Minimax Rates. Homology Inference
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1 Minimax Rates for Homology Inference Don Sheehy Joint work with Sivaraman Balakrishan, Alessandro Rinaldo, Aarti Singh, and Larry Wasserman
2 Something like a joke.
3 Something like a joke. What is topological inference?
4 Something like a joke. What is topological inference? It s when you infer the topology of a space given only a finite subset.
5 Something like a joke. What is topological inference? It s when you infer the topology of a space given only a finite subset.
6 We add geometric and statistical hypotheses to make the problem well-posed. Geometric Assumption: The underlying space is a smooth manifold M. Statistical Assumption: The points are drawn i.i.d. from a distribution derived from M.
7 We add geometric and statistical hypotheses to make the problem well-posed. Geometric Assumption: The underlying space is a smooth manifold M. Statistical Assumption: The points are drawn i.i.d. from a distribution derived from M.
8 We add geometric and statistical hypotheses to make the problem well-posed. Geometric Assumption: The underlying space is a smooth manifold M. Statistical Assumption: The points are drawn i.i.d. from a distribution derived from M.
9 We add geometric and statistical hypotheses to make the problem well-posed. Geometric Assumption: The underlying space is a smooth manifold M. Statistical Assumption: The points are drawn i.i.d. from a distribution derived from M.
10
11 Input: n points from a d-manifold M in D-dimensions.
12 Input: n points from a d-manifold M in D-dimensions. Output: The homology of M.
13 Input: n points from a d-manifold M in D-dimensions. Output: The homology of M. Upper bound: What is the worst case complexity?
14 Input: n points from a d-manifold M in D-dimensions. Output: The homology of M. Upper bound: What is the worst case complexity? Lower Bound: What is the worst case complexity of the best possible algorithm?
15 sampled i.i.d. Input: n points from a d-manifold M in D-dimensions. Output: The homology of M. Upper bound: What is the worst case complexity? Lower Bound: What is the worst case complexity of the best possible algorithm?
16 sampled i.i.d. distribution supported on Input: n points from a d-manifold M in D-dimensions. Output: The homology of M. Upper bound: What is the worst case complexity? Lower Bound: What is the worst case complexity of the best possible algorithm?
17 sampled i.i.d. distribution supported on Input: n points from a d-manifold M in D-dimensions. with noise Output: The homology of M. Upper bound: What is the worst case complexity? Lower Bound: What is the worst case complexity of the best possible algorithm?
18 sampled i.i.d. distribution supported on Input: n points from a d-manifold M in D-dimensions. an estimate of with noise Output: The homology of M. Upper bound: What is the worst case complexity? Lower Bound: What is the worst case complexity of the best possible algorithm?
19 sampled i.i.d. distribution supported on Input: n points from a d-manifold M in D-dimensions. an estimate of with noise Output: The homology of M. Upper bound: What is the worst case complexity? Lower Bound: What is the worst case complexity of the best possible algorithm? probability of giving a wrong answer
20 sampled i.i.d. distribution supported on Input: n points from a d-manifold M in D-dimensions. an estimate of with noise Output: The homology of M. Upper bound: What is the worst case complexity? Lower Bound: What is the worst case complexity of the best possible algorithm? probability of giving a wrong answer The Goal: Matching Bounds (asymptotically)
21 Minimax risk is the error probability of the best estimator on the hardest examples.
22 Minimax risk is the error probability of the best estimator on the hardest examples. Minimax Risk: R n =inf Ĥ sup Q Q Q n (Ĥ H(M))
23 Minimax risk is the error probability of the best estimator on the hardest examples. Minimax Risk: R n =inf Ĥ sup Q Q Q n (Ĥ H(M)) the best estimator
24 Minimax risk is the error probability of the best estimator on the hardest examples. Minimax Risk: R n =inf Ĥ sup Q Q Q n (Ĥ H(M)) the best estimator the hardest distribution
25 Minimax risk is the error probability of the best estimator on the hardest examples. Minimax Risk: R n =inf Ĥ sup Q Q Q n (Ĥ H(M)) the best estimator the hardest distribution product distribution
26 Minimax risk is the error probability of the best estimator on the hardest examples. Minimax Risk: R n =inf Ĥ sup Q Q the best estimator the hardest distribution Q n (Ĥ H(M)) product distribution the true homology
27 Minimax risk is the error probability of the best estimator on the hardest examples. Minimax Risk: R n =inf Ĥ sup Q Q the best estimator the hardest distribution Q n (Ĥ H(M)) product distribution the true homology Sample Complexity: n(ɛ) =min{n : R n ɛ}
28 We assume manifolds without boundary of bounded volume and reach.
29 We assume manifolds without boundary of bounded volume and reach. Let M be the set of compact d-dimensional Riemannian manifolds without boundary such that
30 We assume manifolds without boundary of bounded volume and reach. Let M be the set of compact d-dimensional Riemannian manifolds without boundary such that 1 M ball D (0, 1)
31 We assume manifolds without boundary of bounded volume and reach. Let M be the set of compact d-dimensional Riemannian manifolds without boundary such that 1 M ball D (0, 1) 2 vol(m) c d
32 We assume manifolds without boundary of bounded volume and reach. Let M be the set of compact d-dimensional Riemannian manifolds without boundary such that 1 M ball D (0, 1) 2 3 vol(m) c d The reach of M is at most τ.
33 We assume manifolds without boundary of bounded volume and reach. Let M be the set of compact d-dimensional Riemannian manifolds without boundary such that 1 M ball D (0, 1) 2 3 vol(m) c d The reach of M is at most τ. Let P be the set of probability distributions supported over M Mwith densities bounded from below by a constant a.
34 We consider 4 different noise models. Noiseless Clutter Tubular Additive
35 We consider 4 different noise models. Noiseless Clutter Q = P Tubular Additive
36 We consider 4 different noise models. Noiseless Clutter Q = P Q =(1 γ)u + γp P P U is uniform on ball(0, 1) Tubular Additive
37 We consider 4 different noise models. Noiseless Clutter Q = P Q =(1 γ)u + γp P P U is uniform on ball(0, 1) Tubular Let Q M,σ be uniform on M σ. Additive Q = {Q M,σ : M M}
38 We consider 4 different noise models. Noiseless Clutter Q = P Q =(1 γ)u + γp P P U is uniform on ball(0, 1) Tubular Let Q M,σ be uniform on M σ. Q = {Q M,σ : M M} Additive Q = {P Φ:P P} Φ is Gaussian with σ τ or Φ has Fourier transform bounded away from 0 and τ is fixed.
39 Le Cam s Lemma is a powerful tool for proving minimax lower bounds.
40 Le Cam s Lemma is a powerful tool for proving minimax lower bounds. Lemma. Let Q be a set of distributions. Let θ(q) take values in a metric space (X, ρ) for Q Q. For any Q 1,Q 2 Q, inf ˆθ sup Q Q E Q n [ ] ρ(ˆθ, θ(q)) 1 8 ρ(θ(q 1),θ(Q 2 ))(1 TV(Q 1,Q 2 )) 2n
41 Le Cam s Lemma is a powerful tool for proving minimax lower bounds. Lemma. Let Q be a set of distributions. Let θ(q) take values in a metric space (X, ρ) for Q Q. For any Q 1,Q 2 Q, inf ˆθ sup Q Q E Q n [ ] ρ(ˆθ, θ(q)) 1 8 ρ(θ(q 1),θ(Q 2 ))(1 TV(Q 1,Q 2 )) 2n For homology, use the trivial metric. ρ(x, y) = { 0 if x = y 1 if x y
42 Le Cam s Lemma is a powerful tool for proving minimax lower bounds. Lemma. Let Q be a set of distributions. Let θ(q) take values in a metric space (X, ρ) for Q Q. For any Q 1,Q 2 Q, inf ˆθ sup Q Q E Q n [ ] ρ(ˆθ, θ(q)) 1 8 ρ(θ(q 1),θ(Q 2 ))(1 TV(Q 1,Q 2 )) 2n For homology, use the trivial metric. ρ(x, y) = { 0 if x = y 1 if x y inf Ĥ sup Q Q Q n (Ĥ H(M)) 1 8 (1 TV(Q 1,Q 2 )) 2n
43 Le Cam s Lemma is a powerful tool for proving minimax lower bounds. Lemma. Let Q be a set of distributions. Let θ(q) take values in a metric space (X, ρ) for Q Q. For any Q 1,Q 2 Q, inf ˆθ sup Q Q E Q n [ ] ρ(ˆθ, θ(q)) 1 8 ρ(θ(q 1),θ(Q 2 ))(1 TV(Q 1,Q 2 )) 2n For homology, use the trivial metric. ρ(x, y) = { 0 if x = y 1 if x y R n = inf Ĥ sup Q Q Q n (Ĥ H(M)) 1 8 (1 TV(Q 1,Q 2 )) 2n
44 The lower bound requires two manifolds that are geometrically close but topologically distinct.
45 The lower bound requires two manifolds that are geometrically close but topologically distinct. B =ball d (0, 1 τ) A = B \ ball d (0, 2τ)
46 The lower bound requires two manifolds that are geometrically close but topologically distinct. B =ball d (0, 1 τ) A = B \ ball d (0, 2τ) M 1 = (B τ ) M 2 = (A τ )
47 The lower bound requires two manifolds that are geometrically close but topologically distinct. B =ball d (0, 1 τ) A = B \ ball d (0, 2τ) M 1 = (B τ ) M 2 = (A τ ) The overlap
48 It suffices to bound the total variation distance.
49 It suffices to bound the total variation distance. Total Variation Distance: TV(Q 1,Q 2 )=sup A Q 1 (A) Q 2 (A) a max{vol(m 1 \ M 2 ), vol(m 2 \ M 1 )} C d aτ d
50 It suffices to bound the total variation distance. Total Variation Distance: Minimax Risk: TV(Q 1,Q 2 )=sup A Q 1 (A) Q 2 (A) a max{vol(m 1 \ M 2 ), vol(m 2 \ M 1 )} C d aτ d R n 1 8 (1 TV(Q 1,Q 2 ) 2n 1 8 (1 C daτ d ) 2n 1 8 e 2C daτ d n
51 It suffices to bound the total variation distance. Total Variation Distance: Minimax Risk: TV(Q 1,Q 2 )=sup A Q 1 (A) Q 2 (A) a max{vol(m 1 \ M 2 ), vol(m 2 \ M 1 )} C d aτ d R n 1 8 (1 TV(Q 1,Q 2 ) 2n 1 8 (1 C daτ d ) 2n 1 8 e 2C daτ d n Sampling Rate: n(ɛ) ( 1 τ ) d log 1 ε
52 The upper bound uses a union of balls to estimate the homology of M.
53 The upper bound uses a union of balls to estimate the homology of M.
54 The upper bound uses a union of balls to estimate the homology of M.
55 The upper bound uses a union of balls to estimate the homology of M. 1 Take a union of balls.
56 The upper bound uses a union of balls to estimate the homology of M. 1 Take a union of balls. Compute the homology of the 2 resulting Cech complex.
57 The upper bound uses a union of balls to estimate the homology of M. 1 Take a union of balls. Compute the homology of the 2 resulting Cech complex.
58 The upper bound uses a union of balls to estimate the homology of M. 0 1 Denoise the data. Take a union of balls. Compute the homology of the 2 resulting Cech complex.
59 The upper bound uses a union of balls to estimate the homology of M. 0 1 Denoise the data. Take a union of balls. Compute the homology of the 2 resulting Cech complex.
60 The upper bound uses a union of balls to estimate the homology of M. 0 1 Denoise the data. Take a union of balls. Compute the homology of the 2 resulting Cech complex.
61 The upper bound uses a union of balls to estimate the homology of M. 0 1 Denoise the data. Take a union of balls. Compute the homology of the 2 resulting Cech complex.
62 The upper bound uses a union of balls to estimate the homology of M. 0 1 Denoise the data. Take a union of balls. Compute the homology of the 2 resulting Cech complex.
63 The upper bound uses a union of balls to estimate the homology of M. 0 1 Denoise the data. Take a union of balls. Compute the homology of the 2 resulting Cech complex. To prove: The density is bounded from below near M and from above far from M.
64 Many fundamental problems are still open.
65 Many fundamental problems are still open. 1 Is the reach the right parameter?
66 Many fundamental problems are still open. 1 Is the reach the right parameter? 2 What about manifolds with boundary?
67 Many fundamental problems are still open. 1 Is the reach the right parameter? 2 What about manifolds with boundary? 3 Homotopy equivalence?
68 Many fundamental problems are still open. 1 Is the reach the right parameter? 2 What about manifolds with boundary? 3 Homotopy equivalence? 4 How to choose parameters?
69 Many fundamental problems are still open. 1 Is the reach the right parameter? 2 What about manifolds with boundary? 3 Homotopy equivalence? 4 How to choose parameters? 5 Are there efficient algorithms?
70 Thank you.
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