Analysis of Dominating Manifolds and Asymptotic Behavior.

Analysis of Dominating Manifolds and Asymptotic Behavior

Proposition and Proof

Main Proposition

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

1. Define 𝐼(𝐵𝑡) in terms of an infimum

2. Use change of variables and asymptotic expansion

3. Show that minimization occurs when most 𝑢𝑖 are small

4. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

5. Use parameterization from Lemma 8.2.2

6. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

7. Define 𝐼(𝐵𝑡) in terms of an infimum

8. Use change of variables and asymptotic expansion

9. Show that minimization occurs when most 𝑢𝑖 are small

10. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

11. Use parameterization from Lemma 8.2.2

12. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

13. Define 𝐼(𝐵𝑡) in terms of an infimum

14. Use change of variables and asymptotic expansion

15. Show that minimization occurs when most 𝑢𝑖 are small

16. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

17. Use parameterization from Lemma 8.2.2

18. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

19. Define 𝐼(𝐵𝑡) in terms of an infimum

20. Use change of variables and asymptotic expansion

21. Show that minimization occurs when most 𝑢𝑖 are small

22. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

23. Use parameterization from Lemma 8.2.2

24. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

25. Define 𝐼(𝐵𝑡) in terms of an infimum

26. Use change of variables and asymptotic expansion

27. Show that minimization occurs when most 𝑢𝑖 are small

28. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

29. Use parameterization from Lemma 8.2.2

30. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

31. Define 𝐼(𝐵𝑡) in terms of an infimum

32. Use change of variables and asymptotic expansion

33. Show that minimization occurs when most 𝑢𝑖 are small

34. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

35. Use parameterization from Lemma 8.2.2

36. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

37. Define 𝐼(𝐵𝑡) in terms of an infimum

38. Use change of variables and asymptotic expansion

39. Show that minimization occurs when most 𝑢𝑖 are small

40. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

41. Use parameterization from Lemma 8.2.2

42. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

43. Define 𝐼(𝐵𝑡) in terms of an infimum

44. Use change of variables and asymptotic expansion

45. Show that minimization occurs when most 𝑢𝑖 are small

46. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

47. Use parameterization from Lemma 8.2.2

48. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

49. Define 𝐼(𝐵𝑡) in terms of an infimum

50. Use change of variables and asymptotic expansion

51. Show that minimization occurs when most 𝑢𝑖 are small

52. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

53. Use parameterization from Lemma 8.2.2

54. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

55. Define 𝐼(𝐵𝑡) in terms of an infimum

56. Use change of variables and asymptotic expansion

57. Show that minimization occurs when most 𝑢𝑖 are small

58. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

59. Use parameterization from Lemma 8.2.2

60. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

61. Define 𝐼(𝐵𝑡) in terms of an infimum

62. Use change of variables and asymptotic expansion

63. Show that minimization occurs when most 𝑢𝑖 are small

64. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

65. Use parameterization from Lemma 8.2.2

66. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

67. Define 𝐼(𝐵𝑡) in terms of an infimum

68. Use change of variables and asymptotic expansion

69. Show that minimization occurs when most 𝑢𝑖 are small

70. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

71. Use parameterization from Lemma 8.2.2

72. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

73. Define 𝐼(𝐵𝑡) in terms of an infimum

74. Use change of variables and asymptotic expansion

75. Show that minimization occurs when most 𝑢𝑖 are small

76. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

77. Use parameterization from Lemma 8.2.2

78. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

79. Define 𝐼(𝐵𝑡) in terms of an infimum

80. Use change of variables and asymptotic expansion

81. Show that minimization occurs when most 𝑢𝑖 are small

82. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

83. Use parameterization from Lemma 8.2.2

84. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

85. Define 𝐼(𝐵𝑡) in terms of an infimum

86. Use change of variables and asymptotic expansion

87. Show that minimization occurs when most 𝑢𝑖 are small

88. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

89. Use parameterization from Lemma 8.2.2

90. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

91. Define 𝐼(𝐵𝑡) in terms of an infimum

92. Use change of variables and asymptotic expansion

93. Show that minimization occurs when most 𝑢𝑖 are small

94. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

95. Use parameterization from Lemma 8.2.2

96. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

97. Define 𝐼(𝐵𝑡) in terms of an infimum

98. Use change of variables and asymptotic expansion

99. Show that minimization occurs when most 𝑢𝑖 are small

100. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

101. Use parameterization from Lemma 8.2.2

102. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

103. Define 𝐼(𝐵𝑡) in terms of an infimum

104. Use change of variables and asymptotic expansion

105. Show that minimization occurs when most 𝑢𝑖 are small

106. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

107. Use parameterization from Lemma 8.2.2

108. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

109. Define 𝐼(𝐵𝑡) in terms of an infimum

110. Use change of variables and asymptotic expansion

111. Show that minimization occurs when most 𝑢𝑖 are small

112. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

113. Use parameterization from Lemma 8.2.2

114. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

115. Define 𝐼(𝐵𝑡) in terms of an infimum

116. Use change of variables and asymptotic expansion

117. Show that minimization occurs when most 𝑢𝑖 are small

118. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

119. Use parameterization from Lemma 8.2.2

120. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

121. Define 𝐼(𝐵𝑡) in terms of an infimum

122. Use change of variables and asymptotic expansion

123. Show that minimization occurs when most 𝑢𝑖 are small

124. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

125. Use parameterization from Lemma 8.2.2

126. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

127. Define 𝐼(𝐵𝑡) in terms of an infimum

128. Use change of variables and asymptotic expansion

129. Show that minimization occurs when most 𝑢𝑖 are small

130. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

131. Use parameterization from Lemma 8.2.2

132. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

133. Define 𝐼(𝐵𝑡) in terms of an infimum

134. Use change of variables and asymptotic expansion

135. Show that minimization occurs when most 𝑢𝑖 are small

136. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

137. Use parameterization from Lemma 8.2.2

138. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

139. Define 𝐼(𝐵𝑡) in terms of an infimum

140. Use change of variables and asymptotic expansion

141. Show that minimization occurs when most 𝑢𝑖 are small

142. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

143. Use parameterization from Lemma 8.2.2

144. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

145. Define 𝐼(𝐵𝑡) in terms of an infimum

146. Use change of variables and asymptotic expansion

147. Show that minimization occurs when most 𝑢𝑖 are small

148. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

149. Use parameterization from Lemma 8.2.2

150. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

151. Define 𝐼(𝐵𝑡) in terms of an infimum

152. Use change of variables and asymptotic expansion

153. Show that minimization occurs when most 𝑢𝑖 are small

154. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

155. Use parameterization from Lemma 8.2.2

156. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

157. Define 𝐼(𝐵𝑡) in terms of an infimum

158. Use change of variables and asymptotic expansion

159. Show that minimization occurs when most 𝑢𝑖 are small

160. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

161. Use parameterization from Lemma 8.2.2

162. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

163. Define 𝐼(𝐵𝑡) in terms of an infimum

164. Use change of variables and asymptotic expansion

165. Show that minimization occurs when most 𝑢𝑖 are small

166. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

167. Use parameterization from Lemma 8.2.2

168. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

169. Define 𝐼(𝐵𝑡) in terms of an infimum

170. Use change of variables and asymptotic expansion

171. Show that minimization occurs when most 𝑢𝑖 are small

172. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

173. Use parameterization from Lemma 8.2.2

174. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

175. Define 𝐼(𝐵𝑡) in terms of an infimum

176. Use change of variables and asymptotic expansion

177. Show that minimization occurs when most 𝑢𝑖 are small

178. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

179. Use parameterization from Lemma 8.2.2

180. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

181. Define 𝐼(𝐵𝑡) in terms of an infimum

182. Use change of variables and asymptotic expansion

183. Show that minimization occurs when most 𝑢𝑖 are small

184. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

185. Use parameterization from Lemma 8.2.2

186. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

187. Define 𝐼(𝐵𝑡) in terms of an infimum

188. Use change of variables and asymptotic expansion

189. Show that minimization occurs when most 𝑢𝑖 are small

190. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

191. Use parameterization from Lemma 8.2.2

192. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

193. Define 𝐼(𝐵𝑡) in terms of an infimum

194. Use change of variables and asymptotic expansion

195. Show that minimization occurs when most 𝑢𝑖 are small

196. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

197. Use parameterization from Lemma 8.2.2

198. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

199. Define 𝐼(𝐵𝑡) in terms of an infimum

200. Use change of variables and asymptotic expansion

201. Show that minimization occurs when most 𝑢𝑖 are small

202. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

203. Use parameterization from Lemma 8.2.2

204. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

205. Define 𝐼(𝐵𝑡) in terms of an infimum

206. Use change of variables and asymptotic expansion

207. Show that minimization occurs when most 𝑢𝑖 are small

208. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

209. Use parameterization from Lemma 8.2.2

210. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

211. Define 𝐼(𝐵𝑡) in terms of an infimum

212. Use change of variables and asymptotic expansion

213. Show that minimization occurs when most 𝑢𝑖 are small

214. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

215. Use parameterization from Lemma 8.2.2

216. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

217. Define 𝐼(𝐵𝑡) in terms of an infimum

218. Use change of variables and asymptotic expansion

219. Show that minimization occurs when most 𝑢𝑖 are small

220. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

221. Use parameterization from Lemma 8.2.2

222. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

223. Define 𝐼(𝐵𝑡) in terms of an infimum

224. Use change of variables and asymptotic expansion

225. Show that minimization occurs when most 𝑢𝑖 are small

226. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

227. Use parameterization from Lemma 8.2.2

228. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

229. Define 𝐼(𝐵𝑡) in terms of an infimum

230. Use change of variables and asymptotic expansion

231. Show that minimization occurs when most 𝑢𝑖 are small

232. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

233. Use parameterization from Lemma 8.2.2

234. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

235. Define 𝐼(𝐵𝑡) in terms of an infimum

236. Use change of variables and asymptotic expansion

237. Show that minimization occurs when most 𝑢𝑖 are small

238. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

239. Use parameterization from Lemma 8.2.2

240. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

241. Define 𝐼(𝐵𝑡) in terms of an infimum

242. Use change of variables and asymptotic expansion

243. Show that minimization occurs when most 𝑢𝑖 are small

244. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

245. Use parameterization from Lemma 8.2.2

246. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

247. Define 𝐼(𝐵𝑡) in terms of an infimum

248. Use change of variables and asymptotic expansion

249. Show that minimization occurs when most 𝑢𝑖 are small

250. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

251. Use parameterization from Lemma 8.2.2

252. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

253. Define 𝐼(𝐵𝑡) in terms of an infimum

254. Use change of variables and asymptotic expansion

255. Show that minimization occurs when most 𝑢𝑖 are small

256. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

257. Use parameterization from Lemma 8.2.2

258. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

259. Define 𝐼(𝐵𝑡) in terms of an infimum

260. Use change of variables and asymptotic expansion

261. Show that minimization occurs when most 𝑢𝑖 are small

262. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

263. Use parameterization from Lemma 8.2.2

264. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

265. Define 𝐼(𝐵𝑡) in terms of an infimum

266. Use change of variables and asymptotic expansion

267. Show that minimization occurs when most 𝑢𝑖 are small

268. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

269. Use parameterization from Lemma 8.2.2

270. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

271. Define 𝐼(𝐵𝑡) in terms of an infimum

272. Use change of variables and asymptotic expansion

273. Show that minimization occurs when most 𝑢𝑖 are small

274. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

275. Use parameterization from Lemma 8.2.2

276. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

277. Define 𝐼(𝐵𝑡) in terms of an infimum

278. Use change of variables and asymptotic expansion

279. Show that minimization occurs when most 𝑢𝑖 are small

280. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

281. Use parameterization from Lemma 8.2.2

282. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

283. Define 𝐼(𝐵𝑡) in terms of an infimum

284. Use change of variables and asymptotic expansion

285. Show that minimization occurs when most 𝑢𝑖 are small

286. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

287. Use parameterization from Lemma 8.2.2

288. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

289. Define 𝐼(𝐵𝑡) in terms of an infimum

290. Use change of variables and asymptotic expansion

291. Show that minimization occurs when most 𝑢𝑖 are small

292. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

293. Use parameterization from Lemma 8.2.2

294. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

295. Define 𝐼(𝐵𝑡) in terms of an infimum

296. Use change of variables and asymptotic expansion

297. Show that minimization occurs when most 𝑢𝑖 are small

298. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

299. Use parameterization from Lemma 8.2.2

300. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

301. Define 𝐼(𝐵𝑡) in terms of an infimum

302. Use change of variables and asymptotic expansion

303. Show that minimization occurs when most 𝑢𝑖 are small

304. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

305. Use parameterization from Lemma 8.2.2

306. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

307. Define 𝐼(𝐵𝑡) in terms of an infimum

308. Use change of variables and asymptotic expansion

309. Show that minimization occurs when most 𝑢𝑖 are small

310. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

311. Use parameterization from Lemma 8.2.2

312. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

313. Define 𝐼(𝐵𝑡) in terms of an infimum

314. Use change of variables and asymptotic expansion

315. Show that minimization occurs when most 𝑢𝑖 are small

316. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

317. Use parameterization from Lemma 8.2.2

318. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

319. Define 𝐼(𝐵𝑡) in terms of an infimum

320. Use change of variables and asymptotic expansion

321. Show that minimization occurs when most 𝑢𝑖 are small

322. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

323. Use parameterization from Lemma 8.2.2

324. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

325. Define 𝐼(𝐵𝑡) in terms of an infimum

326. Use change of variables and asymptotic expansion

327. Show that minimization occurs when most 𝑢𝑖 are small

328. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

329. Use parameterization from Lemma 8.2.2

330. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

331. Define 𝐼(𝐵𝑡) in terms of an infimum

332. Use change of variables and asymptotic expansion

333. Show that minimization occurs when most 𝑢𝑖 are small

334. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

335. Use parameterization from Lemma 8.2.2

336. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

337. Define 𝐼(𝐵𝑡) in terms of an infimum

338. Use change of variables and asymptotic expansion

339. Show that minimization occurs when most 𝑢𝑖 are small

340. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

341. Use parameterization from Lemma 8.2.2

342. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Assume 𝐽1 is nonempty. For 𝑦 ∈ ∂𝐵𝑡 with 𝐼(𝑦)≤ 𝐼(𝐵𝑡)+ 𝑂(1) as 𝑡 → ∞:

• 𝑦 is in a 𝑂(1)-neighborhood of points 𝑞𝜖,𝑗,𝑡 for some 𝑗 ∈ 𝐽1 and 𝜖 ∈ {−1,1}

• As 𝑡 → ∞:

𝐼(𝑞𝜖,𝑗,𝑡) = 𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝐶𝑗,𝑗 + 𝑜(1)

Proof Outline

343. Define 𝐼(𝐵𝑡) in terms of an infimum

344. Use change of variables and asymptotic expansion

345. Show that minimization occurs when most 𝑢𝑖 are small

346. Analyze the function 𝐼 near 𝑞𝜖,𝑗,𝑡

347. Use parameterization from Lemma 8.2.2

348. Obtain the final expression for 𝐼(𝑞𝜖,𝑗,𝑡)

Key Observations

• If 𝑥 ∈ ∂𝐵𝑡 and 𝐼(𝑥)= 𝐼(𝐵𝑡)+ 𝑜(loglog𝑡), then 𝑥 is in a 𝑜(loglog𝑡)-neighborhood of

some 𝑞𝜖,𝑗,𝑡

• Define 𝛾1= max1≤𝑗≤𝑑𝐶𝑗,𝑗

• Asymptotic formula for 𝐼(𝐵𝑡):

𝐼(𝐵𝑡)=𝛼

2log𝑡 − 1

2loglog√𝑡 − log(𝐾𝑠,𝛼𝛼𝛼/22√𝜋) + log(2𝜋)𝑑/2 −𝛼

2log𝛾1+ 𝑜(1)

Dominating Manifold

Candidate for Dominating Manifold

Initial candidate: {𝑞𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Issue with Initial Candidate

The points 𝑞𝜖,𝑗,𝑡 do not lie on the same sphere centered at the origin, which is required for a

base manifold.

Adjusted Candidate

• Define 𝜌𝑡 as the radius of the sphere 𝐼(𝐵𝑡)

• 𝜌𝑡=√𝐼(𝐵𝑡)+ 𝑑log(2𝜋)

• Define 𝑟𝜖,𝑗,𝑡 = 𝜌𝑡𝑞𝜖,𝑗,𝑡/|𝑞𝜖,𝑗,𝑡|

• New candidate: 𝐷𝐵𝑡= {𝑟𝜖,𝑗,𝑡: 𝑗 ∈ 𝐽1, 𝜖 ∈ {−1,1}}

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

Conclusion

The adjusted candidate 𝐷𝐵𝑡 moves the points 𝑞𝜖,𝑗,𝑡 along the normal flow until they reach

the level line 𝐼(𝐵𝑡). This coincides with the Euclidean projection on the sphere when

working with the normal distribution.

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