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w ͜ͷลͷܭࢉΛղੳతʹͰ͖Δͱݴ͏͜ͱ͕ॏཁͳϙΠϯτ PT is the parallel transport on hyperbolic space, then hPT⌫!µ(v), PT⌫!µ(v0)iL = hv, v0iL. The explicit formula for the parallel transport on the Lorentz model (Figure 2(b)) is given by: PT⌫!µ(v) = v + hµ ↵⌫, viL ↵ + 1 (⌫ + µ), (3) where ↵ = h⌫, µiL. The inverse parallel transport PT 1 ⌫!µ simply carries the vector in TµHn back to T⌫Hn along the geodesic. That is, v = PT 1 ⌫!µ (u) = PTµ!⌫(u). (4) Exponential map and inverse exponential map Finally, we will describe a function that maps a vector in a tangent space to its surface. According to the basic theory of differential geometry, ev- ery u 2 TµHn determines a unique maximal geodesic µ : [0, 1] ! Hn with µ(0) = µ and ˙µ(0) = u. Ex- ponential map expµ : TµHn ! Hn is a map defined by expµ (u) = µ(1), and we can use this map to project a vector v in TµHn onto Hn in a way that the distance from µ to destination of the map coincides with kvkL, the metric bolic space, we need to be able to map the point back to t tangent space, on which the distribution is initially define We, therefore, need to be able to compute the inverse of t exponential map, which is also called logarithm map, well. Solving eq. (13) for u, we can obtain the inverse exponenti map as u = exp 1 µ (z) = arccosh(↵) p ↵2 1 (z ↵µ), ( where ↵ = hµ, ziL. See Appendix A.1 for further detai 3. Pseudo-Hyperbolic Gaussian 3.1. Construction Finally, we are ready to provide the construction of o wrapped gaussian distribution G(µ, ⌃) on Hyperbolic spa with µ 2 Hn and positive definite ⌃. In the language of the differential geometry, our strateg can be re-described as follows: 1. Sample a vector ˜ v from the Gaussian distributio N(0, ⌃) defined over Rn. 2. Interpret ˜ v as an element of Tµ0 Hn ⇢ Rn+1 by rewr ing ˜ v as v = [0, ˜ v]. model (Figure 2(b)) is given by: PT⌫!µ(v) = v + hµ ↵⌫, viL ↵ + 1 (⌫ + where ↵ = h⌫, µiL. The inverse paralle PT 1 ⌫!µ simply carries the vector in TµHn bac along the geodesic. That is, v = PT 1 ⌫!µ (u) = PTµ!⌫(u). Exponential map and inverse exponential ma Finally, we will describe a function that maps a tangent space to its surface. According to the basic theory of differential ge ery u 2 TµHn determines a unique maxima µ : [0, 1] ! Hn with µ(0) = µ and ˙µ(0) ponential map expµ : TµHn ! Hn is a map expµ (u) = µ(1), and we can use this map t vector v in TµHn onto Hn in a way that the dis µ to destination of the map coincides with kvkL H1 (red) and its tangent space TµH1 (blue). (b) Parallel transport carries ng k · kL . (c) Exponential map projects the u 2 Tµ (blue) to z 2 Hn (red). measured on the surface of Hn coincides with kukL. nt space of µ0 Hn con- p hv, viL = port n, the par- ap PT⌫!µ in T⌫Hn el manner words, if pace, then he Lorentz µ), (3) norm of v. For hyperbolic space, this map (Figure 2(c)) is given by z = expµ (u) = cosh (kuk L )µ+sinh (kuk L ) u kuk L . (5) As we can confirm with straightforward computation, this exponential map is norm preserving in the sense that d`(µ, expµ (u)) = arccosh hµ, expµ (u)iL = kukL. Now, in order to evaluate the density of a point on hyper- bolic space, we need to be able to map the point back to the tangent space, on which the distribution is initially defined. We, therefore, need to be able to compute the inverse of the exponential map, which is also called logarithm map, as well. Solving eq. (13) for u, we can obtain the inverse exponential map as u = exp 1 µ (z) = arccosh(↵) p (z ↵µ), (6) ng the geodesic. That is, v = PT 1 ⌫!µ (u) = PTµ!⌫(u). (4) ponential map and inverse exponential map nally, we will describe a function that maps a vector in a gent space to its surface. cording to the basic theory of differential geometry, ev- y u 2 TµHn determines a unique maximal geodesic : [0, 1] ! Hn with µ(0) = µ and ˙µ(0) = u. Ex- nential map expµ : TµHn ! Hn is a map defined by pµ (u) = µ(1), and we can use this map to project a ctor v in TµHn onto Hn in a way that the distance from o destination of the map coincides with kvkL, the metric 3. Pseudo-Hyperbolic G 3.1. Construction Finally, we are ready to pro wrapped gaussian distribution with µ 2 Hn and positive de In the language of the differ can be re-described as follow 1. Sample a vector ˜ v fro N(0, ⌃) defined over R 2. Interpret ˜ v as an elemen ing ˜ v as v = [0, ˜ v]. v = PT 1 ⌫!µ (u) = PTµ!⌫(u). (4) ntial map and inverse exponential map we will describe a function that maps a vector in a space to its surface. ng to the basic theory of differential geometry, ev- 2 TµHn determines a unique maximal geodesic 1] ! Hn with µ(0) = µ and ˙µ(0) = u. Ex- al map expµ : TµHn ! Hn is a map defined by ) = µ(1), and we can use this map to project a v in TµHn onto Hn in a way that the distance from tination of the map coincides with kvkL, the metric 3.1. Construction Finally, we are ready to provide t wrapped gaussian distribution G(µ, with µ 2 Hn and positive definite ⌃ In the language of the differential can be re-described as follows: 1. Sample a vector ˜ v from the N(0, ⌃) defined over Rn. 2. Interpret ˜ v as an element of Tµ ing ˜ v as v = [0, ˜ v]. where ↵ = h⌫, µiL. The inverse parallel transport PT 1 ⌫!µ simply carries the vector in TµHn back to T⌫Hn along the geodesic. That is, v = PT 1 ⌫!µ (u) = PTµ!⌫(u). (4) Exponential map and inverse exponential map Finally, we will describe a function that maps a vector in a tangent space to its surface. According to the basic theory of differential geometry, ev- ery u 2 TµHn determines a unique maximal geodesic µ : [0, 1] ! Hn with µ(0) = µ and ˙µ(0) = u. Ex- ponential map expµ : TµHn ! Hn is a map defined by expµ (u) = µ(1), and we can use this map to project a vector v in TµHn onto Hn in a way that the distance from µ to destination of the map coincides with kvkL, the metric u = exp 1 µ (z) = p where ↵ = hµ, ziL. See App 3. Pseudo-Hyperbolic G 3.1. Construction Finally, we are ready to prov wrapped gaussian distribution G with µ 2 Hn and positive defi In the language of the differe can be re-described as follows 1. Sample a vector ˜ v from N(0, ⌃) defined over Rn 2. Interpret ˜ v as an element ing ˜ v as v = [0, ˜ v]. where ↵ = h⌫, µiL. The inverse parallel transport PT 1 ⌫!µ simply carries the vector in TµHn back to T⌫Hn along the geodesic. That is, v = PT 1 ⌫!µ (u) = PTµ!⌫(u). (4) Exponential map and inverse exponential map Finally, we will describe a function that maps a vector in a tangent space to its surface. According to the basic theory of differential geometry, ev- ery u 2 TµHn determines a unique maximal geodesic µ : [0, 1] ! Hn with µ(0) = µ and ˙µ(0) = u. Ex- ponential map expµ : TµHn ! Hn is a map defined by expµ (u) = µ(1), and we can use this map to project a vector v in TµHn onto Hn in a way that the distance from µ to destination of the map coincides with kvkL, the metric ↵ where ↵ = hµ, ziL. See Appendix 3. Pseudo-Hyperbolic Gauss 3.1. Construction Finally, we are ready to provide th wrapped gaussian distribution G(µ, ⌃ with µ 2 Hn and positive definite ⌃ In the language of the differential g can be re-described as follows: 1. Sample a vector ˜ v from the N(0, ⌃) defined over Rn. 2. Interpret ˜ v as an element of Tµ0 ing ˜ v as v = [0, ˜ v]. d its tangent space TµH1 (blue). (b) Parallel transport carries c) Exponential map projects the u 2 Tµ (blue) to z 2 Hn (red). the surface of Hn coincides with kukL. norm of v. For hyperbolic space, this map (Figure 2(c)) is given by z = expµ (u) = cosh (kuk L )µ+sinh (kuk L ) u kuk L . (5) As we can confirm with straightforward computation, this exponential map is norm preserving in the sense that d`(µ, expµ (u)) = arccosh hµ, expµ (u)iL = kukL. Now, in order to evaluate the density of a point on hyper- bolic space, we need to be able to map the point back to the tangent space, on which the distribution is initially defined. We, therefore, need to be able to compute the inverse of the exponential map, which is also called logarithm map, as tangent space TµH1 (blue). (b) Parallel transport carries ponential map projects the u 2 Tµ (blue) to z 2 Hn (red). surface of Hn coincides with kukL. orm of v. For hyperbolic space, this map (Figure 2(c)) is iven by z = expµ (u) = cosh (kuk L )µ+sinh (kuk L ) u kuk L . (5) s we can confirm with straightforward computation, this xponential map is norm preserving in the sense that `(µ, expµ (u)) = arccosh hµ, expµ (u)iL = kukL. ow, in order to evaluate the density of a point on hyper- olic space, we need to be able to map the point back to the ngent space, on which the distribution is initially defined. We, therefore, need to be able to compute the inverse of the xponential map, which is also called logarithm map, as ell. olving eq. (13) for u, we can obtain the inverse exponential map as u = exp 1 µ (z) = arccosh(↵) p ↵2 1 (z ↵µ), (6) here ↵ = hµ, ziL. See Appendix A.1 for further details. . Pseudo-Hyperbolic Gaussian a) One-dimensional Lorentz model H1 (red) and its tangent space TµH1 (blue). (b) Parallel transport carries reen) to u 2 Tµ (blue) while preserving k · kL . (c) Exponential map projects the u 2 Tµ (blue) to z 2 Hn (red). e between µ and expµ (u) which is measured on the surface of Hn coincides with kukL. an be literally thought of as the tangent space of hyperboloid sheet at µ. Note that Tµ0 Hn con- Rn+1 with v0 = 0, and kvkL := p hv, viL = ansport and inverse parallel transport n arbitrary pair of point µ, ⌫ 2 Hn, the par- ort from ⌫ to µ is defined as a map PT⌫!µ n to TµHn that carries a vector in T⌫Hn geodesic from ⌫ to µ in a parallel manner anging its metric tensor. In other words, if parallel transport on hyperbolic space, then ), PT⌫!µ(v0)iL = hv, v0iL. formula for the parallel transport on the Lorentz ure 2(b)) is given by: ⌫!µ(v) = v + hµ ↵⌫, viL ↵ + 1 (⌫ + µ), (3) = h⌫, µiL. The inverse parallel transport mply carries the vector in TµHn back to T⌫Hn eodesic. That is, v = PT 1 ⌫!µ (u) = PTµ!⌫(u). (4) norm of v. For hyperbolic space, this map (Figure 2(c)) is given by z = expµ (u) = cosh (kuk L )µ+sinh (kuk L ) u kuk L . (5) As we can confirm with straightforward computation, this exponential map is norm preserving in the sense that d`(µ, expµ (u)) = arccosh hµ, expµ (u)iL = kukL. Now, in order to evaluate the density of a point on hyper- bolic space, we need to be able to map the point back to the tangent space, on which the distribution is initially defined. We, therefore, need to be able to compute the inverse of the exponential map, which is also called logarithm map, as well. Solving eq. (13) for u, we can obtain the inverse exponential map as u = exp 1 µ (z) = arccosh(↵) p ↵2 1 (z ↵µ), (6) where ↵ = hµ, ziL. See Appendix A.1 for further details. 3. Pseudo-Hyperbolic Gaussian 3.1. Construction
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