entropy based on sentence embeddings in high-dimensional metric space. We also introduce A2C to select such a diversified training subset efficiently. Our experiments achieve up to +23.8 accuracy increase (38.0{\%} relatively) in sentiment analysis, -44.7 perplexity decrease (37.9{\%} relatively) in language modeling, and consistent improvements in named entity recognition over various domains. In particular, our method outperforms both domain adaptation and generalization baselines without using any target domain knowledge.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/info13010003
