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Hilbert Space of Stationary Ergodic Processes Ishanu chattopadhyay University of Chicago D3M Workshop ICDM 2017

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  • Hilbert Space of Stationary Ergodic Processes

    Ishanu chattopadhyay

    University of Chicago

    D3M Workshop ICDM 2017

  • 2

    Angle?

  • WhyAngles

    ● Intrinsic Geometry● Riemannian Structure● Projections, optimizations

    3

  • Classification Prediction

    Features

    Samples

    4

  • Time Series

    5

  • Time Series

    6

  • Time Series

    ● Prediction● Classification● Understanding

    7

  • Time Series

    8

  • Time Series

    9

  • Time Series

    10

  • Stochastic Processes

    11

  • 12

  • Random Walk

    13

  • Random Walk

    14

  • Probabilistic Finite Automata

    15

  • Learning Probabilistic Finite Automata

    I. Chattopadhyay and H. Lipson , "Abductive learning of quantized stochastic processes with probabilistic finite automata.", Philosophical Transactions of The Royal Society A, Vol. 371(1984), Feb 2013, pp 20110543.

    16

  • Probabilistic Finite Automata

    17

  • Probabilistic Finite Automata

    Ergodic stationaryFinite Valued process

    18

  • Algebraic Structures on Model Space

    Abelian Group19

  • Zero Machine

    20

  • Algebraic Structures on Model Space

    Multiplication by scalars

    21

  • Algebraic Structures on Model Space

    Multiplication by scalars

    System of Equations

    22

  • Data smashing

    I. Chattopadhyay and H. Lipson , "Data Smashing: Uncovering Lurking Order In Data", Royal Society Interface, 11: 20140826

    23

  • where X,Y are probabilistic finite automata

    24

  • TangentsNormalsInner Product

    25

  • TangentsNormalsInner Product

    iid processes26

  • ConstructionFor prob.vectors

    27

  • Inner Product

    28

  • Geodesicspace

    29

  • GeodesicSpace:

    Every noise corruption is a spiral

    30

  • Resistance To Noise Corruption

    31

  • Identify Intrinsic geometry of dataNew Classification Algorithms

    32


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