Gradient ascent curves
$begingroup$
Let $f:R^2rightarrow R$ be continuously differentiable $m$ times. Pick $(x,y) in R^2$. Define the "gradient ascent curve from $(x,y)$ on $f$" to be the limit of the path followed by performing gradient ascent:
$$(x_t, y_t) = (x_{t-1}, y_{t-1}) + h*nabla f(x_{t-1}, y_{t-1}) (1)$$
...as the step size $h rightarrow 0$. In other words, if $V_h$ is the sequence defined by $(1)$ for $tin mathbf{N}$, the "gradient ascent curve" is $lim_{hrightarrow 0} V_h$.
If $nabla f$ is continuous, surely this curve is continuous? Does it have a nice parameterization?
More importantly, does it have a name already, and if so, where can I learn more about it?
gradient-descent
asked Dec 12 '18 at 5:01
ScottScott
16211
$endgroup$
add a comment |
$begingroup$
Let $f:R^2rightarrow R$ be continuously differentiable $m$ times. Pick $(x,y) in R^2$. Define the "gradient ascent curve from $(x,y)$ on $f$" to be the limit of the path followed by performing gradient ascent:
$$(x_t, y_t) = (x_{t-1}, y_{t-1}) + h*nabla f(x_{t-1}, y_{t-1}) (1)$$
...as the step size $h rightarrow 0$. In other words, if $V_h$ is the sequence defined by $(1)$ for $tin mathbf{N}$, the "gradient ascent curve" is $lim_{hrightarrow 0} V_h$.
If $nabla f$ is continuous, surely this curve is continuous? Does it have a nice parameterization?
More importantly, does it have a name already, and if so, where can I learn more about it?
gradient-descent
asked Dec 12 '18 at 5:01
ScottScott
16211
$endgroup$
2
$begingroup$
You might want to search "gradient flow".
$endgroup$
– user1101010
Dec 12 '18 at 5:11
add a comment |
$begingroup$
Let $f:R^2rightarrow R$ be continuously differentiable $m$ times. Pick $(x,y) in R^2$. Define the "gradient ascent curve from $(x,y)$ on $f$" to be the limit of the path followed by performing gradient ascent:
$$(x_t, y_t) = (x_{t-1}, y_{t-1}) + h*nabla f(x_{t-1}, y_{t-1}) (1)$$
...as the step size $h rightarrow 0$. In other words, if $V_h$ is the sequence defined by $(1)$ for $tin mathbf{N}$, the "gradient ascent curve" is $lim_{hrightarrow 0} V_h$.
If $nabla f$ is continuous, surely this curve is continuous? Does it have a nice parameterization?
More importantly, does it have a name already, and if so, where can I learn more about it?
gradient-descent
asked Dec 12 '18 at 5:01
ScottScott
16211
$endgroup$
Let $f:R^2rightarrow R$ be continuously differentiable $m$ times. Pick $(x,y) in R^2$. Define the "gradient ascent curve from $(x,y)$ on $f$" to be the limit of the path followed by performing gradient ascent:
$$(x_t, y_t) = (x_{t-1}, y_{t-1}) + h*nabla f(x_{t-1}, y_{t-1}) (1)$$
...as the step size $h rightarrow 0$. In other words, if $V_h$ is the sequence defined by $(1)$ for $tin mathbf{N}$, the "gradient ascent curve" is $lim_{hrightarrow 0} V_h$.
If $nabla f$ is continuous, surely this curve is continuous? Does it have a nice parameterization?
More importantly, does it have a name already, and if so, where can I learn more about it?
gradient-descent
gradient-descent
asked Dec 12 '18 at 5:01
ScottScott
16211
asked Dec 12 '18 at 5:01
ScottScott
16211
asked Dec 12 '18 at 5:01
ScottScott
16211
asked Dec 12 '18 at 5:01
ScottScott
16211
asked Dec 12 '18 at 5:01
ScottScott
16211
16211
2
$begingroup$
You might want to search "gradient flow".
$endgroup$
– user1101010
Dec 12 '18 at 5:11
add a comment |
2
$begingroup$
You might want to search "gradient flow".
$endgroup$
– user1101010
Dec 12 '18 at 5:11
2
2
$begingroup$
You might want to search "gradient flow".
$endgroup$
– user1101010
Dec 12 '18 at 5:11
$begingroup$
You might want to search "gradient flow".
$endgroup$
– user1101010
Dec 12 '18 at 5:11
add a comment |
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$begingroup$
You might want to search "gradient flow".
$endgroup$
– user1101010
Dec 12 '18 at 5:11