Spectral LPM: An Optimal Locality-Preserving Mapping using the Spectral (not Fractal) Order

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Spectral LPM An Optimal Locality-Preserving
Mapping using the Spectral (not Fractal) Order

• Mohamed F. Mokbel
• Walid G. Aref
• Ananth Grama

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Locality-Preserving Mapping

• Multi-dimensional data is more difficult to
  process than one-dimensional data.
• A mapping function f is required to map the
  multi-dimensional space into the one dimensional
  space.
• Locality-Preservation is a desirable property for
  the mapping function f.
• Mapping data from the multi-dimensional space
  into the one-dimensional space is considered
  locality-preserving if the points that are nearby
  in the multi-dimensional space are nearby in the
  one-dimensional space.

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Applications of Locality-Preserving Mapping

• Range Queries, nearest-neighbor queries
• Multi-dimensional spatial join.
• Multi-dimensional indexing.
• R-Tree Packing
• Spatial Access Method.
• Declustering
• Memory Management
• GIS
• Disk Scheduling
• Traveling Salesman problem
• Parallel processing.

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Distance-Preserving or Locality-Preserving

• An optimal Distance-Preserving mapping algorithm
  maps the multi-dimensional space into the
  one-dimensional space such that the distance
  between each pair of points in the
  multi-dimensional space is preserved in the
  one-dimensional space.
• A point P in the D-dimensional space has 2D
  neighbors with Manhattan distance M1. Mapping P
  and its neighbors into the one-dimensional space
  allows only two neighbors to have M1. Thus, the
  distance between 2(D-1) of the points cannot be
  preserved.
• An optimal Distance-Preserving mapping is
  infeasible, what about Locality-Preserving
  mapping ?

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The Good Mapping

• Divide the 2D neighbors into two equal groups.
• Map the first group to the left of P, and the
  second group to the right of P.
• The same algorithm is applied for points with
  Manhattan distance M gt1
• An optimal Locality-Preserving mapping with
  respect to P.
• What about Q and R..?

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The Bad Mapping (Fractals)

• Fractals divide the space into a number of
  fragments.
• Once a fractal starts to visit points from a
  certain fragment, no other fragment is visited
  until the current one is completely exhausted.
• Fractals perform a local optimization based on
  the current fragment.
• Boundary Effect problem. Two points Pi and Pj lie
  on the boundaries of two different fragments and
  Pi-Pj1. However, Pi and Pj will be very far
  from each other in the one-dimensional space.

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How bad is the Fractals ?
Peano Gray Hilbert
P1-P2 6 5 11
P3-P4 22 47 43

• Things become even worse with the increase of the
  grid size

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Can we avoid the Boundary Effect in Fractals?

• NO, the boundary problem is a property of the
  fractals Man77.
• Only two attempts to avoid the boundary effect
• SZM98 uses different space-filling curves for
  the same data. If two points lie on the boundary
  of one space-filling curve, they will not be in
  the boundary of the other space-filling curve.
• LLL01 uses multiple shifted copies of the
  Hilbert SFC. If two points are in the boundary of
  one copy of the Hilbert SFC, they will not be in
  the boundary of another copy.
• The main idea is
• By using more than one SFC, we can hopefully get
  better results

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The Optimal Mapping (Spectral LPM)

• Unlike Fractals, Spectral LPM achieves global
  optimization where all multi-dimensional points
  are taken into account when performing the
  mapping.
• Spectral LPM does not favor any set of points
  over the others.
• Spectral LPM is proved to be globally optimal
  with respect to all multi-dimensional points.

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Overview of the Spectral Algorithms

• Spectral algorithms are based on the spectral
  theory which relates matrix to its eigenvalues
  and eigenvectors.
• A milestone in Spectral algorithms is due to
  Fiedler Fie75 who proposed using the
  eigenvalues and eigenvectors of the Laplacian
  matrix L(G) instead of the Adjacency matrix A(G).
• Spectral Algorithms have been widely used in
• Graph Partitioning
• Data Clustering
• Linear labeling of a graph.
• Up to the authors knowledge, the use of spectral
  mapping to support similarity search queries is a
  novel application

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The Spectral LPM Algorithm

• Algorithm Spectral Locality-Preserving Mapping
  (Spectral LPM)
• Input P, a set of multi-dimensional points.
• Output S, a linear order of the set P.
• Model the set of multi-dimensional points P as a
  graph G(V,E) such that each point Pi?P is
  represented by a vertex vi?V, and there is an
  edge (vi,vj) ?E iff Pi-Pj1.
• Compute the graph Laplacian matrix L(G)
  D(G)-A(G).
• Compute the second smallest eigenvalue ?2 and its
  corresponding eigenvector X2 of L(G).
• For each i1?n, assign the value xi to vi, and
  hence to Pi.
• The linear order S of P is the order of the
  assigned values of Pis.
• Return S
• End

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Example of The Spectral Mapping
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The Optimality of the Spectral Mapping

• Definition
• A vector X (x1,x2,,xn) that represents the n
  one-dimensional values of n multi-dimensional
  points represented as a graph G(V,E) is
  considered to provide the global optimal
  locality-preserving mapping from the
  multi-dimensional space into the one-dimensional
  space if X satisfies the following optimization
  problem

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The Optimality of the Spectral Mapping

• The optimization problem in the optimality
  definition is equivalent to

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The Optimality of the Spectral Mapping

• Theorem Fiedler, 1973
• The solution of the optimization problem
• is the second smallest eigenvalue ?2 and its
  corresponding eigenvector X2

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Extensibility of the Spectral LPM

• Spectral LPM can change the way of constructing
  the graph G.
• Spectral LPM can incorporate any number of
  additional constraints

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Extensibility of the Spectral LPM

• Spectral LPM can model the multi-dimensional
  points as a weighted graph. The weight w of an
  edge e(v1,v2) represents the priority of mapping
  v1 and v2 to nearby locations in the
  one-dimensional space. In this case the objective
  function will be
• The proof of optimality of the Spectral LPM is
  valid regardless of the graph type. Spectral LPM
  is optimal for the chosen graph type.

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Experimental Results

• If the Manhattan distance between any two points
  Pi, Pj in the multi-dimensional space is MD, then
  what is the Manhattan distance M1 between the
  same two points in the one-dimensional space? The
  lower M1 the better the locality-preserving
  mapping.
• For any multi-dimensional range query, what is
  the difference between the minimum and the
  maximum one-dimensional values of the points that
  lie inside the range query? The smaller the
  difference the better the locality-preserving
  mapping.
• The answers of these questions should have
• Bounded worst-case
• Dimension Independence
• Location Independence

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Bounded Worst-Case (Nearest-Neighbor Queries)
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Bounded Worst-Case (Range Queries)
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Dimension Independence
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Dimension Independence (Range Queries)
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Location Independence
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Summary

• The need of a locality-preserving mapping from
  the multi-dimension space into the
  one-dimensional space is needed for a variety of
  applications.
• We argue against the use of fractals as a basis
  for locality-preserving mapping.
• The Spectral LPM algorithm is proposed as an
  optimal locality-preserving mapping that depends
  on the spectral properties of the
  multi-dimensional points.
• The optimality proof of the Spectral LPM
  algorithm is provided.
• Unlike Fractals, the Spectral LPM algorithm can
  be extended in several ways.
• We provide experimental evidence that Spectral
  LPM is superior to the fractal-based algorithms

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References

• Man77 B. Mandelbrot. Fractal Geometry of
  Nature. W. H. Freeman, NY, 1977.
• LLL01 S. Liao, M. Lopez, and S. Leutenegger.
  High Dimensional Similarity search with
  space-filling curves., ICDE 2001.
• SZM98 J. Shepherd, X. Zhu, and N. Megiddo. A
  fast indexing method for multidimensional nearest
  neighbor search. Proc.of SPIE, Storage and
  Retrieval for Image and Video Databases, 1998.
• Ste73 L.A. Steen. Highlights in the history of
  spectral theory. American Mathematically Monthly,
  80(4), 1973.
• Fie73 M. Fiedler. Algebraic Connectivity of
  Graphs. Czechoslovak Mathematical Journal 23(98),
  1973.
• Fie75 M. Fiedler. A property of eigenvectors of
  nonnegative symmetric matrices and its
  application to graph theory. Czechoslovak
  Mathematical Journal 25(100), 1975.