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Introduction to Topological Data Analysis Persistent Homology Norm - - PowerPoint PPT Presentation

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis Introduction to Topological Data Analysis Persistent Homology Norm Matloff University of California, Davis Introduction to Topological Broad Overview

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Introduction to Topological Data Analysis

Persistent Homology Norm Matloff University of California, Davis

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Broad Overview

  • Determine “what is connected to what” in dataset.

Definition of connected depends on the application and the ingenuity of the analyst. (Note this.)

  • Do this in each of a sequence of steps.
  • Each step produces some kind of data summarizing
  • connectivity. The data is collectively called a filtration.
  • Use that output data as features, e.g. to do classification.
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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Image Classification Example

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Image Classification Example

  • The famous MNIST data, hand-drawn digits. Determine

what digit it is, by analyzing the pixels (28 × 28).

  • Not just greyscale, but mainly black-and-white. Here I’ll

look only a pixels > 192 level.

  • For simplicity, I’ll first use a somewhat nonstandard (and

new-ish) TDA method.

  • May or may not be better than other methods.
  • But is simple, easy to explain and draw.
  • Just an example.
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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Crucial need for Dimension Reduction

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Crucial need for Dimension Reduction

  • In MNIST case, we are predicting digit from 282 = 784

features.

  • 784 way too large: (a) Overfitting. (b) Horrendous

computation needs.

  • So, we need to convert the existing 784 features to a

smaller number (dimension reduction). But how?

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Dimension Reduction Methods for Images

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Dimension Reduction Methods for Images

  • Principal Components Analysis (PCA)
  • A traditional approach. Project the data from R784 to, say,

R50, using eigenanalysis.

  • Plug into logit, maybe with polynomial terms (my polyreg

package).

  • Convolutional Neural Networks (CNNs)
  • Currently most fashionable.
  • Not new! The “C” part of CNN is just traditional image

smoothing, breaking the image into small tiles, and then e.g. finding the median pixel intensity in each tile. E.g. in MNIST, take 4×4 tiles, so now have 72 = 49 predictors.

  • Geometric methods:
  • Runs statistics (counts of how many consecutive vertical or

horizontal pixels are black, etc.).

  • TDA.
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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

A ’6’

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

A ’6’

Filtration plan:

  • Draw a series of horizontal lines.
  • See how many components are formed in the figure by a

line.

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

A ’6’

0 components

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

A ’6’

1 component (2 adjacent pixels)

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

A ’6’

3 components (2 adj. pixels, then 1 and 1)

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

A ’6’

3 components (2 adj. pixels, then 1 and 1)

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Birth, Death Times

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Birth, Death Times

Then as the red line is moved upward, will mostly have 3 components for a while, then 1.

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Birth, Death Times

Then as the red line is moved upward, will mostly have 3 components for a while, then 1. We talk about birth and death times. E.g. the first 3-component line is “born” at line 17 and “dies” at line 25.

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

A ’7’

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

A ’7’

A 1-component line will be born early on, then persist for a long time.

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

A ’7’

A 1-component line will be born early on, then persist for a long time. Then we may get a 2-component birth, not long-lived.

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

’6’ vs. ’7’

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

’6’ vs. ’7’

digit pattern ’6’ 3 comps., then 1 ’7’ 1 comp., then 2

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

’6’ vs. ’7’

digit pattern ’6’ 3 comps., then 1 ’7’ 1 comp., then 2

  • So, easy to distinguish ’6’ and ’7’ via BD data, right?
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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

’6’ vs. ’7’

digit pattern ’6’ 3 comps., then 1 ’7’ 1 comp., then 2

  • So, easy to distinguish ’6’ and ’7’ via BD data, right?
  • But what if the top bar of a ’7’ is angled slightly up, not

down?

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

’6’ vs. ’7’

digit pattern ’6’ 3 comps., then 1 ’7’ 1 comp., then 2

  • So, easy to distinguish ’6’ and ’7’ via BD data, right?
  • But what if the top bar of a ’7’ is angled slightly up, not

down? Then only have a 1-comp.

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

’6’ vs. ’7’

digit pattern ’6’ 3 comps., then 1 ’7’ 1 comp., then 2

  • So, easy to distinguish ’6’ and ’7’ via BD data, right?
  • But what if the top bar of a ’7’ is angled slightly up, not

down? Then only have a 1-comp.

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

A Second Opinion

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

A Second Opinion

Solution: “Get a second opinion”: Collect vertical-bar BD data. digit pattern ’6’ mainly 3 comps. ’7’ mainly 2 comps.

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

A Second Opinion

Solution: “Get a second opinion”: Collect vertical-bar BD data. digit pattern ’6’ mainly 3 comps. ’7’ mainly 2 comps. So, our new features could be the two sets of BD data, horizontal and vertical sweeps.

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Not Out of the Woods Yet

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Not Out of the Woods Yet

Not so simple. For instance:

  • Anomalous BDs: Sometimes have fainter pixels than our

192 threshold. E.g. line 20 in the ’6’ had a gap. Causes an incorrect birth/death.

  • Vectorization: Different images for the same digit have

different numbers of BD data. But ML methods require the feature vector to have a constant number of features from one data point to another (in this case one image to another).

  • Orientation:The above filtration scheme largely assumed:
  • Mainly black-and-white image, not even greyscale (e.g.

Fashion MNIST).

  • Image has a notion of left-right, up-down.
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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Possible Solutions: Anomalous BDs

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Possible Solutions: Anomalous BDs

  • Ignore row 20 in the BD calculation.
  • Ignore any row/column that would create a short-lived

component (D - B = 1 or 2, say).

  • But what if they are real?
  • Maybe do BD at each of several pixel intensity thresholds,

e.g. 64, 128, 192.

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Possible Solutions: Vectorization

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Possible Solutions: Vectorization

  • Say have 35-row images. The possible (B,D) grid is

(i, j) : 1 ≤ i < j ≤ 35). For each image, calculate the count of (B,D) pairs at each grid point, as the red horizontal line moves up. Do the same for the red vertical

  • lines. That data, placed in a vector, is now the feature

vector for this image.

  • For a large, detailed image, the above method may need

voluminous computation and/or lead to overfitting. Some analysts devise their own ad hoc method. E.g. Garside (2019) compute a vector consisting of the number of pixels, average lifetime, area under the persistence function, and four measures based on polygons drawn in the graph of persistence.

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Possible Solutions: Orientation

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Possible Solutions: Orientation

Lots of filtration methods don’t assume the image has a left and right, up and down. E.g. “topographic” method (described next).

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

“Topographic” Filtration

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

“Topographic” Filtration

  • Here the thresholding on pixel intensity is the filtration,

rather than an add-on as above.

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

“Topographic” Filtration

  • Here the thresholding on pixel intensity is the filtration,

rather than an add-on as above.

  • Imagine a 3-D representation of image. X, Y dims. are

image row, column, then Z is the pixel intensity. Looks like mountain peaks above the (X,Y) plane.

  • Instead of a red line, we now have a red plane, above and

parallel to the (X,Y).

  • Initially, all nonzero pixels are above the red plane. But as

it moves higher, the pixels with lower intensities begin to drop out, thus creating BD data.

  • No implied notion of left-right, up-down.
  • Again, 3 sets of BD data for RGB.
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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

(Vietoris-)Rips Filtrations

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

(Vietoris-)Rips Filtrations

  • Draw a red circle around each data point, same radius for

all.

  • The filtration consists of drawing an increasing sequence
  • f radii.
  • Points in overlapping circles are considered to be in the

same component.

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

An ’I’

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

An ’I’

  • radius 0.2
  • 8 components
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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

An ’I’

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

An ’I’

  • radius 0.6
  • 1 component
  • the 8 components died at 0.5, the 1 component was born

at 0.5

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

An ’L’

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

An ’L’

  • I took the ’I’ and just bent it; linear distance between

points still 1.0

  • but now there will be a birth at 0.5(0.5

√ 2) = 0.35, not 0.5

  • originally 8 components, then 7, then 1
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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

An ’L’

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

An ’L’

Radius 0.4:

  • 2

4 6 8 10 y

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Rips Senses Angles!

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Rips Senses Angles!

The point: Rips filtration does more than topology; it does

  • geometry. (Math: curvature)
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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Vectorization

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Introduction to Topological Data Analysis Norm Matloff University of California, Davis

Vectorization