# Gradients

For a function $f:\mathcal M→ℝ$ the Riemannian gradient $\operatorname{grad}f(x)$ at $x∈\mathcal M$ is given by the unique tangent vector fulfilling

\[\langle \operatorname{grad}f(x), ξ\rangle_x = D_xf[ξ],\quad \forall ξ ∈ T_x\mathcal M,\]

where $D_xf[ξ]$ denotes the differential of $f$ at $x$ with respect to the tangent direction (vector) $ξ$ or in other words the directional derivative.

This page collects the available gradients.

`Manopt.forward_logs`

— Method```
Y = forward_logs(M,x)
forward_logs!(M, Y, x)
```

compute the forward logs $F$ (generalizing forward differences) occurring, in the power manifold array, the function

\[F_i(x) = \sum_{j ∈ \mathcal I_i} \log_{x_i} x_j,\quad i ∈ \mathcal G,\]

where $\mathcal G$ is the set of indices of the `PowerManifold`

manifold `M`

and $\mathcal I_i$ denotes the forward neighbors of $i$. This can also be done in place of `ξ`

.

**Input**

`M`

– a`PowerManifold`

manifold`x`

– a point.

**Ouput**

`Y`

– resulting tangent vector in $T_x\mathcal M$ representing the logs, where $\mathcal N$ is thw power manifold with the number of dimensions added to`size(x)`

. The computation can be done in place of`Y`

.

`Manopt.grad_L2_acceleration_bezier`

— Method```
grad_L2_acceleration_bezier(
M::AbstractManifold,
B::AbstractVector{P},
degrees::AbstractVector{<:Integer},
T::AbstractVector,
λ,
d::AbstractVector{P}
) where {P}
```

compute the gradient of the discretized acceleration of a composite Bézier curve on the `Manifold`

`M`

with respect to its control points `B`

together with a data term that relates the junction points `p_i`

to the data `d`

with a weight $λ$ compared to the acceleration. The curve is evaluated at the points given in `pts`

(elementwise in $[0,N]$), where $N$ is the number of segments of the Bézier curve. The summands are `grad_distance`

for the data term and `grad_acceleration_bezier`

for the acceleration with interpolation constrains. Here the `get_bezier_junctions`

are included in the optimization, i.e. setting $λ=0$ yields the unconstrained acceleration minimization. Note that this is ill-posed, since any Bézier curve identical to a geodesic is a minimizer.

Note that the Beziér-curve is given in reduces form as a point on a `PowerManifold`

, together with the `degrees`

of the segments and assuming a differentiable curve, the segments can internally be reconstructed.

**See also**

`grad_acceleration_bezier`

, `cost_L2_acceleration_bezier`

, `cost_acceleration_bezier`

.

`Manopt.grad_TV`

— Function```
X = grad_TV(M, λ, x[, p=1])
grad_TV!(M, X, λ, x[, p=1])
```

Compute the (sub)gradient $\partial F$ of all forward differences occurring, in the power manifold array, i.e. of the function

\[F(x) = \sum_{i}\sum_{j ∈ \mathcal I_i} d^p(x_i,x_j)\]

where $i$ runs over all indices of the `PowerManifold`

manifold `M`

and $\mathcal I_i$ denotes the forward neighbors of $i$.

**Input**

`M`

– a`PowerManifold`

manifold`x`

– a point.

**Ouput**

- X – resulting tangent vector in $T_x\mathcal M$. The computation can also be done in place.

`Manopt.grad_TV`

— Method```
X = grad_TV(M, (x,y)[, p=1])
grad_TV!(M, X, (x,y)[, p=1])
```

compute the (sub) gradient of $\frac{1}{p}d^p_{\mathcal M}(x,y)$ with respect to both $x$ and $y$ (in place of `X`

and `Y`

).

`Manopt.grad_TV2`

— Function`grad_TV2(M::PowerManifold, q[, p=1])`

computes the (sub) gradient of $\frac{1}{p}d_2^p(q_1,q_2,q_3)$ with respect to all $q_1,q_2,q_3$ occurring along any array dimension in the point `q`

, where `M`

is the corresponding `PowerManifold`

.

`Manopt.grad_TV2`

— Function```
Y = grad_TV2(M, q[, p=1])
grad_TV2!(M, Y, q[, p=1])
```

computes the (sub) gradient of $\frac{1}{p}d_2^p(q_1, q_2, q_3)$ with respect to all three components of $q∈\mathcal M^3$, where $d_2$ denotes the second order absolute difference using the mid point model, i.e. let

\[\mathcal C = \bigl\{ c ∈ \mathcal M \ |\ g(\tfrac{1}{2};q_1,q_3) \text{ for some geodesic }g\bigr\}\]

denote the mid points between $q_1$ and $q_3$ on the manifold $\mathcal M$. Then the absolute second order difference is defined as

\[d_2(q_1,q_2,q_3) = \min_{c ∈ \mathcal C_{q_1,q_3}} d(c, q_2).\]

While the (sub)gradient with respect to $q_2$ is easy, the other two require the evaluation of an `adjoint_Jacobi_field`

.

`Manopt.grad_acceleration_bezier`

— Method```
grad_acceleration_bezier(
M::AbstractManifold,
B::AbstractVector,
degrees::AbstractVector{<:Integer}
T::AbstractVector
)
```

compute the gradient of the discretized acceleration of a (composite) Bézier curve $c_B(t)$ on the `Manifold`

`M`

with respect to its control points `B`

given as a point on the `PowerManifold`

assuming C1 conditions and known `degrees`

. The curve is evaluated at the points given in `T`

(elementwise in $[0,N]$, where $N$ is the number of segments of the Bézier curve). The `get_bezier_junctions`

are fixed for this gradient (interpolation constraint). For the unconstrained gradient, see `grad_L2_acceleration_bezier`

and set $λ=0$ therein. This gradient is computed using `adjoint_Jacobi_field`

s. For details, see ^{[BergmannGousenbourger2018]}. See `de_casteljau`

for more details on the curve.

**See also**

`cost_acceleration_bezier`

, `grad_L2_acceleration_bezier`

, `cost_L2_acceleration_bezier`

.

`Manopt.grad_distance`

— Function```
grad_distance(M,y,x[, p=2])
grad_distance!(M,X,y,x[, p=2])
```

compute the (sub)gradient of the distance (squared), in place of `X`

.

\[f(x) = \frac{1}{p} d^p_{\mathcal M}(x,y)\]

to a fixed point `y`

on the manifold `M`

and `p`

is an integer. The gradient reads

\[ \operatorname{grad}f(x) = -d_{\mathcal M}^{p-2}(x,y)\log_xy\]

for $p\neq 1$ or $x\neq y$. Note that for the remaining case $p=1$, $x=y$ the function is not differentiable. In this case, the function returns the corresponding zero tangent vector, since this is an element of the subdifferential.

**Optional**

`p`

– (`2`

) the exponent of the distance, i.e. the default is the squared distance

`Manopt.grad_intrinsic_infimal_convolution_TV12`

— Method`grad_u, grad_v = grad_intrinsic_infimal_convolution_TV12(M, f, u, v, α, β)`

compute (sub)gradient of the intrinsic infimal convolution model using the mid point model of second order differences, see `costTV2`

, i.e. for some $f ∈ \mathcal M$ on a `PowerManifold`

manifold $\mathcal M$ this function computes the (sub)gradient of

\[E(u,v) = \frac{1}{2}\sum_{i ∈ \mathcal G} d_{\mathcal M}(g(\frac{1}{2},v_i,w_i),f_i) + \alpha \bigl( β\mathrm{TV}(v) + (1-β)\mathrm{TV}_2(w) \bigr),\]

where both total variations refer to the intrinsic ones, `grad_TV`

and `grad_TV2`

, respectively.

- BergmannGousenbourger2018
Bergmann, R. and Gousenbourger, P.-Y.: A variational model for data fitting on manifolds by minimizing the acceleration of a Bézier curve. Frontiers in Applied Mathematics and Statistics (2018). doi 10.3389/fams.2018.00059, arXiv: 1807.10090