721 research outputs found
Anomaly and Change Detection in Graph Streams through Constant-Curvature Manifold Embeddings
Mapping complex input data into suitable lower dimensional manifolds is a
common procedure in machine learning. This step is beneficial mainly for two
reasons: (1) it reduces the data dimensionality and (2) it provides a new data
representation possibly characterised by convenient geometric properties.
Euclidean spaces are by far the most widely used embedding spaces, thanks to
their well-understood structure and large availability of consolidated
inference methods. However, recent research demonstrated that many types of
complex data (e.g., those represented as graphs) are actually better described
by non-Euclidean geometries. Here, we investigate how embedding graphs on
constant-curvature manifolds (hyper-spherical and hyperbolic manifolds) impacts
on the ability to detect changes in sequences of attributed graphs. The
proposed methodology consists in embedding graphs into a geometric space and
perform change detection there by means of conventional methods for numerical
streams. The curvature of the space is a parameter that we learn to reproduce
the geometry of the original application-dependent graph space. Preliminary
experimental results show the potential capability of representing graphs by
means of curved manifold, in particular for change and anomaly detection
problems.Comment: To be published in IEEE IJCNN 201
Rigid Transformations for Stabilized Lower Dimensional Space to Support Subsurface Uncertainty Quantification and Interpretation
Subsurface datasets inherently possess big data characteristics such as vast
volume, diverse features, and high sampling speeds, further compounded by the
curse of dimensionality from various physical, engineering, and geological
inputs. Among the existing dimensionality reduction (DR) methods, nonlinear
dimensionality reduction (NDR) methods, especially Metric-multidimensional
scaling (MDS), are preferred for subsurface datasets due to their inherent
complexity. While MDS retains intrinsic data structure and quantifies
uncertainty, its limitations include unstabilized unique solutions invariant to
Euclidean transformations and an absence of out-of-sample points (OOSP)
extension. To enhance subsurface inferential and machine learning workflows,
datasets must be transformed into stable, reduced-dimension representations
that accommodate OOSP.
Our solution employs rigid transformations for a stabilized Euclidean
invariant representation for LDS. By computing an MDS input dissimilarity
matrix, and applying rigid transformations on multiple realizations, we ensure
transformation invariance and integrate OOSP. This process leverages a convex
hull algorithm and incorporates loss function and normalized stress for
distortion quantification. We validate our approach with synthetic data,
varying distance metrics, and real-world wells from the Duvernay Formation.
Results confirm our method's efficacy in achieving consistent LDS
representations. Furthermore, our proposed "stress ratio" (SR) metric provides
insight into uncertainty, beneficial for model adjustments and inferential
analysis. Consequently, our workflow promises enhanced repeatability and
comparability in NDR for subsurface energy resource engineering and associated
big data workflows.Comment: 30 pages, 17 figures, Submitted to Computational Geosciences Journa
Information retrieval and mining in high dimensional databases
This dissertation is composed of two parts. In the first part, we present a framework for finding information (more precisely, active patterns) in three dimensional (3D) graphs. Each node in a graph is an undecoraposable or atomic unit and has a label. Edges are links between the atomic units. Patterns are rigid substructures that may occur in a graph after allowing for an arbitrary number of whole-structure rotations and translations as well as a small number (specified by the user) of edit operations in the patterns or in the graph. (When a pattern appears in a graph only after the graph has been modified, we call that appearance approximate occurrence. ) The edit operations include relabeling a node, deleting a node and inserting a node. The proposed method is based on the geometric hashing technique, which hashes node-triplets of the graphs into a 3D table and compresses the label-triplets in the table. To demonstrate the utility of our algorithms, we discuss two applications of them in scientific data mining. First, we apply the method to locating frequently occurring motifs in two families of proteins pertaining to RNA-directed DNA Polymerase and Thymidylate Synthase, and use the motifs to classify the proteins. Then we apply the method to clustering chemical compounds pertaining to aromatic, bicyclicalkanes and photosynthesis. Experimental results indicate the good performance of our algorithms and high recall and precision rates for both classification and clustering. We also extend our algorithms for processing a class of similarity queries in databases of 3D graphs.
In the second part of the dissertation, we present an index structure, called MetricMap, that takes a set of objects and a distance metric and then maps those objects to a k-dimensional pseudo-Euclidean space in such a way that the distances among objects are approximately preserved. Our approach employs sampling and the calculation of eigenvalues and eigenvectors. The index structure is a useful tool for clustering and visualization in data intensive applications, because it replaces expensive distance calculations by sum-of-square calculations. This can make clustering in large databases with expensive distance metrics practical.
We compare the index structure with another data mining index structure, FastMap, proposed by Faloutsos and Lin, according to two criteria: relative error and clustering accuracy. For relative error, we show that (i) FastMap gives a lower relative error than MetrieMap for Euclidean distances, (ii) MetricMap gives a lower relative error than Fast Map for non-Euclidean distances (i.e., general distance metrics), and (iii) combining the two reduces the error yet further. A similar result is obtained when comparing the accuracy of clustering. These results hold for different data sizes. The main qualitative conclusion is that these two index structures capture complenleiltary information about distance metrics and therefore can be used together to great benefit. The net effect is that multi-day computations can be done in minutes.
We have implemented the proposed algorithms and the MetricMap index structure into a toolkit. This toolkit will be useful for data mining, visualization, and approximate retrieval in scientific, multimedia and high dimensional databases
Disturbance Grassmann Kernels for Subspace-Based Learning
In this paper, we focus on subspace-based learning problems, where data
elements are linear subspaces instead of vectors. To handle this kind of data,
Grassmann kernels were proposed to measure the space structure and used with
classifiers, e.g., Support Vector Machines (SVMs). However, the existing
discriminative algorithms mostly ignore the instability of subspaces, which
would cause the classifiers misled by disturbed instances. Thus we propose
considering all potential disturbance of subspaces in learning processes to
obtain more robust classifiers. Firstly, we derive the dual optimization of
linear classifiers with disturbance subject to a known distribution, resulting
in a new kernel, Disturbance Grassmann (DG) kernel. Secondly, we research into
two kinds of disturbance, relevant to the subspace matrix and singular values
of bases, with which we extend the Projection kernel on Grassmann manifolds to
two new kernels. Experiments on action data indicate that the proposed kernels
perform better compared to state-of-the-art subspace-based methods, even in a
worse environment.Comment: This paper include 3 figures, 10 pages, and has been accpeted to
SIGKDD'1
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