Spherical Designs and Determinantal Point Processes Masatake HIRAO ( 平尾 将剛 ) (Aichi Prefectural University, Japan) ——————————————— 2015 April 22 Workshop on Spherical Design and Numerical Analysis@SJTU
1. Motivation Problem in Geostatistics We want to investigate, e.g., temperature, GH gas, magnetic field, ... Where do we have to observe these data? via Wikipedia To challenge this optimal sampling problem, we study by combining - “exact-type” cubature formula (Numerical analysis) - Euclidean/spherical or combinatorial design (Combinatorics) • Hirao-Sawa-Jimbo(’15), Sawa-Hirao(15+) Today, instead of the two above techniques, we focus on - Determinantal point process(DPP) (Probability theory) - Quasi-Monte Carlo(QMC) design (“robust-type” cubature formula)
• I believe that good approximation points are also good observation points for many statistical methods (e.g., regression analysis, kernel methods, ...).
1. Motivation 2. Spherical design and problem 3. Determinatal point process 4. QMC design and main results
Spherical design X ⊂ S d , | X | < ∞ A set X is called a spherical t -design of S d , Definition if the following equation holds 1 ∫ ∀ f ∈ P t ( S d ) , ∑ f ( x ) = x ∈ S d f ( x ) dν ( x ) , | X | x ∈ X where ν is the normalized surface meas. of S d . Note. Spherical designs give a good approximation for other spaces. Ex. (spherical design of S 2 ) 8 1 1 ∫ ∀ f ∈ P 3 ( R 3 ) . ∑ f ( x i ) = S 2 f ( x ) dν ( x ) , | S 2 | 8 i =1
N 1 1 ∫ ∀ f ∈ P t ( R 3 ) . ∑ f ( x i ) = S 2 f ( x ) dν ( x ) , | S 2 | N i =1 ⋆ t = 5 ⋆ t = 3 ⋆ t = 2 N = 8 N = 20 N = 4 N = 6 N = 12 Theorem (Bondarenko et al., ’11). Given t , there always exists a spherical t -design of S d with N ≍ t d ( ← lower bound order)
Regression analysis on S d & our problems f 1 ( x ) , . . . , f m ( x ): a basis of P e ( S d ) ( m := dim( P e ( S d ))) We consider a full regression model of degree e : x ∈ S d . Y ( x ) = θ 1 f 1 ( x ) + · · · + θ m f m ( x ) + ϵ ( x ) , θ 1 , . . . , θ m : unknown true parameters, ϵ ( x ): a random error. Ex. { Y l,k | k = 1 , . . . , Z ( d, l ) } : a basis of Harm l ( S d ). Z ( d,l ) e ∑ ∑ Y ( x ) = θ k,l Y k,l ( x ) + ϵ ( x ) . l =0 k =1 ⋆ A function f ∈ L 2 ( S d ) admits the Fourier expansion Z ( d,l ) ∞ ∫ ∑ ∑ ˆ ˆ f ( x ) = f k,l Y k,l ( x ) , f k,l := S d f ( x ) Y k,l ( x ) dν ( x ) . l =0 k =1 Thus, we want to apply to higher degrees and higher dimension cases.
Theorem (e.g., Neumaier-Seidel, ’92). An experimental design S d is optimal of degree e , iff it is a spherical 2 e -design (with repeated points allows). Note. In order to estimate θ 1 , . . . , θ m , a set of spherical design is “good” observation points. Problem How we can construct a spherical t -design with sample size N ≍ t d for large t ? • It is not easy (for me) to construct such a design. Thus, we try to construct “approximate” spherical t -design by using DPP. Namely, we use random scattered points on S d .
3. Determinantal point process(DPP) Macchi (1975): position of an electron in quantum mechanics. . . . Peres-Vir´ ag(2005), Krishnapur (2009), Hough et al.(2009) . . .
Example. Poisson and Ginibre point process Note. R ⊂ R n or C n Q := Q ( R ): Z ≥ 0 -valued Radon meas., i.e., Q ∋ ξ = ∑ ( x i ∈ R ) i δ x i • point process: Q -valued random variable (ex. Poisson, Cox, Hard- Core, ...)
Poisson point process Ginibre point process 6 10 15 14 23 4 34 12 27 19 21 30 42 44 2 15 61 18 9 29 28 61 53 57 38 68 5 41 89 57 20 77 62 77 78 64 25 59 80 59 92 50 19 39 75 16 81 67 11 83 39 29 90 54 64 63 2 79 97 21 82 89 91 52 96 60 11 3 79 1 49 75 37 1 92 58 30 40 98 67 43 87 72 83 14 69 100 69 18 46 87 99 88 70 58 32 22 23 35 20 22 45 43 5 72 94 41 85 50 9 91 66 48 36 47 97 55 26 70 27 84 44 86 4 86 78 33 46 93 32 63 13 96 73 17 6 8 31 12 42 94 95 73 84 74 62 65 85 74 88 52 93 100 56 53 81 35 8 65 71 82 51 51 40 7 68 76 54 3 95 28 47 13 66 45 60 56 71 33 24 37 48 31 16 36 76 38 17 26 25 55 7 49 80 98 24 90 99 10 34 DPP → there is an effective repulsion among particles.
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