Questions tagged [hessian-matrix]
The Hessian matrix of function is used to second derivative test when $f$ has a critical point $x$. If the Hessian is positive definite at $x,ドル then $f$ attains a local minimum at $x$. If the Hessian is negative definite at $x,ドル then $f$ attains a local maximum at $x$. If the Hessian has both positive and negative eigenvalues then $x$ is a saddle point for $f$.
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Two functions having the same zero set
If $f$ and $g$ are $C^2$ functions $\mathbb{R}^2\to\mathbb{R}$ having the same zero set $\cal C$.
I want to ask whether
$$\kappa_f:= \frac{\text{Hess}_f(t,t)}{\|\nabla f\|}$$
is equal to (up to a sign)...
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If the second partial derivatives are matrices, can I plug them into the a new matrix and call new matrix the Hessian $H$ to show $H \succeq 0$?
$\def\diag{\operatorname{diag}} \Lambda^{-1}=\diag\left(\dfrac{1}{\lambda_i}\right).$
$\lambda_i \geq 0$
$w_i \in \mathbb{R}$
$C>0, C \in \mathbb{R}_{++}$
$\mathbf w$ is length $d$ and so is lambda....
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The meaning of the elements of the inverse of the Jacobian (and Hessian)
Page 147, equation (3.9.9, of "The Finite Element Method Linear Static and Dynamic Finite Element Analysis" by Thomas J. R. Hughes contains the following formula
$$
\begin{bmatrix}
\dfrac{\...
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Is there a smooth function having any given number of extrema and saddles?
For which triples $(l,s,h) \in \mathbf{Z}^3_{\geq 0}$ does there exist an example of a smooth function $f\colon \mathbf{R}^2 \to \mathbf{R}$ that has $l$ local minimums (low points), $s$ saddle points,...
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Is there vocabulary for critical points of multivariable functions at which the second partial derivative test fails?
For a smooth multivariable function $f \colon \mathbf{R}^2 \to \mathbf{R},ドル a critical point of $f$ is any $(a,b)$ at which $\nabla f(a,b) = \mathbf{0}$. Whether a critical point corresponds to a ...
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Why is the Hessian of the negative log-likelihood in multinomial logistic regression positive definite?
I am studying the convexity properties of the negative log-likelihood in multinomial logistic regression.
Let me briefly set up the notation:
We have a dataset
$$
D = \{(x_n, y_n)\}_{n=1}^N, \quad ...
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Why does the product of the Hessian and the transformation determinant only involve first derivatives?
In Leonard Dickson's Modern Algebraic Theories (p. 3), the Hessian $h$ of a function $f$ is introduced, where
the elements of the ith row are
$\frac{\partial^2}{\partial x_i \partial x_1}, \frac{\...
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Index notation for mixed partial derivatives – $f_{xy}=(f_x)_y$ or $f_{xy}=(f_y)_x$?
The mixed partial derivative $\frac{\partial^{2} f}{\partial x ,円 \partial y} := \frac{\partial}{\partial x}\!\left( \frac{\partial f}{\partial y} \right)$ is obtained by first differentiating with ...
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Is the hessian of a smooth planar projective complex cubic again smooth?
I have a planar complex projective cubic, let’s call it $F$. I’ve proven that it’s nonsingular and I’m now asked to prove that $D=\det(H(F))$ is again a smooth cubic. ($H$ is the hessian matrix of $F$....
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Characterization of positive definite matrices and convexity
Let $f\colon \mathbb{R}^d \to \mathbb{R}$ be twice continuously differentiable. In the theory of Hamilton–Jacobi–Bellman or convex analysis in general one can encounter conditions on data like
$$(\...
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How do I find and classify a stationary point of the function 5ドルx^3 - 3yx - 6y^3 - 2$ using Newton's method and the Hessian eigenvalues?
Setup to the problem:
We are going to determine the stationary points of the function
5ドルx^3 - 3yx - 6y^3 - 2$
in the region $-1 \leq x \leq 1, \ -1 \leq y \leq 1$.
Calculate the gradient $\nabla f(\...
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How does this proof connect to the claim?
This proof was taken from the book Multivariable Mathematics by Theodore Shifrin. How does the definition of continuity in the second part of the proof connect to the claim that $$f(\mathbf a+\mathbf ...
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Gradient and Hessian of a pseudo-quadratic form
Problem formulation
I would like to generalize the following well-known and super-nice formulas for the gradient $\nabla J(x)$ and Hessian $\nabla^2 J(x)$ of a quadratic cost $J(x)\triangleq x'Qx$
\...
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Hessian matrix and gradient vector for the regularized quadratic function
Let $\mathcal{l}$ be the following quadratic loss function: $\frac{1}{2} \theta ^t H \theta$ where H is Hessian matrix, $\theta \in R^d$ is the parameter vector. If we define the regularized version ...
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Effect of gradient regularization on the Eigenvalues of Hessian matrix of the regularized loss function
Assume $\mathcal{l}$ is a standard loss function for training a neural network with differentiability, convexity, and Lipschitz continuity assumptions. Let's say the largest eigenvalue of the Hessian ...