An Interesting Question

An Interesting Question
How generally applicable is Backwards approach to PCA? An Attractive Answer: James Damon, UNC Mathematics Key Idea: Express Backwards PCA as Nested Series of Constraints

General View of Backwards PCA
Define Nested Spaces via Constraints E.g. SVD 𝑆 𝑘 = 𝑥 :𝑥= 𝑗=1 𝑘 𝑐 𝑗 𝑢 𝑗 Now Define: 𝑆 𝑘−1 = 𝑥∈ 𝑆 𝑘 : 𝑥, 𝑢 𝑘 =0 Constraint Gives Nested Reduction of Dim’n

Vectors of Angles Vectors of Angles as Data Objects 𝜃 1 ⋮ 𝜃 𝑑 ∈ 𝑆 1 𝑑
𝜃 1 ⋮ 𝜃 𝑑 ∈ 𝑆 1 𝑑 Slice space 𝑆 1 𝑑 with hyperplanes???? (ala Principal Nested Spheres)

Vectors of Angles E.g. 𝑑=2, Data w/ “Single Mode of Var’n”
Best Fitting Planar Slice gives Bimodal Dist’n Special Thanks to Eduardo García-Portugués

Torus Space 𝑺 𝟏 × 𝑺 𝟏 Try To Fit A Geodesic Challenge: Can Get
Arbitrarily Close

Torus Space 𝑺 𝟏 × 𝑺 𝟏 Fit Nested Sub-Manifold

PNS Main Idea Data Objects: 𝑋 1 ,⋯, 𝑋 𝑛 ∈ 𝑆 𝑑 ⊆ ℝ 𝑑+1 Where 𝑆 𝑑 is a 𝑑 dimensional manifold Consider a nested series of sub-manifolds: 𝒮= 𝑆 0 ,⋯, 𝑆 𝑑−1 where for 𝑗=0,⋯,𝑑−1 𝑑𝑖𝑚 𝑆 𝑗 =𝑗 and 𝑆 𝑗 ⊆ 𝑆 𝑗+1 Goal: Fit all of 𝒮 simultaneously to 𝑋 1 ,⋯ 𝑋 𝑛

General Background Call each 𝑆 𝑗 a stratum,
so 𝒮 is a manifold stratification To be fit to 𝑋 1 ,⋯ 𝑋 𝑛 New Approach: Simultaneously fit 𝒮= 𝑆 0 ,⋯, 𝑆 𝑑−1 Nested Submanifold (NS)

Projection Notation For 𝑗=0,⋯,𝑑−1 let 𝑃 (𝑘) denote the telescoping projection onto 𝑆 𝑘 I.e. for X∈ 𝑆 𝑑 𝑃 (𝑘) = 𝑃 𝑘 𝑃 𝑘+1 ⋯ 𝑃 𝑑−1 𝑋 Note: This projection is fundamental to Backwards PCA methods

PNS Components For a given 𝒮, represent a point 𝑋∈ 𝑆 𝑑
by its Nested Submanifold components: 𝑐 1 𝑋 ,⋯, 𝑐 𝑑 (𝑋) where for 𝑗=1,⋯,𝑑 𝑐 𝑗 𝑋 = 𝑃 𝑗 𝑋 − 𝑃 𝑗−1 (𝑋) In the sense that “𝐴−𝐵” means the shortest geodesic arc between 𝐴 & 𝐵

Nested Submanifold Fits
Simultaneous Fit Criteria? Based on Stratum-Wise Sums of Squares For 𝑗=1,⋯,𝑑 define 𝑆𝑆 𝑗 = 𝑖=1 𝑛 𝑑 𝑃 𝑗 𝑋 𝑖 , 𝑃 𝑗−1 𝑋 𝑖 2 Uses “lengths” of NS Components: 𝑐 𝑗 𝑋 = 𝑃 𝑗 𝑋 − 𝑃 𝑗−1 (𝑋)

NS Components in ℝ 2 NS Candidate 2 (Shifted 𝑆 0 to Sample Mean)
Note: Both 𝑆𝑆 1 & 𝑆𝑆 2 Decrease

NS Components in ℝ 2 NS based On PC1 Note: 𝑆𝑆 1 ↑ 𝑆𝑆 2 ↓
𝑆𝑆 1 ↑ 𝑆𝑆 2 ↓ Yet 𝑆𝑆 𝑆𝑆 2 is Constant (Pythagorean Thm)

NS Components in ℝ 2 NS based On PC2 Note: 𝑆𝑆 1 ↓ 𝑆𝑆 2 ↑
𝑆𝑆 1 ↓ 𝑆𝑆 2 ↑ 𝑆𝑆 1 + 𝑆𝑆 2 is Constant (Pythagorean Thm)

NS Components in ℝ 2 NS Candidate 1

NS Components in ℝ 2 NS Candidate 2 0.3× 𝑆𝑆 × 𝑆𝑆 2 ↓

NS Components in ℝ 2 NS based On PC1 0.3× 𝑆𝑆 × 𝑆𝑆 2 ↓

NS Components in ℝ 2 NS based On PC2 0.3× 𝑆𝑆 × 𝑆𝑆 2 ↑

Simultaneously fit 𝒮= 𝑆 0 ,⋯, 𝑆 𝑑−1 Simultaneous Fit Criterion? 𝑤 1 𝑆𝑆 1 + 𝑤 2 𝑆𝑆 2 +⋯+ 𝑤 𝑑 𝑆𝑆 𝑑 Above Suggests Want: 𝑤 1 < 𝑤 2 <⋯< 𝑤 𝑑 Works for Euclidean PCA (?)

Simultaneous Fit Criterion? 𝑤 1 𝑆𝑆 1 + 𝑤 2 𝑆𝑆 2 +⋯+ 𝑤 𝑑 𝑆𝑆 𝑑 Above Suggests Want: 𝑤 1 < 𝑤 2 <⋯< 𝑤 𝑑 Important Predecessor Pennec (2016) AUC Criterion: 𝑤 𝑗 ∝(𝑗−1)

Pennec’s Area Under the Curve
100% Based on Scree Plot 𝑆𝑆 1 𝑆𝑆 2 𝑆𝑆 3 𝑆𝑆 4 1 2 3 4 Component Index

100% Based on Scree Plot Cumulative 𝑆𝑆 1 𝑆𝑆 2 𝑆𝑆 3 𝑆𝑆 4 1 2 3 4 Component Index

100% Based on Scree Plot Cumulative Area = 𝑆𝑆 2 +2 𝑆𝑆 3 +3 𝑆𝑆 4 𝑆𝑆 1 𝑆𝑆 2 𝑆𝑆 3 𝑆𝑆 4 1 2 3 4 Component Index

Torus Space 𝑺 𝟏 × 𝑺 𝟏 Fit Nested Sub-Manifold Choice of 𝑤 1 & 𝑤 2 in:
𝑤 1 𝑆𝑆 1 + 𝑤 2 𝑆𝑆 2 ???

(maybe OK for low rank approx.)
Torus Space 𝑺 𝟏 × 𝑺 𝟏 ×⋯× 𝑺 𝟏 Tiled (−𝜋,𝜋] 𝑑 embedding is complicated (maybe OK for low rank approx.) Instead Consider Nested Sub-Torii Work in Progress with Garcia, Wood, Le Key Factor: Important Modes of Variation

OODA Big Picture New Topic: Curve Registration Main Reference: Srivastava et al (2011)

Collaborators Anuj Srivastava (Florida State U.)
Wei Wu (Florida State U.) Derek Tucker (Florida State U.) Xiaosun Lu (U. N. C.) Inge Koch (U. Adelaide) Peter Hoffmann (U. Adelaide) J. O. Ramsay (McGill U.) Laura Sangalli (Milano Polytech.)

Context Functional Data Analysis Curves as Data Objects Toy Example:

Context Functional Data Analysis Curves as Data Objects Toy Example:
How Can We Understand Variation?

Functional Data Analysis
Insightful Decomposition

Insightful Decomposition Horiz’l Var’n

Insightful Decomposition Vertical Variation Horiz’l Var’n

(even mathematical formulation)
Challenge Fairly Large Literature Many (Diverse) Past Attempts Limited Success (in General) Surprisingly Slippery (even mathematical formulation)

Challenge (Illustrated)
Thanks to Wei Wu

Appropriate Mathematical Framework? Vertical Variation Horiz’l Var’n

Landmark Based Shape Analysis
Approach: Identify objects that are: Translations Rotations Scalings of each other Mathematics: Equivalence Relation Results in: Equivalence Classes Which become the Data Objects

Landmark Based Shape Analysis
Equivalence Classes become Data Objects a.k.a. “Orbits” Mathematics: Called “Quotient Space” , , , , , ,

Curve Registration What are the Data Objects? Vertical Variation
Horiz’l Var’n

Curve Registration What are the Data Objects? Consider “Time Warpings” 𝛾: 0,1 → [0,1] (smooth) More Precisely: Diffeomorphisms

Curve Registration Diffeomorphisms 𝛾: 0,1 → [0,1] 𝛾 is 1 to 1
𝛾: 0,1 → [0,1] 𝛾 is 1 to 1 𝛾 is onto (thus 𝛾 is invertible) Differentiable 𝛾 −1 is Differentiable

Time Warping Intuition
Elastically Stretch & Compress Axis

Elastically Stretch & Compress Axis 𝛾 x =x (identity)

Elastically Stretch & Compress Axis 𝛾 x

Curve Registration Say curves 𝑓 1 (𝑥) and 𝑓 2 (𝑥) are equivalent, 𝑓 1 ≈ 𝑓 2 When ∃𝛾 so that 𝑓 1 𝛾 𝑥 = 𝑓 1 ∘𝛾 𝑥 = 𝑓 2 (𝑥)

Curve Registration Toy Example: Starting Curve, 𝑓 𝑥

Curve Registration Toy Example: Equivalent Curves, 𝑓 𝑥

Curve Registration Toy Example: Warping Functions

Curve Registration Toy Example: Non-Equivalent Curves Cannot Warp Into Each Other

Data Objects I Equivalence Classes of Curves (parallel to Kendall shape analysis)

Data Objects I Equivalence Classes of Curves (Set of All Warps of Given Curve) Notation: 𝑓 = 𝑓∘𝛾: 𝛾∈Γ for a “representor” 𝑓 𝑥

Data Objects I Equivalence Classes of Curves (Set of All Warps of Given Curve) Next Task: Find Metric on Space of Curves

Metrics in Curve Space Find Metric on Equivalence Classes Start with Warp Invariant Metric on Curves & Extend

Metrics in Curve Space Traditional Approach to Curve Registration:
Align curves, say 𝑓 1 and 𝑓 2 By finding optimal time warp, 𝛾, so: 𝑖𝑛𝑓 𝛾 𝑓 1 − 𝑓 2 ∘𝛾 Vertical var’n: PCA after alignment Horizontal var’n: PCA on 𝛾s

Metrics in Curve Space Problem: Don’t have proper metric Since: 𝑑 𝑓 1 , 𝑓 2 ≠𝑑 𝑓 2 , 𝑓 1 Because: 𝑖𝑛𝑓 𝛾 𝑓 1 − 𝑓 2 ∘𝛾 ≠ 𝑖𝑛𝑓 𝛾 𝑓 2 − 𝑓 1 ∘𝛾

Metrics in Curve Space 𝑖𝑛𝑓 𝛾 𝑓 1 − 𝑓 2 ∘𝛾 ≠ 𝑖𝑛𝑓 𝛾 𝑓 2 − 𝑓 1 ∘𝛾
𝑖𝑛𝑓 𝛾 𝑓 1 − 𝑓 2 ∘𝛾 ≠ 𝑖𝑛𝑓 𝛾 𝑓 2 − 𝑓 1 ∘𝛾 Thanks to Xiaosun Lu

Metrics in Curve Space 𝑖𝑛𝑓 𝛾 𝑓 1 − 𝑓 2 ∘𝛾 ≠ 𝑖𝑛𝑓 𝛾 𝑓 2 − 𝑓 1 ∘𝛾 Note:
𝑖𝑛𝑓 𝛾 𝑓 1 − 𝑓 2 ∘𝛾 ≠ 𝑖𝑛𝑓 𝛾 𝑓 2 − 𝑓 1 ∘𝛾 Note: Very Different L2 norms Thanks to Xiaosun Lu

Metrics in Curve Space Solution: Look for Warp Invariant Metric 𝑑 Where: 𝑑 𝑓 1 , 𝑓 2 =𝑑 𝑓 1 ∘𝛾, 𝑓 2 ∘𝛾

Metrics in Curve Space 𝑑 𝑓 1 , 𝑓 2 =𝑑 𝑓 1 ∘𝛾, 𝑓 2 ∘𝛾 I.e. Have “Parallel” Representatives Of Equivalence Classes

Metrics in Curve Space Warp Invariant Metric 𝑑 Developed in context of: Likelihood Geometry Fisher – Rao Metric: 𝑑 𝐹𝑅 𝑓 1 , 𝑓 2 = 𝑑 𝐹𝑅 𝑓 1 ∘𝛾, 𝑓 2 ∘𝛾

Metrics in Curve Space Fisher – Rao Metric: Computation Based on Square Root Velocity Function (SRVF) 𝑞 𝑓 𝑡 = 𝑓 𝑡 𝑓 𝑡 Signed Version Of Square Root Derivative Where 𝑓 𝑡 = 𝜕 𝜕𝑡 𝑓(𝑡)

Metrics in Curve Space Square Root Velocity Function (SRVF) 𝑞 𝑓 𝑡 = 𝑓 𝑡 𝑓 𝑡 𝑓 𝑡 =𝑓 𝑡 𝑞 𝑓 𝑠 𝑞 𝑓 𝑠 𝑑𝑠

Metrics in Curve Space Fisher – Rao Metric: Computation Based on SRVF: 𝑑 𝐹𝑅 𝑓 1 , 𝑓 2 = 𝑞 1 − 𝑞 2 2 So work with SRVF, Since much easier to compute.

Metrics in Curve Space Why square roots? Thanks to Xiaosun Lu

Metrics in Curve Space Why square roots?

Metrics in Curve Space Why square roots? Dislikes Pinching Focusses Well On Peaks of Unequal Height

Metrics in Curve Space Note on SRVF representation: 𝑑 𝐹𝑅 𝑓 1 , 𝑓 2 = 𝑞 1 − 𝑞 2 2 Can show: Warp Invariance 𝑑 𝐹𝑅 𝑓 1 , 𝑓 2 = 𝑑 𝐹𝑅 𝑓 1 ∘𝛾, 𝑓 2 ∘𝛾 Follows from Jacobean calculation

Metrics in Curve Quotient Space
Above was Invariance for Individual Curves Now extend to: Equivalence Classes of Curves I.e. Orbits as Data Objects I.e. Quotient Space

Metrics in Curve Quotient Space
Define Metric on Equivalence Classes: For 𝑓 1 & 𝑓 2 , i.e. 𝑞 1 & 𝑞 2 𝑑 𝑓 1 , 𝑓 2 = 𝑖𝑛𝑓 𝛾∈Γ 𝑞 1 − 𝑞 2 ∘𝛾 Independent of Choice of 𝑓 1 & 𝑓 2 By Warp Invariance

Mean in Curve Quotient Space
Benefit of a Metric: Allows Definition of a “Mean” Fréchet Mean Geodesic Mean Barycenter Karcher Mean

Given Equivalence Class Data Objects: 𝑓 1 , 𝑓 2 , ⋯, 𝑓 𝑛 The Karcher Mean is: 𝜇 = 𝑎𝑟𝑔𝑚𝑖𝑛 𝑞 𝑖=1 𝑛 𝑑 𝑞 , 𝑞 𝑖 2

The Karcher Mean is: 𝜇 = 𝑎𝑟𝑔𝑚𝑖𝑛 𝑞 𝑖=1 𝑛 𝑑 𝑞 , 𝑞 𝑖 2 Intuition: Recall, for Euclidean Data Minimizer = Conventional 𝑋

Next Define “Most Representative” Choice of 𝜇 𝑛 As Representer of 𝜇

“Most Representative” 𝜇 𝑛 in 𝜇 Given a candidate 𝜇 Consider warps to each 𝑞 𝑖 Choose 𝜇 𝑛 to make Karcher mean of warps = Identity (under Fisher Rao metric)

“Most Representative” 𝜇 𝑛 in 𝜇 Thanks to Anuj Srivastava

Toy Example – (Details Later)
Estimated Warps (Note: Represented With Karcher Mean At Identity)

“Most Representative” 𝜇 𝑛 in 𝜇 Terminology: The “Template Mean”

More Data Objects Final Curve Warps:
Warp Each Data Curve, 𝑓 1 , ⋯, 𝑓 𝑛 To Template Mean, 𝜇 𝑛 Denote Warp Functions 𝛾 1 , ⋯, 𝛾 𝑛 Gives (Roughly Speaking): Vertical Components 𝑓 1 ∘ 𝛾 1 , ⋯, 𝑓 𝑛 ∘ 𝛾 𝑛 (Aligned Curves) Horizontal Components 𝛾 1 , ⋯, 𝛾 𝑛 Data Objects I

Data Objects II Final Curve Warps: Warp Each Data Curve, 𝑓 1 , ⋯, 𝑓 𝑛 To Template Mean, 𝜇 𝑛 Denote Warp Functions 𝛾 1 , ⋯, 𝛾 𝑛 Gives (Roughly Speaking): Vertical Components 𝑓 1 ∘ 𝛾 1 , ⋯, 𝑓 𝑛 ∘ 𝛾 𝑛 (Aligned Curves) Horizontal Components 𝛾 1 , ⋯, 𝛾 𝑛 ~ Kendall’s Shapes

Warp Each Data Curve, 𝑓 1 , ⋯, 𝑓 𝑛 To Template Mean, 𝜇 𝑛 Denote Warp Functions 𝛾 1 , ⋯, 𝛾 𝑛 Gives (Roughly Speaking): Vertical Components 𝑓 1 ∘ 𝛾 1 , ⋯, 𝑓 𝑛 ∘ 𝛾 𝑛 (Aligned Curves) Horizontal Components 𝛾 1 , ⋯, 𝛾 𝑛 Data Objects III ~ Chang’s Transfo’s

Computation Several Variations of Dynamic Programming Done by Eric Klassen, Wei Wu

Toy Example Raw Data

Toy Example Raw Data Both Horizontal And Vertical Variation

Toy Example Conventional PCA Projections

Toy Example Conventional PCA Projections Power Spread Across Spectrum

Toy Example Conventional PCA Scores

Toy Example Conventional PCA Scores Views of 1-d Curve Bending Through 4 Dim’ns’

Toy Example Conventional PCA Scores Patterns Are “Harmonics” In Scores

Toy Example Scores Plot Shows Data Are “1” Dimensional So Need Improved PCA Decomp.

Visualization Vertical Variation:
PCA on Aligned Curves, 𝑓 1 ∘ 𝛾 1 , ⋯, 𝑓 𝑛 ∘ 𝛾 𝑛 Projected Curves

Toy Example Aligned Curves (Clear 1-d Vertical Var’n)

Toy Example Aligned Curve PCA Projections All Var’n In 1st Component

Visualization Horizontal Variation: PCA on Warps, 𝛾 1 , ⋯, 𝛾 𝑛
Projected Curves

Toy Example Estimated Warps

Toy Example Warps, PC Projections

Toy Example Warps, PC Projections Mostly 1st PC

Toy Example Warps, PC Projections Mostly 1st PC, But 2nd Helps Some

Toy Example Warps, PC Projections Rest is Not Important

Toy Example Horizontal Var’n Visualization Challenge: (Complicated) Warps Hard to Interpret Approach: Apply Warps to Template Mean (PCA components)

Toy Example Warp Compon’ts (+ Mean) Applied to Template Mean

Participant Presentations
Xi Yang Multi-View Weighted Network Hang Yu Introduction to multiple kernel learning Zhipeng Ding Fast Predictive Simple Geodesic Regression

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